Atomistic modeling of BN nanofillers for mechanical and thermal properties: a review

Abstract

Due to their exceptional mechanical properties, thermal conductivity and a wide band gap (5–6 eV), boron nitride nanotubes and nanosheets have promising applications in the field of engineering and biomedical science. Accurate modeling of failure or fracture in a nanomaterial inherently involves coupling of atomic domains of cracks and voids as well as a deformation mechanism originating from grain boundaries. This review highlights the recent progress made in the atomistic modeling of boron nitride nanofillers. Continuous improvements in computational power have made it possible to study the structural properties of these nanofillers at the atomistic scale.

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Rajesh Kumar

Mr Rajesh Kumar is a PhD candidate in the department of mechanical and industrial engineering at the Indian Institute of Technology-Roorkee. His research area focuses on atomistic modelling of BN nanofillers and their hybrids.

Avinash Parashar

Dr Avinash Parashar received his PhD in mechanical engineering (2012) from the University of Alberta. He worked as an attached scientist with the Computational Reactor Physics laboratory in Atomic Energy of Canada Limited. Currently, he is working as an assistant professor in the department of Mechanical and Industrial Engineering at the Indian Institute of Technology-Roorkee. His research interest includes atomistic modeling of nanofillers and nanocomposites.

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1. Introduction

Boron nitride (BN) is a lab-grown binary compound consisting of an equal number of boron (B) and nitrogen (N) atoms. BN crystallizes either as a hexagonal layered structure or as a tetrahedral linked structure, similar to that of graphite and diamond respectively. Due to its white color and slippery properties hexagonal boron nitride (h-BN) is also known as white graphite or graphitic boron nitride.^1,2 In 1995, BN was first synthesized as boron nitride nanotubes (BNNTs) and in continuation to that work, BN nanomeshes were further developed.³

Similar to carbon based nanofillers e.g. graphene and carbon nanotubes; BN nanotubes (BNNTs) and nanosheets (BNNSs) also have a hexagonal closed packed atomic configuration.^4–6 BN nanofillers (BNNT or BNNS) exist in the hexagonal form with alternatively placed boron (B) and nitrogen (N) atoms as illustrated in Fig. 1. Despite similar atomic configurations, and comparable mechanical and thermal properties,^7,8 BN nanofillers are good electrical insulators^1,9,10 as compared to the corresponding carbon based nanofillers.

Fig. 1 Atomic configuration of (a) single layer BNNS and (b) single-walled BNNT. Red and blue dots correspond to boron and nitrogen atoms, respectively.

Due to its wide band gap, h-BN is preferred over graphene for specific applications such as electronic packaging.^11–14 In addition to exceptional mechanical properties,^15–26 thermal conductivity²⁷ and electrical insulating properties,^1,9,10 BN nanofillers are also highly stable against oxidation even at very high temperatures such as 800 °C.²⁸ At such high working temperatures, BN nanofillers are considered chemically and structurally more stable as compared to the corresponding carbon based nanostructures.^28,29 The thermal conductivity of BN nanofillers has been reported to be in the range of 1700–2000 W m⁻¹ K⁻¹.²⁷ In addition to these exceptional properties, BN nanofillers have also shown deep ultraviolet photon emission,³⁰ and a good amount of piezoelectricity.^31–33

Due to the above mentioned exceptional properties, h-BN nanofillers have found applications in many areas such as nano-sensors,^34–39 bio-sensors,^34–37,39 biomaterials^40–42 and in developing corrosion-resistant and thermally stable composites.^43–49 The bio-compatibility, biosafety and high hemo/histocompatibility of BN nanofillers are encouraging researchers to use them as therapeutic agents to treat neuro-genetic disorders.^50–55 The neutron absorption capacity of these nanofillers has also found applications in the field of cancer treatment such as cerebral glioblastoma multiforme.⁵⁶ Adsorption of hydrogen on the surface of BNNTs and BNNSs^57–59 also opens the door for hydrogen storage at ambient temperatures. Some other prominent applications of BN nanofillers include optoelectronics^60–66 and tribology.^67–69 Due to their excellent sorption capabilities for a wide range of oils, solvents and dyes, their application as a water purifier has also been proposed.⁷⁰

Despite the availability of high yield synthesis techniques,⁷¹ commercial production of these nanofillers is still in an immature state, due to the limited availability of techniques for characterizing these nanofillers. Due to challenges associated with the characterization of these BN nanofillers, limited research has been published in this area as compared to that of carbon based nanofillers. A year wise comparative chart for the number of publications in the field of respective nanofillers is plotted in Fig. 2.

Fig. 2 A year wise graphical comparison of research papers published and containing respective phrases i.e. BNNT, BNNS, CNT & graphene in their title. Statistics based on Scopus database, as of June 15^th, 2015.

There are a number of methods that are currently being used for the production of h-BN nanofillers. Monolayer to few-layers of h-BN flakes can be exfoliated from bulk BN crystals either by mechanical cleavage/exfoliation^72,73 or by a chemical-solution-derived method.⁷⁴ Other synthesis methods for BN nanofillers include chemical vapor deposition (CVD),^75–79 sonication,^80,81 ball milling,^81,82 high-energy electron beam irradiation,^83,84 reaction of boric acid and urea,^85,86 a metal-catalyst-free approach,⁸⁷ chemical blowing (a mass production technique that relies on generating large bubbles of BNH or BNCH from a precursor ammonia borate compound),^88,89 using substitution reaction,^90,91via micro-fluidization⁹² and via laser ablation.^93–96 The electrical insulating properties that single out BN nanofillers from other nanofillers have been extensively studied by researchers. In 1994, Blase et al.⁹ predicted a constant band gap for BN nanotubes and this was further explored by Watanabe et al.¹⁰ to establish the fact that BN nanotubes have a direct-band gap in the ultra-violet region. Zhang et al.⁹⁷ predicted the stability and electronic structures of BNNTs with diameters below 0.4 nm. In their work on BNNTs numerical based techniques have been used to study the effect of nanotube diameter, chirality and multiwall BN structures on the band gap or electrical insulation. These exceptional properties, along with a diversified field of applications have aroused keen interest in the researchers to explore more about these nanofillers. Starting from 1994, researchers have been trying to study BN nanofillers using different techniques such as experiments, finite element based continuum models, molecular dynamics based atomistic models, and higher fidelity numerical simulations such as density functional theory (DFT).

So far, only a limited number of review papers have been published on BN nanofillers, and most of them are focused on the generalized topics. Atomistic modeling techniques are emerging as a focused area of research for researchers investigating the behavior of these nanofillers. The aim of the present article is to address the current state of the art techniques for simulating the mechanical as well as thermal behavior of BNNSs and BNNTs with the help of atomistic modeling based approaches. The authors have also attempted to illustrate the challenges associated with each kind of atomistic modeling technique.

2. Atomistic modeling techniques

During the last couple of years, several experimental and modeling techniques have been employed by the researchers to characterize or predict the mechanical and thermal behaviors of BN nanofillers. Despite the fact that experimental techniques are costly and time consuming, these experiments are still considered as a realistic tool for characterizing the behavior of materials. Transmission electron microscopy (TEM)^{15,16,23,24,85,98} and atomic force microscopy (AFM) based techniques^15,16,73,99 are usually employed in the experimental characterization of nano materials. In addition to these common techniques, X-ray based scattering has also been employed for estimating the five elastic moduli of h-BN.¹⁰⁰

In addition to the cost and time involved, experimental techniques also have some limitations in estimating the localized mechanical or thermal behavior of these nanofillers at the atomistic level. Due to these limitations associated with experimental techniques, computational techniques based on finite element, DFT and MD are emerging as effective tools for predicting the mechanical as well as thermal behavior of these nanofillers. Atomistic simulations are not substituting experiments but are complementing them at the nanoscale.

2.1. Structural mechanics based approach

The theoretical approaches, mentioned above, can be further classified as atomistic and continuum based approaches. The atomistic approach includes classical molecular dynamics and density functional theory, whereas the continuum mechanics based approach mainly consists of classical continuum mechanics and shell modeling.^101–103 The structural mechanics based approach developed and proposed by Li and Chou¹⁰⁴ is considered to be a pioneer research in the field of atomistic modeling of nanofillers. Their model was based on the notion that the nanofillers (e.g. CNT) are geometrical space frame structures and can be studied with the help of structural mechanics. In the structural mechanics based approach atomic nuclei are treated as material points. The motion of atomic nuclei is regulated by a force field and this force field is generally expressed as the steric potential energy which depends on the relative position of atomic nuclei. In 2005, Tserpes and Papanikos¹⁰⁵ implemented the concept of Li and Chou¹⁰⁴ in the finite element method (FEM). In their FEM based approach, bonds are simulated as load carrying elements, and atoms as joints or nodes for connecting these elements as illustrated in Fig. 3. In FEM based approaches the material properties or the stiffness matrix is generated by establishing a linkage between molecular and continuum mechanics.

Fig. 3 Schematic of the space frame model for CNT (reproduced with permission from ref. 105).

Harmonic, Morse and Lennard Jones LJ 6–12 (van der Waals) like interatomic potentials are commonly employed in FEM based models for generating the stiffness matrix. These interatomic potentials account for all the bonded as well as non-bonded interactions. These structural mechanics based modeling techniques have also been extended by researchers for the modeling of BN nanofillers.^106–108 In 2011, Boldrin et al.¹⁰⁶ extended the space frame models for the simulation of h-BN nanosheets. They employed molecular mechanics in conjunction with finite element to study the effective mechanical properties of h-BN nanosheets. In addition to these space frame based atomistic models, the continuum based lattice approach has also been employed in conjunction with the Tersoff–Brenner potential for estimating the Young's modulus and Poisson's ratio of boron nitride crystals.¹⁰⁹ Numerical techniques, based on the sub-hexagonal lattice-sums,¹¹⁰ have been further extended by Green et al.¹¹¹ to study the elastic behavior of h-BN. Finite element based continuum models are considered suitable for predicting the average or the global properties of these nanofillers.

2.2. Density functional theory (DFT)

Density Functional Theory (DFT) is a quantum mechanics based higher fidelity simulation technique, and is common among the researchers for estimating the atomic level properties of materials in physics, materials science and chemistry. In DFT, atomic nuclei are usually treated as frozen while electrons are considered to be moving in the electrostatic field generated by the nuclei (Born–Oppenheimer approximation). In DFT based atomistic simulations, electron density is used as its central part to calculate the total energy of the system. This energy E[ρ] is a functional of the electron energy density ‘ρ’ and is expressed as:

E[ρ] = T[ρ] + E_H[ρ] + E_xc[ρ] + E_ext[ρ] + E_zz (1)

where T[ρ], E_H[ρ], E_xc[ρ], E_ext[ρ] and E_zz are energy contributions from the kinetic energy of non-interacting electrons, the Hartree energy, the exchange–correlation energy, the external potential (here due to the potential of nuclei) and ion energy due to interactions among nuclei, respectively. The Kohn–Sham method is the most widely used DFT technique in which rather than finding a solution for electron wave functions a solution for electron density functional is sought. Based on the Kohn–Sham theory, a system of many interacting particles is converted into an equivalent fictitious system of non-interacting particles. Kohn–Sham (KS) equations for this new system model are given by eqn (2) and (3).whereand is known as Kohn–Sham or effective potential, ϕ_i are the Kohn–Sham one-electron orbitals whereas electron density ρ(r) is given by:

Due to Kohn–Sham formulations, the multi-electron problem is reduced to one seeking a set of solutions to one electron problems. For solving the equations, an initial guess of electron density “ρ” is made and is used for further calculations. For solving the Kohn–Sham equations, two types of basis sets “atomic orbitals” or “plane wave” are usually used by researchers. In the first case, a linear combination of atomic orbitals (LCAO), such as Gaussian orbitals, is used. LCAO is especially suitable for a study of molecules or systems with strong directional bonding. But the non-orthogonal nature of these orbitals makes the calculations complex and creates practical inconvenience in a way that there is no systematic way that improves accuracy if the number of basis sets is increased. The other common basis set i.e. plane wave, is good for periodic systems (such as solids) but can also be adapted for finite size systems, such as atoms or molecules, with a supercell approach where large periodicity of the simulation box is ensured. The plane wave basis sets are usually used for valence spaces in conjunction with pseudopotentials.

Before solving Kohn–Sham equations, all the energy terms in the equations need to be evaluated. Except for exchange–correlation energy, all other energy terms can be estimated easily. The exchange–correlation energy is usually estimated by using some approximations such as local density approximation (LDA) [(ref. 112) (PW92)–(ref. 113) (PZ81)], generalized gradient approximation (GGA) [(ref. 114) (PBE)–(ref. 115) (LYP)] etc. LDA is the simplest one and neglects the gradients in electron density. Despite being simple, LDA gives quite good results even with this simple approach. Other approximations include meta-GGA [(ref. 116) (TPSS)], hybrid functional [(ref. 117 and 118) (PBE0)] and some meta-hybrid functionals and the choice of one out of them is application specific.

For solid state systems, a suitable method is to use the pseudopotential (PP) approach. In this method, interactions between core electrons and valence electrons are treated with a pseudopotential function. The pseudopotential approach is based on the consideration that core electrons do not participate significantly in the bonding process and can be neglected while solving for the electronic structure of the electrons. On the other hand, valence electrons, which are fewer in number, are treated with plane-wave basis sets. Although higher in number, plane-waves are used in solids because they are cheap to compute. Pseudopotentials may be an empirical pseudopotential or an ab initio pseudopotential. Ab initio pseudopotentials based on local solutions of wave functions near the atomic nuclei are commonly preferred over the empirical pseudopotentials. However, in more accurate DFT implementations, the non-local form of pseudopotentials is used.

The pseudopotential approach is much faster than all-electron calculations because tightly-bound core electrons are not directly taken into the account for calculations. After solving KS equations, the output electron density obtained is used to calculate the total ground state energy of the whole system. This procedure is repeated until a self-consistent solution is achieved. This optimized total ground state energy is further used to estimate the other structural properties of the material. An overview of the DFT process has been shown in Fig. 4. However, solving KS equations for construction of the density matrix from the Fock/Kohn–Sham matrix and then solving for density functional is an expensive process which makes DFT a computationally intensive technique. However, another approach based on DFT, which is faster with surprisingly good accuracy, is the tight-binding (TB) method in which valence electrons are treated by being ‘tightly bound’ to their nucleus and atomic orbitals are not altered during the bonding process. The tight-binding model is based on a set of approximations obtained from quantum mechanics and hence is less computationally intensive. Even though this method is less accurate as compared to ab initio DFT it is sufficiently accurate for the structures to which it is fitted. The method is usually applied to covalent systems such as semi-conductors (Si, Ge) and to carbon. In DFT calculations, mechanical properties are estimated using strain energy (U) obtained by the following equation:

U = U_ε − U_ε=0 (5)

where, U_ε and U_ε=0 are the potential energy of the strained and unstrained atomistic system respectively.

Fig. 4 An overview chart of the iterative DFT process.

The value of strained energy is further processed to calculate the Young's modulus (Y in N m⁻²) and 2D Young's modulus (Y_S in N m⁻¹) as illustrated in eqn (6) and (7) respectively:

Y = (∂²U/∂ε²)/V (6)

Y_S = (∂²U/∂ε²)/S (7)

Here, V is the volume and S is the surface area of the system.

With BN nanofillers, DFT and other DFT based approaches have also been used by researchers to estimate the elastic as well as other structural properties.^{17–20,25,119–123} Kudin and Scuseria¹⁷ employed DFT with Gaussian type orbitals and PBE¹¹⁴ approximations to study the elastic behavior of BNNTs. They predicted an average value of nearly 810 GPa for the Young's modulus of the nanotubes which was in close agreement with tight-binding calculations of Hernández et al.¹²⁴ Later on, Suryavanshi et al.²⁴ also reported a value of nearly 722 GPa for the Young's modulus of BNNTs using experimental techniques. In 2009, Sahin et al.¹⁸ used the DFT framework to investigate the elastic and other properties of BN nanosheets. A combination of first-principles plane-wave calculations and projector augmented wave potentials¹²⁵ was used to ensure accuracy and computation speed. They predicted a value of nearly 810 GPa (assuming a thickness of 0.33 nm for BNNS) for the Young's modulus which was close to the value reported by Bosak et al.¹⁰⁰ using inelastic X-ray scattering measurements. Some more research studies^19,20,126 performed in the DFT framework also predicted values of the Young's modulus to be nearly close to those reported in the studies mentioned previously in this paragraph. A combination of DFT and quasi-harmonic approximation (QHA)¹²⁷ was successfully applied by researchers¹²⁸ to evaluate the effect of temperature on the mechanical behavior of BN nanosheets. It was found that the Young's modulus of boron nanosheets and nanotubes increases with an increase in the temperature up to 800 °C. After this temperature, the rate of increase was slow and the temperature tended toward a constant value.

Although, the DFT based approach has been successfully applied for the atomistic modeling and predicting of properties of materials quite accurately yet it has certain limitations associated with it. As already mentioned generally it is considered computationally intensive and can only be performed with a limited number of atoms. Tight-binding methods are considered good from a computational point of view but these have issues of transferability to other systems and even have the issue of number of atoms. Moreover, DFT based approaches are very useful for developing new empirical interatomic potentials for empirical methods like molecular dynamics. These empirical potentials are considered as the back bone for the less computational intensive molecular dynamics based simulations.

2.3. Molecular dynamics (MD)

As discussed above, FEM and DFT based approaches have been applied pretty much successfully in atomistic modeling, but still have certain limitations. As compared to FEM and DFT based atomistic modeling techniques, the molecular dynamics (MD) based approach is emerging as yet another alternative for performing atomistic simulations.

MD is a widely used computer based technique for the atomistic simulation of atoms/molecules in the context of N-body simulation or many body systems.¹²⁹ In MD based simulations atoms are treated as classical particles. The atomic position and potential energy of the system help in computing the atomic forces that are further used in Newton's equation of motion. The relationship derived from the Newton's second law of motion is used for updating the position, velocity and acceleration of each atom in the system at an integrated time step. The fundamental equation solved in molecular dynamics based simulation is given as:

m_αa_α = F_α = −(∂E/∂r_α) (8)

where α = 1, 2, ⋯, N.

Here, N is the total number of atoms; m_α, r_α and F_α are the mass, the position and the time dependent force acting on them due to external agents, respectively. The potential energy ‘E’ consists of an internal part (E^int) that accounts for the interaction between the atoms and an external part (E^ext) that accounts for external fields and constraints. The potential energy (E) and hence force F_α are the functions of position and velocity vectors of the atoms. Statistical mechanics based approaches are commonly used over the time averages to derive macroscopic properties such as pressure, temperature, stresses etc.

In MD based simulations, position and velocity vectors are updated after each time step, usually by using any of the numerical integration algorithms such as Verlet algorithm, Velocity-Verlet algorithm,¹³⁰ Leapfrog algorithm,¹³¹ and Beeman's algorithm.¹³² But the Velocity-Verlet algorithm (an extension of Verlet algorithm) is the most widely used integrating scheme in MD based simulations. The position vector ‘r’ at any time t and after an increment Δt, for any atom involved in the simulation can be expanded as a Taylor series expansion as given by eqn (9).

r(t + Δt) ≈ r(t) + ṙ(t)Δt + (̈(t)Δt²)/2 + … (9)

The updated velocity vector after time ‘t + Δt’ can be easily obtained with the help of the Velocity-Verlet equation as:

v(t + Δt) = v(t) + (F(t + Δt) + F(t)/2m)Δt (10)

MD simulations are carried out under different types of ‘Ensembles’. An ensemble is a collection of all possible different microstates of a system which have an identical macroscopic state. Micro-canonical (NVE), canonical (NVT) and isobaric–isothermal (NPT) ensembles are the three most commonly used ensembles in MD simulations.

Most of the interatomic potentials employed in MD based simulations are either derived or developed empirically with the help of experiments or higher fidelity numerical simulations. Interatomic potentials are mathematical expressions for estimating the potential energy (E) of the system of atoms. Commonly used interatomic potentials are Tersoff,^133–136 LJ,¹³⁷ the embedded atom method (EAM),¹³⁸ REBO¹³⁹ and AIREBO.¹⁴⁰ Tersoff type^133–136 interatomic potentials are commonly used for simulating the mechanical and thermal behaviors of BN nanofillers. The Tersoff potential can be expressed mathematically as given by eqn (11) and (12):

V_ij = f_c(r_ij)[a_ijf_R(r_ij) + b_ijf_A(r_ij)]. (12)

Here, E is the total energy of the system, which is decomposed, for convenience, into site energy E_i and bond energy V_ij. The indices i and j run over the atoms of the system, and r_ij is the distance between atoms i and j. The functions f_R and f_A represent the repulsive and attractive parts of the pair potential. The functions f_c and b_ij are the cutoff function and the bond-order function respectively.

Significant progress has been made in the development of interatomic potentials for estimating the behavior of BN nanofillers with improved accuracy. For the modeling of BN nanofillers in a MD based environment, most of the researchers have employed either Tersoff or a modified form of the Tersoff potential.^21,141,142 A harmonic force field has also been employed by a couple of researchers for managing the atomic interactions in BN nanofillers.^22,143 The Tersoff potential that was initially introduced in ref. 133 has been further modified by Brenner¹⁴⁴ to include additional terms for the inherent over-binding of radicals and for non-local effects. In addition to Brenner,¹⁴⁴ various researchers have also proposed optimized Tersoff potential parameters for improving the accuracy of atomistic simulations. Sekkal et al.¹⁴⁵ proposed Tersoff parameters for boron by fitting the bulk properties of cubic BN. In their simulations, BN was treated as a single entity, hence, an identical set of parameters was proposed for both boron and nitrogen. Tersoff potential parameters proposed by Sekkal et al.¹⁴⁵ are limited to BN interactions only, and cannot be extended to study interactions of boron or nitrogen with other atoms. These potential parameters¹⁴⁵ have been successfully employed by many researchers for predicting the structural and thermal properties of BN nanofillers. Sevik et al.¹⁴⁶ proposed parameters for the Tersoff potential that effectively capture the experimental phonon dispersion in BN. Due to the capability of these parameters to effectively capture phonon dispersion, they are considered more suitable for estimating the lattice thermal conductivity of BN based nanostructures. Matsunaga et al.^147–149 proposed separate sets of parameters for boron, nitrogen and carbons atoms. This set of potential parameters is considered more suitable for reproducing experimentally measured lattice constants, bulk modulus and the binding energy of cubic boron nitride. In addition to these parameters, Albe et al.^150,151 also proposed Tersoff potential parameters for bulk BN materials to simulate ion-beam deposition of BN thin films. One big advantage with parameters provided by Albe et al.^150,151 is that they account for the sp² structure of BN bonding. The parameters proposed by Sekkal et al.,¹⁴⁵ Sevik et al.,¹⁴⁶ Matsunaga et al.,^147–149 and Albe et al.^150,151 have been summarized in Table 1.

Table 1 Compilation of parameters for the Tersoff type potential proposed by different researchers

Parameter	Sekkal et al.^145	Sevik et al.^146	Matsunaga et al.^147 (χi−j = 1.1593)		Albe et al.^150
	B–N	B–N	B–B	N–N	B–N
m	3.0	3.0	3.0	3.0	3.0
Γ	1.0	1.0	1.0	1.0	1.0
λ3	0.0	0.0	0.0	0.0	1.9925
c	38 049	25 000	0.52629	79 934	1092.9287
d	4.384	4.3484	0.001587	134.32	12.38
h	−0.57058	−0.89	0.5	−0.9973	−0.5413
n	0.72751	0.72751	3.9929	12.4498	0.364 53367
β	0.00000015724	0.0000001257	0.0000016	0.10562	0.000011134
λ2	2.22	2.199	1.5856	2.5115	2.784247207
B	346.7	340	183.49	219.45	3613.431337
R	1.95	1.95	1.95	2.15	2.0
D	0.2	0.05	0.15	0.15	0.1
λ1	3.468	3.568	1.9922	5.7708	2.998355817
A	1393.6	1380	277.02	11 000	4460.833973

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Due to accurate parameterization that helps in achieving accurate fit with the DFT calculations and experimental results, Tersoff type potentials have attracted more attention among the researchers for simulating the bonded interactions. On the other hand, reactive force field (ReaxFF) potentials are common among researchers for simulating the complex atomic structures of hydrocarbons.^152–154 In addition to hydrocarbons, ReaxFF has also been applied for BN nanofillers and will be discussed further in the next section. For ReaxFF potentials, parameters are obtained from fittings to the results of quantum chemical analysis. Similar to other empirical non-reactive potentials, the total energy of the system is derived from various partial energy contributions (bonded as well as non-bonded) as expressed in eqn (13).

E_system = E_bond + E_over + E_under + E_val + E_pen + E_tors + E_conj + E_vdWaals + E_Columb (13)

ReaxFF type potentials are reported to be more suitable for simulating bond-breaking and bond-forming mechanisms. The fundamental difference between ReaxFF and most of the other empirical interatomic potentials is that ReaxFF does not employ any fixed connectivity for the bonded interaction. In ReaxFF type potentials the bond order is directly calculated from the instantaneous interatomic distance r_ij, which keeps on updating during the simulation. This bond order is responsible for the creation and dissociation of bonds during the simulation. The mathematical expression for calculating the bond order with respective details can be found in the papers.^152–154

In MD based simulations, strain calculations are performed with the help of initial (L_i) and final (L_f) lengths of the simulation box as expressed by eqn (14). The concept of stress in a material or a body is well known at the continuum level and is usually referred to as the Cauchy stress tensor. However, at the atomistic level the principles of continuum are not applicable, and hence an alternative stress referred to as virial stress is estimated at the atomistic level. Virial stress is the most commonly used definition of stress in discrete particle systems. Virial stress is the summation of two components. The first component depends upon the mass and velocity of the atoms in the MD simulations, whereas the second component depends upon the interatomic forces and atomic positions. The mathematical expression for virial stress as the summation of two components is given by eqn (15).

where i and j denote indices in the Cartesian coordinate system; α and β are the atomic indices; m^α and v^α are the mass and velocity of atom α; r_αβ is the distance between atoms α and β; and φ^α is the atomic volume of atom α.

The strain energy based concepts, as discussed in the DFT section, are also applicable in MD simulations for estimating the Young's modulus and 2D Young's moduli values of the system as per eqn (5)–(7). In addition to predicting the mechanical behavior of an atomistic system, the MD based simulations have also been employed for predicting the thermal behavior of the system. Thermal conductivity (k) of a simulated atomistic configuration can be estimated by using the mathematical expressions given below:

Here, in eqn (16) and (17), refers to the temperature gradient along the direction of heat flux (J), A is the cross sectional area, h and c refer to the hot and cold atoms, and N_transfer and t_transfer are the summation time and the number of exchanges respectively.

3. MD based simulations for BN nanofillers

As described in previous sections atomistic modeling techniques are emerging as a potential tool for predicting the mechanical and thermal behaviors of BN nanofillers. MD based simulations are considered sufficiently accurate and less computationally intensive as compared to FEM and DFT based approaches, respectively. In this part of the article, the authors have made an attempt to review the current state of art techniques for simulating the mechanical as well as thermal behavior of BN nanofillers with the help of molecular dynamics.

3.1. Mechanical behavior

The mechanical behavior of BN nanofillers has been widely investigated by many researchers. Among various mechanical properties, Young's modulus is considered as the most widely predicted or estimated property of any material. Mechanical properties of BN nanofillers (BNNS or BNNT) predicted or estimated by using different means have been compiled in Tables 2 and 3. It can be inferred from these tables that researchers have simulated BN nanofillers both as an isotropic as well as an in-plane orthotropic material (arm-chair and zig-zag configurations). While treating BNNS as an isotropic material,^{18–20,75,100,106,109,111,155–157} researchers have predicted a range of Young's modulus starting from 0.79 TPa to 0.97 TPa (assuming 0.33 nm as the thickness of a BNNS). Similarly, for BNNTs, the Young's modulus values with isotropic material properties range from 0.77 TPa to 1.2 TPa.^{16,23,24,109,142,158} In addition to the Young's modulus, researchers have also predicted the 2D Young's modulus for BN nanofillers as tabulated in Table 3. At the atomistic level, these nanofillers are not isotropic in nature and usually have in-plane orthotropic properties that are dependent on the chirality vector. Based on the alignment of the bonds with the loading direction, BN nanofillers (BNNT or BNNS) can be further classified as zig-zag and armchair configurations. The Young's modulus (Y) as well as 2D Young's modulus (Y_S) for zig-zag and armchair directions has also been summarized in Tables 2 and 3, respectively.^{17,21,22,26,124,126,143,159–161} It can be inferred from these tables that the 2D Young’ modulus (Y_S) predicted through atomistic simulations and experimental techniques lies in the range of 200 N m⁻¹ to 500 N m⁻¹.⁷⁵

Table 2 Mechanical properties of BN nanofillers from different studies with different methods

Method	Reference	Nanofiller	Young's modulus (TPa)	Fracture stress (GPa)	Fracture strain (%)	Poisson's ratio
AC stands for Armchair and ZZ stands for Zigzag.
Experimental	Chopra et al.^23	BNNT	1.22 ± 0.24	—	—	—
	Suryavanshi et al.^24	BNNT	0.722	—	—	—
	Wei et al.^158	BNNT	0.895	33	3	—
	Arenal et al.^16	BNNT	1.11 ± 0.17	—	—	—
	Bosak et al.^100	BNNS	0.811	—	—	—
DFT	Hernández et al.^124	BNNT	0.884^ZZ	—	—	0.23–0.27
			0.912^AC
	Fakhrabad et al.^25	BNNT	0.821^AC
			0.764^AC	—	—	—
	Ohba et al.^155	BNNS	0.952	—	—	—
	Mirnezhad et al.^156	BNNS	0.829	—	—	0.21
	Zhang et al.^142	BNNT	0.850
MD	Verma et al.^26	BNNT	0.982^ZZ	—	—	0.13^ZZ
			1.111^AC			0.16^AC
		BNNS	1.004^ZZ	—	—	—
			1.111^AC
	Mortazavi et al.^159	BNNS	0.800^ZZ	165	30	—
			0.850^AC
	Zhao et al.^160	BNNS	0.693^ZZ	120	28	—
			0.740^AC
	Han et al.^157	BNNS	0.881	133	33	—
Structural mechanics & others	Eun-Suok Oh^109	BNNS	0.977	—	—	0.17
		BNNT	0.950
	Boldrin et al.^106	BNNS	0.797	—	—	0.21
	Green et al.^111	BNNS	0.803	—	—	—

---

Table 3 Mechanical properties of BN nanofillers from different studies with different methods

Method	Reference	Nanofiller	Young's modulus (N m^−1)	Fracture stress (N m^−1)	Fracture strain (%)	Poisson's ratio
AC stands for Armchair and ZZ stands for Zigzag.
Experimental	Li et al.^75	BNNS	200–500	—	22	—
DFT	Kudin et al.^17	BNNT	266^ZZ	—	—	0.21
			267^AC
	Sahin et al.^18	BNNS	267	—	—	0.21
	Topsakal et al.^126	BNNS	258^AC	—	—	—
	Andrew et al.^19	BNNS	276	—	—	0.22
	Peng et al.^20	BNNS	278	—	—	—
MD	Minh-Quy Le^21	BNNS	263^ZZ	36^ZZ	23^ZZ	—
			253^AC	30^AC	17^AC
	Minh-Quy Le^22	BNNS	284^AC & ZZ	—	—	—
		BNNT	283^AC & ZZ
	Minh-Quy Le^143	BNNS	296^ZZ	—	—	—
			286^AC
	Le and Nguyen^161	BNNS	258^ZZ	38^ZZ	26^ZZ	—
			251^AC	36^AC	26^AC
			257^ZZ	31^ZZ	17^ZZ	—
			249^AC	30^AC	18^AC
			257^ZZ	35^ZZ	21^ZZ	—
			251^AC	32^AC	20^AC
			258^ZZ	31^ZZ	17^ZZ	—
			251^AC	33^AC	22^AC
Structural mechanics	Minh-Quy Le^143	BNNS	332	—	—	—

---

Le²¹ performed MD simulations using Tersoff-potential parameters provided by Sevik et al.¹⁴⁶ to predict mechanical properties of BNNSs. In this study, the author extended the small cut-off distance to a large cut-off distance and obtained lower values of fracture stress and strains as compared with earlier studies.¹⁵⁹ The fracture strain predicted by Le²¹ was in good agreement with DFT calculations performed by Topsakal et al.¹²⁶ Molecular dynamics based simulations has been performed by Verma et al.²⁶ to estimate the Young's modulus and shear modulus of BN nanofillers. In their computational work, interatomic interactions between boron and nitrogen atoms were estimated with the help of Tersoff-type potentials^134,135 with parameters obtained from the work of Sekkal et al.¹⁴⁵ The potential parameters provided by Sekkal et al.¹⁴⁵ were slightly modified by Verma et al. to obtain the best fit for the bond length and cohesive energy of the h-BN sheet. Verma et al.²⁶ predicted the values of the Young's modulus for BNNT as 1.11 TPa and 0.98 TPa in the armchair and zig-zag directions respectively. Nearly same values of Young's modulus were predicted for BNNSs i.e. 1.11 TPa and 1.0 TPa in the arm-chair and zig-zag directions, respectively. They further validated their computational model with the help of experimental results reported in the work of Chopra et al.²³ It was concluded in the research work of Verma et al.²⁶ that the Young's modulus of small diameter BNNTs increases with an increase in the diameter and attains a maximum value of 1.11 TPa at a radius of 0.95 nm, and then falls with a further increase in the diameter as illustrated in Fig. 5 (left). On the other hand, shear modulus decreases with an increase in the diameter of the BNNT and attains a constant value at a higher tube diameter as shown in Fig. 5 (right).

Fig. 5 Variation of Young's modulus (left figure) and shear strain (right) with respect to the radius of BNNT (reproduced with permission from ref. 26).

Zhang et al.¹⁴² studied the intrinsic twisting of helical BNNTs with the help of the tight-binding MD [TBMD] method, and estimated Young's and shear modulus values, as 0.850 and 0.370 TPa, respectively. In 2012, Mortazavia and his research team¹⁵⁹ performed MD based simulations to study the tensile behavior of BNNSs. In their computational work, the Tersoff potential was used in conjunction with optimized parameters obtained from the work of Matsunaga et al.^147,149 It was predicted in their computational work that BN nanostructures failed in a brittle manner, and had their Young's modulus in the range of 0.8–0.85 TPa.

MD based studies^162,163 have also been performed to investigate the effect of grafted carboxyl groups on the elastic properties of BNNTs. Yuan and Liew¹⁶² studied the effects of grafting carboxyl groups on the elastic properties of BNNTs. It was predicted in their computational work that the Young's moduli of nanotubes started deteriorating with the attachment of a carboxyl group. The zig-zag configuration of BNNTs was reported to be affected the most as compared to the armchair configuration.

It was further concluded in the research of Yuan and Liew¹⁶² that the reduction in the elastic modulus for the zig-zag direction was least affected by the quantity of attached carboxyl groups, whereas the reduction in the armchair direction depends on the quantity of the chemical group. Recently in 2014, Ghazizadeh et al.¹⁶³ conducted a MD based study on the mechanical properties of hydrogenated BNNTs. MD simulations were performed in conjunction with the universal force field (UFF) and Dreiding potentials.^164–166a It was concluded in their research that the Young's modulus of BNNTs showed a decreasing trend when hydrogen atoms were bonded with all available nitrogen atoms.

3.2. Buckling behavior

In 2001, Srivastava et al.^166b investigated the stiffness and plasticity of BNNTs. They studied the implications of rotated BN bonds on the buckling behavior of zig-zag nanotubes, and reported anisotropic strain release followed by anisotropic plastic buckling for axially compressed BNNTs. In their research paper,^166b they also proposed a “skin-effect” model for smart nano composite materials that can localize structural damage towards the skin or the surface side of the material. Nejad et al.¹⁶⁷ also investigated the buckling behavior of open-ended boron nitride nanotubes. They performed MD based simulations with the help of optimized Tersoff potential parameters obtained from the work of Liao et al.¹⁶⁸ In their conclusion, it was reported that hydrogen physisorption decreased the buckling stability of BNNT by 10% in the temperature range of 300–3000° K as compared to pristine BNNTs. Hai-Jun Shen¹⁶⁹ performed MD based simulations to compare the effect of radial compression on BNNTs and CNTs. The author predicted a comparable radial compressive stiffness for both CNTs and BNNTs. However, it was also observed in their simulations that BNNT had lower energy absorbing, load-support and deformation-support capabilities as compared to CNT. Nejad et al.¹⁷⁰ examined the behavior of fixed length BNNTs with varying diameters. MD simulations were performed with the help of Lennard-Jones¹³⁷ and harmonic potentials. It was concluded in their numerical simulations that critical buckling loads exhibited an increasing trend with an increase in the nanotube diameter.

Slotman and Fasolino¹⁷¹ compared the bending rigidity of BNNTs with that of graphene. Simulations were performed with the help of the Tersoff potential¹³⁶ in combination with optimized parameters obtained from the work of Albe et al.¹⁵¹ It was also predicted in their computational model that BNNSs had lower bending rigidity as compared to that of graphene sheets. In 2010, Liao et al.¹⁷² investigated the effects of temperature on the tensile and compressive behavior of BNNTs. They performed MD simulations in conjunction with the Tersoff potential parameters proposed by them after fitting MD simulation results to the one obtained from density functional theory. These potential parameters were suggested to be more suitable for the simulation of nano-sized BNNTs and also for simulating their bond breaking behavior. It was reported in their work that both failure as well as buckling strain decreased with an increase in temperature. It was further noted from their computational model that BNNTs failed in a chain like mode without any obvious yielding in tension, as shown in Fig. 6 whereas no chain formation was observed under compressive loading. Shokuhfar and his team¹⁷³ investigated the effect of temperature (up to 3000° K) on the buckling behavior of BNNTs using potential parameters provided by Albe et al.¹⁵¹ It was predicted in their computational model that higher modeling temperatures reduced the value of critical buckling load as well as the critical buckling strain. A similar kind of study on BNNSs was performed by Yuan and Liew.¹⁷⁴ They performed their investigation with the help of MD simulations in conjunction with the universal force field (UFF).¹⁶⁴ A higher value of buckling strain was predicted in the MD simulations for the zig-zag configuration of BN nanofillers as compared to the armchair one.

Fig. 6 Deformation morphology of a BNNT subjected to tensile loading with a chain (adapted from ref. 172, simulations, with similar conditions as in ref. 172, were performed by authors of this paper to reproduce this image).

3.3. Defects in BN nanofillers

BN nanofillers prepared experimentally are not flawless and usually contain some native defects such as vacancies, dislocation, Stone Wales defects etc. The existence of defects in BN nanofillers has already been established by experimental as well as theoretical studies.^175,176 The following section reviews MD based studies conducted on defective BN nanofillers and the effects of these defects on their mechanical and thermal properties.

Griebel et al.¹⁷⁷ used MD based simulations in conjunction with the Parrinello-Rahman¹⁷⁸ approach to study the impact of vacancy defects and functionalization on the Young's modulus of BNNTs. In their numerical work, the Tersoff potential in conjunction with parameters provided by Matsunaga et al.¹⁴⁷ was used to estimate interactions between boron and nitrogen. In the original parameters proposed by Matsunaga et al.¹⁴⁷ the possibility of bond formation between same atoms was excluded by only accounting for repulsive forces. But in the work of Griebel et al.¹⁷⁷ the potential parameters were adapted to deal with B–B and N–N bonds as well. In order to account for homo-elemental bond formation the attraction term was also included in the potential function.

In their work, it was predicted that the Young's modulus of BNNTs starts deteriorating with an increase in concentration of vacancy defects. This reduction in the Young's modulus was found to be independent of the tube-chirality. In order to better visualize the mode of failure with single defects in BN nanotubes with different chiralities, snapshots from the simulation box at different levels of strain are shown in Fig. 7. Griebel and his team¹⁷⁷ also investigated the effect of functionalization on the Young's modulus of BNNTs, and summarized the effect of functionalization as shown in Fig. 8. A general decrease in the value of Young's modulus up to 2/3^rd of pristine BNNTs can be inferred from the plotted data in Fig. 8. No significant effect of functionalizing was observed in the range of 60% to full functionalization for the zig-zag configuration, whereas a slight increase in the Young's modulus was reported for the armchair configuration between the same ranges of functionalization.

Fig. 7 Stress induced deformation and failure of BNNT with different chiralities. Color scheme refers to the level of stress per atom values from blue over green to red. Four snapshots for each chiral nanotube refer to, no strain, strain before, at and after failure states (reproduced with permission from ref. 177).

Fig. 8 Young's modulus estimated as a function of the relative functionalization degree for (6, m) BN nanotubes (reproduced with permission from ref. 177).

Shokuhfar et al.¹⁷⁹ carried out MD simulations with GROMACS harmonic potential parameters for bonded interactions. In their MD based simulations the harmonic potential parameters were obtained by fitting the potential parameters of Liao et al.¹⁶⁸ to the DFT calculations. In their simulations, the NVT ensemble was enforced to study the effect of vacancy on the critical buckling performance of BNNTs, and it was reported in the conclusion that the B–N-double vacancy was most detrimental to the structural strength, followed by B-vacancy and N-vacancy defects. It was also reported in their work that B-type vacancy was most detrimental to the fracture strain. In order to visualize the failure modes under compressive loading, snapshots from the simulations of BNNTs at the time of failure are shown in Fig. 9. It can be inferred from the snapshots that the failure morphology of BNNTs is chirality dependent.

Fig. 9 Defective BNNT after compressive buckling (a) (10,10), (b) (5,5), (c) (7,7) (reproduced with permission from ref. 179).

Le and Nguyen¹⁶¹ examined the effects of vacancies and Stone–Wales (SW) defects on the tensile properties of h-BN nanosheets. The Tersoff potential in conjunction with parameters obtained from Sevik et al.¹⁴⁶ was employed to simulate atomistic interactions. In order to avoid the overestimation of the maximum force to break the bond the cutoff function was removed from the potential function (f_c(r_ij) = 1). The notion that overestimation of maximum forces can be avoided by removing the cutoff function has been proposed by the author with the help of Fig. 10. It can be inferred from their plotted data in Fig. 10 that the potential energy and force reduce smoothly to zero as the bond strain increases with the removed cutoff function. Le and Nguyen¹⁶¹ employed finite element in conjunction with the MD based simulations. The relaxed atomic positions of B and N atoms were determined with the help of MD simulations. It was concluded in their computational model that a centrally located single-defect noticeably reduced the fracture stress and strain values, as compared to those of pristine sheets. The Young's modulus of BNNSs was reported to be unaffected due to the presence of a single vacancy but a di-vacancy had detrimental effects on the values of fracture stress and strain. It was further observed that reduction in fracture properties was dependent on the applied loading direction and orientation of Stone–Wales defects.

Fig. 10 (a) Potential energy vs. bond strain. (b) Force vs. bond strain, on keeping the bond angle value constant according to the Tersoff potential (reproduced with permission from ref. 161).

Recently in 2014, Sarma and his team¹⁸⁰ used the MD based simulations in conjunction with reactive force field potentials (ReaxFF) with parameters taken from the work of Weismiller et al.¹⁵³ It was predicted in their simulations that the point and line defects significantly reduced the tensile strength of BNNTs. Tube diameter and simulation temperature also had a significant effect on the tensile behavior of BNNTs. Similar to the work of Sarma et al.¹⁸⁰ MD based simulations with the Tersoff–Brenner potential¹⁴⁴ were also performed by Krishnan et al.¹⁸¹ to investigate the effects of vacancy and Stone–Wales (SW) defects on the failure behavior of BNNTs. The simulations were performed with the help of potential parameters proposed by Sevik et al.¹⁴⁶ It was concluded in their work¹⁸⁰ that BNNTs in the armchair configuration were highly sensitive to defects, whereas the zig-zag configurations were the least sensitive.

Pristine nanotubes (both armchair as well as zig-zag) failed in a brittle manner whereas in the case of defective nanotubes only the zig-zag configuration of nanotubes underwent brittle failure. The defect-induced plastic behavior was also observed in armchair BNNTs. The activation of different failure mechanisms in the zig-zag and arm-chair configurations of BNNTs resulted in plastic failure for the armchair configuration and brittle failure for the zig-zag configuration. In the armchair configuration of BNNTs, once the bond in pentagon breaks it leads to a failure of bonds oriented at an angle of 30° to the axis of BNNTs as shown in Fig. 11(a). This failure of bonds aligned at an angle leads to the formation of decagon-chains, as shown in Fig. 11(b). This sequence of events in the armchair configuration induces a plastic behavior in BNNTs. This plastic behavior due to SW defects had a strong radius-dependency in the armchair configuration of BNNTs, and turned into brittle behavior with radii greater than 2.5 nm as in the case of (20, 20) BNNTs. In the case of the zig-zag configuration of BNNTs, formation of a decagon due to the failure of the B–B bond is followed by the failure of the N–N bond which in turn is followed by the failure of the N–N bond and then rotation of the bond between B–N. This bond rotation further resulted in the formation of two new B–N bonds. The formation of these two new bonds brought back the pristine form and resulted in a brittle failure as compared to the armchair configuration.

Fig. 11 (a) Breaking of the B–B bond in the 5-7-7-5 defect in the arm chair configuration. (b) Formation of a decagon with the propagation of bond breaking. (c) Breaking of the B–B bond in the zig-zag configuration. (d) Rotation of defective B–N bonds to regain the pristine form (reproduced with permission from ref. 181).

Mortazavi and Cuniberti (2014)¹⁸² examined the properties of polycrystalline BN nanosheets using the Tersoff-potential¹³⁴ in conjunction with optimized parameters proposed by Matsunaga et al.¹⁴⁹ They concluded in their work that a reduction in the grain size ultimately resulted in a gradual decrease in elastic moduli of polycrystalline films. They also suggested that experimentally fabricated polycrystalline BN nanosheets might possess remarkably high mechanical properties. Moon and Hwang¹⁸³ performed MD based simulations using the Tersoff potential^133–136 in combination with parameters proposed by Albe et al.^150,151 to examine the structural and thermal behavior of BNNTs. It was reported in their computational work that the strain energy had a decreasing trend with an increase in the nanotube diameter. The strain energy was found to be proportional to the inverse of the square of the tube diameter. At higher temperatures, Stone–Wales SW defects were found in nanotubes during their thermal treatment. Also the formation energy of SW defects increased with an increase in the diameter of the nanotubes.

3.4. Failure/fracture

In addition to the elastic properties of BN nanofillers, their fracture behavior has also been investigated by many researchers with the help of a MD based platform. In 2011, Liao et al.¹⁶⁸ investigated the deformation of armchair BNNTs on being subjected to axial tensile strains using a modified version of Tersoff potential parameters developed by Albe et al.¹⁵⁰ The modified set of parameters predicted a better bond breaking behavior. A chain like failure mode (as discussed and explained earlier w.r.t. ref. 172), without any noticeable yielding, was reported in their computational work. Wei et al.¹⁸⁴ examined and compared the behavior of CNTs and BNNTs subjected to tensile as well as compressive loading. In their atomistic simulation, it was concluded that under compression, the zig-zag configuration of CNTs as well as BNNTs had a higher Young's modulus as compared to the armchair configuration of the respective nanotubes. It was also concluded from the computational model of Wei et al.¹⁸⁴ that the fracture behavior of CNTs or BNNTs is dependent on the chirality, and the nanotubes with the same chirality had similar deformation patterns regardless of the material.

Tian et al.¹⁸⁵ investigated the impact of shear displacement on the mechanical properties, wrinkling patterns and fracture behavior of h-BN nanosheets. In order to study the wrinkling in a single BN nanosheet, an analytical model was developed with geometrical parameters and boundary conditions as shown in Fig. 12. In Fig. 12, L, H and t are the length, width and thickness of the plane, respectively, E and ν are the Young's modulus and Poisson's ratio, respectively. The schematic of one wrinkle in a nanosheet is shown in Fig. 12, where σ_ξ and σ_η are the applied stresses in ξ and η directions respectively. The mathematical expressions for shear stresses σ_ξ and σ_η are provided in eqn (19), where γ is referred to as δ/H.

Fig. 12 Single wrinkle in a nanosheet (reproduced with permission from ref. 185).

The out of plane displacement ω with respect to the mode shape is given mathematically by eqn (18). In this eqn (18), A and λ refer to the amplitude and wavelength of the wrinkle.

σ_ηκ_η + σ_ξκ_ξ = 0 (20)

In the analytical model proposed by Tian et al.¹⁸⁵ the stress state equilibrium equation is established as eqn (20), where κ_η and κ_ξ are the curvatures along the directions η and ξ, respectively.

It can be inferred from the wrinkling pattern of monolayer BNNS, shown in Fig. 13(a), that wrinkling initiates at a point when in-plane shear displacement (δ) reaches a critical buckling value of 0.08 nm. At critical buckling the wrinkle initiates from the free edges and keeps the central part of the sheet wrinkle-free. Further increase in shear displacement results in a uniform distribution of wrinkle patterns with identical crest and trough as indicated in Fig. 13(c). It can be inferred from the results obtained from the simulations of Tian et al.¹⁸⁵ that the average wrinkling amplitude of the nanosheets increases with an increase in shear displacement, whereas the wrinkling wavelength has a reverse effect as indicated in Fig. 14(a) and (b), respectively.

Fig. 13 Wrinkling pattern of nanosheets with respect to shear displacement δ (reproduced with permission from ref. 185).

Fig. 14 Influence of shear displacement on the wrinkle profile (reproduced with permission from ref. 185).

An important contribution regarding MD based simulations on boron nitride came from Han et al.¹⁵⁷ who studied the effects of temperature, strain rate and crystal orientations on the mechanical properties of BNNSs. The interactions between boron and nitrogen atoms were described by the parameters proposed by Albe et al.^150,151 It was observed that the nanosheets were basically an anisotropic material and any increase in the temperature led to a decrease in the Young's modulus, fracture strength and fracture strain. A reduction in the Young's modulus was observed with an increase in the strain rate, while a reverse effect was observed on the fracture strength of BNNSs. Perim et al.¹⁸⁶ appraised the fracture behavior of defective carbon and boron nitride nanotubes in accordance with MD based simulations in conjunction with the ReaxFF force field.^152–154 The similarities and differences in their (CNT and BNNT) fracture patterns were outlined, under similar stress conditions.

3.5. Chirality & torsion

Krishnan and Ghosh¹⁸⁷ studied the effect of chirality on the elastic properties of single-walled BNNTs subjected to tensile and torsional loading. Their simulations were performed with the help of the Tersoff-potential parameter set provided by Sevik et al.¹⁴⁶ They concluded in their simulation results (as shown in Fig. 15) that with an increase in aspect ratios (length to diameter ratio of a nanotube) the elastic modulus also increases. It can also be inferred from their summarized data in Fig. 15 that at higher aspect ratios the variation in the Young's modulus is comparatively small.

Fig. 15 Elastic modulus of BNNT with respect to the aspect ratio (reproduced with permission from ref. 187).

Recently in 2014, Ansari and Ajori¹⁸⁸ studied the torsional vibration performance of double-wall BNNTs using Tersoff-potential parameters developed by Albe et al.¹⁵⁰ They extended MD simulations in conjunction with continuum modeling to estimate the shear modulus of nanotubes. The torsional frequency was predicted to be varying with geometrical parameters such as length and boundary conditions. Moreover, the torsional frequency of BNNTs was observed to be higher than that of their carbon counterparts i.e. nanotubes (CNT).

Ajori and Ansari¹⁸⁹ employed MD based simulations with a Tersoff-like potential and parameters reported by Albe and Möller¹⁵⁰ to investigate the torsional buckling response of BN nanotubes. In their simulations, (6, 6) and (10, 10) armchair BNNTs with the length varying from 5.13 nm and 15.4 nm were chosen for investigation. They predicted size dependent values of the shear modulus which differed from 0.64 TPa at 5.13 nm and finally converged to a value of 0.75 TPa as the tube length was increased. Critical torque was found to be highly dependent upon the geometrical parameters such as length, chirality and boundary conditions. The critical shear strain was predicted to be independent of chirality.

Minh-Quy Le¹⁹⁰ investigated the dependency of mechanical properties of BN nanoribbons on their size using potential parameters for boron and nitrogen from the work of Sevik et al.¹⁴⁶ In the case of rectangular nanoribbons, tensile properties were found to be significantly affected by the length to width ratios. However, size effects were found to be low in the case of square boron nitride nanosheets as compared to those of rectangular ones. It was also observed that the Young's modulus, fracture stress and fracture strain had an increasing trend with decreasing width for the zigzag configuration of nanoribbons with a fixed length. In the case of armchair BN nanoribbons with a constant length these properties varied slightly as compared to the zig-zag configuration. BN nanoribbons showed higher mechanical strength in the zigzag direction as compared to the armchair direction. Tang et al.¹⁹¹ investigated the effect of the interfacial-structure on the mechanical properties of BN nano-bamboos (BNNB).^192,193 In these simulations, interactions between boron and nitrogen atoms were modeled by the Tersoff-like potential¹³⁴ with parameters taken from the work of Albe et al.¹⁵¹ and the van der Waals interactions at interfaces were modeled with the help of the Lennard-Jones (LJ) potential.¹³⁷ BNNBs with interlocked joint interfacial structures with compressive interfacial stresses, showed higher tensile fracture strength and a Young's modulus up to 8.0 GPa and 225 GPa, respectively. Due to these interlocked joint interfacial structures a shift from the inter-planar sliding mode to the in-plane tensile elongation mode was observed in the deformation mechanism.

3.6. Thermal conductivity

BN nanofillers have exceptionally high thermal conductivities. Molecular dynamics based simulations have the capability to accurately predict the thermal conductivity of these nanofillers under different types of boundary and loading scenarios. Shen¹⁹⁴ studied the variation in thermal conductivity of BNNTs with temperature and tube diameter with the help of Tersoff type potential parameters proposed by Albe et al.¹⁵⁰ It was observed in his computational work that the thermal conductivity had a decreasing trend with an increase in the temperature and tube diameter. Sevik et al.¹⁴⁶ studied the thermal transport properties of BN nanostructures using the parameterized Tersoff-potential under a (NVE) micro-canonical ensemble. Though, BN nanostructures were found to be possessing quite high thermal conductivities, they were lower as compared to their carbon counterparts. But qualitatively, both carbon and BN nanoribbons showed similar behavior with respect to the variation in the width and edge structure (zig-zag and armchair). It was further concluded that thermal conductivities were independent of chirality. Mortazavi and Rémond¹⁵⁹ also evaluated the thermal conductivity of a single-layer of BN nanosheet through MD simulations with parameters for boron and nitrogen taken from the work of Matsunaga et al.^147–149 and predicted a value of 80 W m⁻¹ K⁻¹ for thermal conductivity of the nanosheet.

Kınacı et al.¹⁹⁵ investigated the variation in thermal conductivity (κ) of hybrid graphene/h-BN nanostructures with stripe super-lattices (alternate stripes of graphene and BN) and BN dots embedded in graphene. The parameters for boron, nitrogen and carbon interactions were developed by the author and his team using B–N parameters from Sevik et al.¹⁴⁶ and carbon interactions from the work of Lindsay and Broido.¹⁹⁶ For stripe super-lattices, it was predicted that the interface in the zigzag configuration produced a higher κ in the direction parallel to the interface than in the armchair configuration, while the perpendicular conductivity was less prone to the details of the interface and was limited by the thermal conductivity of h-BN. However, the embedded dot structures having mixed zigzag and armchair interfaces, had a stronger effect on the thermal transport properties as compared to super-lattices. Recently in 2014, Chen et al.¹⁹⁷ examined the thermal conductivity of BN nanoribbons in various transport directions (from 0° to full 30° chirality) using non-equilibrium molecular dynamics based simulations with the use of the Müller-Plathe¹⁹⁸ method and Tersoff-type three-body potential.¹⁹⁵ Edge specularity for boundary scattering (which is closely related to the edge roughness) was also studied with respect to different chiral angles. It was concluded that both thermal conductivities and edge specularity attained local maxima at a chiral angle of 19.11°. Thermal conductivity was predicted to be increasing with an increase in the ribbon length up to a certain critical length value depending upon the chiral angle.

Sevik et al.¹⁹⁹ investigated the effect of boron isotopes on the thermal conductivity of BN nanotubes. It was concluded that thermal conductivity varied in the range of 450–500 W m⁻¹ K⁻¹ and 340–400 W m⁻¹ K⁻¹ for isotopically pure (100% 11B) and natural (80% 11B) BNNTs and h-BN nanosheets, respectively. In addition to these quantitative results, a temperature dependency of the thermal conductivity of BNNTs on isotopes was also studied with the same computational model. It was also concluded from their atomistic model that effects of isotopes were more significant at lower temperatures. The scattering of phonons from isotopes at lower temperatures was associated with the reduction in the thermal transport phenomenon. Sevik et al.¹⁹⁹ further concluded that the effect of isotopes was more pronounced in 3D structures as compared to 1D and 2D structures. This dependency of thermal conductivity on structures was considered to be associated with average isotope–isotope separation, where smaller distances reduced the relaxation length, ultimately reducing the lattice thermal conductivity. Haijun Shen²⁰⁰ examined and compared the thermal conductivities of CNTs and BNNTs, under radial compression, with the help of the Tersoff-potential.¹³⁵ It was predicted in his work that under local as well global compression, the thermal conductivity exhibited a decreasing trend. But global compression had a more profound effect than local compression. In addition to this an increase in temperature resulted in the lowering of thermal conductivity. Mashreghi²⁰¹ also conducted a MD based study on thermal expansion/contraction of armchair BNNTs in various directions i.e. axial, radial and circumferential. Boron and nitrogen interactions were modeled using parameters provided by Albe et al.¹⁵⁰ Coefficients of thermal expansion (CTE) were estimated along the axial direction with different diameters. The CTE in the radial direction were reported to be lower as compared to those in the circumferential direction.

3.7. Hybrids

So far, this review paper has exclusively discussed the current state of art techniques in MD based simulations of BN nanofillers either in the pristine or the defective form. But some progress has also been made in simulating the hybrids of BN nanofillers with polymers. Liew and Yuan²⁰² investigated the structural performance of a double-walled nanotube consisting of a carbon nanotube inside a boron nitride nanotube. Liew and Yuan performed these simulations in the MD framework in combination with the universal force field.^164,203 During their simulations, the nanotubes were subjected to axial compressive loading. It was reported in their simulations that at the annealing temperatures of 3500 K and 4000 K, distortion and critical buckling load of the inner CNTs were dependent on the chiral vector.

It can be inferred from Fig. 16 that in the case of C(5, 5)@BN(10, 10) the outer BNNT starts showing instability at ε = 11.83% while the inner carbon nanotube remains stable till the higher values of strains (12.74%). But in the case of C(5, 5)@C(10, 10), both tubes became unstable simultaneously at lower strain values (9.10%) than in the previous case. Hence it can be concluded from the work of Liew and Yuan²⁰² that BNNT acts as a better protection than a CNT when used as an outer protection tube.

Fig. 16 Coaxial nanotubes: (a) optimized C(5,5)@BN(10,10), (b) optimized C(5,5)@C(10,10) at different axial compressive strains (ε) at the temperature of 300° K (reproduced with permission from ref. 202).

Yuan et al.^204,205 investigated the structural stability and elastic properties of bilayer and trilayer graphene and boron nitride nanosheets by employing MD based simulations in conjunction with the universal force field.¹⁶⁴ Their investigation on multiplayers of nanosheets was based on analyzing and comparing the binding energy, van der Waals interactions between layers and radial distribution function (RDF). In their computational model the full width half maximum (FWHM) of RDF was considered to be closely related to the integrity of the atomistic structure. The structures with sharp peaks and smaller values of FWHM for RDF were predicted to possess better integrity. They performed simulations with bilayers and trilayers of nanosheets, and compared the results with the monolayer of graphene and BNNS. FWHM values obtained from the first, second and third peaks (refer to Table 4) suggested a small deviation between the bilayer and single sheets of BNNS, and also these changes in values of FWHM for BNNS are comparatively lower as compared to graphene. These trends in FWHM between single and bilayers indicate that bilayer BNNS can be considered as a more integrated structure as compared to graphene.²⁰⁴

Table 4 FWHM values of single, bilayer and trilayer sheets of graphene and BNNS

Material	FWHM (1^st peak)	FWHM (2^nd peak)	FWHM (3^rd peak)
Gr (monolayer)	0.00258 nm	0.00221 nm	0.00257 nm
Gr–Gr (bilayer)	0.00236 nm	0.00225 nm	0.00234 nm
Gr–Gr–Gr (trilayer)	0.00231 nm	0.00201 nm	0.00255
BN (monolayer)	0.00301 nm	0.00217 nm	0.0301 nm
BN–BN (bilayer)	0.00302 nm	0.00219 nm	0.00311 nm
BN–Gr–BN (sandwich)	0.00221 nm	0.00184 nm	0.00255 nm

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Young's moduli of the combined bilayer nanosheets were found to be lower than those of the corresponding monolayers for graphene and BNNS. In the case of BN nanosheets, the reduction in the Young's modulus was less as compared to that of graphene nanosheets. Yuan and Liew²⁰⁵ also performed MD simulations to investigate the structural stability and high temperature distortion-resistance of a sandwich structure of graphene and BN nanosheets. The optimal interlayer equilibrium distances between graphene nanosheets (in the case of graphene sandwich) and graphene and BN nanosheets (in the case of the BNNS-graphene sandwich) were reported as 0.347 nm and 0.341 nm, respectively. Equilibrium distance was the spacing between nanosheets at which repulsive and attractive forces are zero. It was further predicted in their work that graphene sandwiched between BN nanosheets proved to be a more integrated structure as compared to monolayer graphene sandwiched between graphene nanosheets in a crystal structure on the basis of FWHM values obtained from the first peak in RDF as provided in Table 4 and illustrated in Fig. 17.

Fig. 17 RDF for monolayer graphene, three layers of graphene and the sandwich structure of the graphene sheet between BNNS (reproduced with permission from ref. 205).

Song and Medhekar²⁰⁶ also studied the hybrid nanostructures of graphene and h-BN nanoribbons. In their MD based simulations, interatomic interactions were modeled with the help of the Tersoff potential¹³⁶ with optimized parameters taken from the work of Matsunaga et al.¹⁴⁹ The thermal transport in the direction perpendicular to the interface was reported to be less affected by the composition. On the other hand, it was also concluded in their work that the thermal transport in these hybrid nanoribbons was significantly affected by the interface scattering and the presence of a low thermal conductivity component i.e. h-BN in the path of the thermal current. However, thermal transport parallel to the interface was limited by the graphene component. Shen Haijun²⁰⁷ investigated the compressive strength of single-walled BNNTs and the hybrid structure of BNNTs embedded into CNTs. The simulations were performed using B–N potential parameters from Albe et al.¹⁵⁰ while non-bonded interactions were treated with the help of LJ potential parameters.¹⁶⁴ It can be inferred from their conclusions that hybrid structures show higher compressive strength as well as melting temperatures as compared to single BNNTs. In 2013, Zhao and Xue¹⁶⁰ examined the mechanical properties of hybrid structures of graphene and h-BN. While B–N, B–C and N–C interactions were modeled by mixing of potential parameters of Matsunaga et al.¹⁴⁸ parameters provided by Lindsay and Broido¹⁹⁶ were used to model carbon–carbon interactions. Hybrid sheets were predicted to show plastic behavior that was absent in pure h-BN and graphene nanosheets. The Young's modulus for hybrid nanosheets showed a decreasing trend with increasing concentrations of BN, irrespective of BN shapes and distribution. DFT based simulations have also been employed to investigate the hybrid structures of BN nanofillers. Zhang et al.²⁰⁸ investigated the electronic and transport properties of fullerene and vanadium based peapods of BNNTs using DFT calculations. Based on their simulation results, BNNT based peapods were predicted to be semiconducting with a narrow bandgap. Further details regarding this behavior of BNNTs can be found in their paper.²⁰⁸

3.8. Nanocomposites

So far, this review paper has exclusively discussed nanostructures such as BN nanotubes, BN nanosheets and hybrid structures of these BN nanofillers with graphene or CNT. But a decent amount of progress has also been made in simulating BN nanofiller based nanocomposites. This section focuses on the MD based simulations of BN nanocomposites. Nasrabadi and Foroutan²⁰⁹ used MD based simulations for investigating the interfacial binding of BN nanotubes with poly[m-phenylenevinylene-co-(2,5-dioctyloxy-p-phenylenevinylene)] PmPV, polystyrene, and polythiophene. In their simulations, MM3²¹⁰ and Lennard Jones force fields have been employed for simulating the bonded and non-bonded interactions (interface), respectively. From their simulations, it can be inferred that the interaction energy of BNNT–polymer composites is strongly influenced by the specific monomer structure and nanotube radius. On the basis of higher values of the interaction energy of BNNT–polymer composites, they suggested that the BNNTs could be more efficient nano-fillers than the CNTs for nano-composite reinforcement applications. The effect of polarization of BNNTs on the interfacial strength was also studied in the same work²⁰⁹ and it was concluded that interfacial strength increased due to polarization of BNNTs.

Fatemi and Foroutan²¹¹ investigated the dispersion of two BNNTs in an aqueous solution of triton X-100 as a surfactant, using MD based simulations in combination with the AMBER99 force field.²¹² The presence of the surfactant resulted in an increase in the surface angle between the two nanotubes and hence more gap was created between two BNNTs leading to the dispersion of the BNNTs. The presence of the surfactant shortened the bond distance between the nitrogen atom of the BNNTs and the hydrogen atom of water molecules. Moreover, the estimated values of the radius of gyration of the surfactant and the interfacial angle between two BNNTs shed more light on the arrangement of the surfactants around the BNNTs with and without the presence of water molecules.

Wang and his team²¹³ applied axial compression on copper-filled BNNTs to study the buckling behavior of BNNTs using the MD framework. The universal force field, with parameters taken from the work of Jung et al.²¹⁴ and Zhou et al.²¹⁵ was used for simulating the atomic interactions between the atoms. It was observed that nanotubes with different chiralities exhibited different deformation patterns and energy loss. Side and top views for different tubes at the buckling stage have been shown in Fig. 18. Fig. 18(a) shows a 49 Cu@BNNT (7, 7) i.e. 49 copper atoms in BNNT (7, 7), Fig. 18(b) is for 49 Cu@BNNT (11, 11), both showing different deformation patterns. Fig. 18(c) and (d) depicted the same atomic configuration i.e. a 195 Cu@BNNT (11, 11) nanotube, but at different simulation steps. It is clear from these two figures that a regular hexahedron like shape evolved before buckling of the tube and at the time of buckling, this shape finally transformed into a square shape. In addition to these deformation patterns, BN nanotubes filled with a larger number of copper atoms were found to be more stable as compared to the nanotubes filled with a smaller number of copper atoms.

Fig. 18 Deformation pattern of (a) 49 Cu@BNNT(7,7), (b) 49Cu@BNNT(11,11), (c) 195CuBNNT(11,11) at a simulation time t, (d) 195Cy@BNNT(11,11) after an increased simulation time t + Δt (reproduced with permission from Wang et al.²¹³).

Bari et al.²¹⁶ carried out MD simulations to study the non-covalent interactions between polyvinyl pyrrolidone (PVP) and isolated BNNSs in a water medium. Interactions between the PVP molecule and water atoms were treated with a general AMBER force field^217,218 while the LJ potential was used to take care of the interactions between the BNNSs and water with parameters from the work of Mayo et al.²¹⁹ It was predicted in their research that in the initial stages of simulation, PVP chains showed a random thermal motion in water, and finally reached within an approachable distance to BNNS, so that, interactions could be possible between BNNS and PVP. It can be inferred from their conclusion that after the chain reaches within close proximity to BNNS, the distance between these two does not change any further. This emphasized that in the presence of water, there exists a strong affinity between the PVP chain and the BNNS. Griebel and Hamaekers²²⁰ examined the elastic properties of BNNT embedded into amorphous silicon–boron–nitride ceramics. The computational work was performed using Parrinello–Rahman's approach^178,221 with potential parameters from the work of Matsunaga et al.¹⁴⁷ The results obtained in their computational model further suggested that out of Si3BN5, Si3B2N6 and Si3B3N7 matrices, Si3B3N7 had the highest Young's modulus with the highest elastic range. Further examination of the properties of a ceramic composite material made of Si3B3N7 as the matrix and BN nanotubes as the filler was performed in the computational work of Griebel and Hamaekers.²²⁰ It was further suggested that the simple macroscopic rule-of-mixture, which just accounts for the volume fraction of the nanotube, could be used for composites with periodically infinitely long BN nanotubes.

Thomas and his team²²² studied the interactions of BNNTs and CNTs embedded in lipid bilayers. They employed MD based simulations in conjunction with CHARMM36 force field parameters^223,224 for investigating the interaction of BNNTs and CNTs with lipid bilayers. It was predicted in their simulations that the stability of BNNTs in cell membranes was comparable to that of functionalized CNTs. It was further observed in their simulations that a narrow BNNT might be blocked at one end by lipid molecules as indicated in Fig. 19(a), while a wider BNNT was only partially blocked by lipid molecules as shown in Fig. 19(b). On the other hand, CNT was reported to be completely occupied by lipid molecules as illustrated in Fig. 19(c). Yuan and Liew²²⁵ examined the strength of a coaxially placed silicon wire (SiNW) inside a BN nanotube. MD based studies were performed in conjunction with the universal force field^147,199 to investigate the effect of the silicon nanowire on the structural strength of BN nanotubes. Two different types of silicon nanowires (SiNWs) were simulated by Yuan and Liew in their atomistic model as shown in Fig. 20. An axial monatomic chain with a helical shell of SiNWs, coaxially aligned inside a BNNT (10, 10) and only an axially aligned monatomic chain of SiNW inside the BNNT (5, 5) are shown in Fig. 20(a), (b) and (c), (d), respectively. It was reported in their work that BNNT and SiNW nanocomposites were more stable as compared to their respective pristine forms. It was also concluded in their research work that the deformation pattern and energy loss varied from form to form, whereas the buckling strain remained the same for the pristine BNNT and BNNT/Si NW composites.

Fig. 19 (a) Blocked lipid at the end of narrow BNNT, (b) partially blocked lipid along with a wider BNNT, (c) completely occupied CNT with the lipid molecules (reproduced with permission from Thomas et al.²²²).

Fig. 20 (a) Axial monatomic and helical shell of silicon nanowires coaxially aligned inside the BNNT (10,10), (b) side view of the atomic configuration shown in part (a). (c) Only a monatomic chain of silicon nanowire inside BNNT (5,5), (d) side view of the atomic configuration shown in part (c) (reproduced with permission from Yuan and Liew²²⁵).

It can be inferred from the preceding review that Tersoff is the most common among the potentials for simulating interatomic interactions in BN nanofillers. In addition to the Tersoff type potential, ReaxFF, Harmonic, LJ, MM3, AMBER and CHARM potentials were also employed by few researchers for the modeling of BN nanofillers. A summary of different types of potentials employed in various MD simulations for estimating the mechanical and thermal behaviors of BN nanofillers is provided in Table 5.

Table 5 Compilation of different types of potentials employed in different molecular dynamics based simulations to predict the mechanical and thermal behaviors of BN nanofillers

Potential parameters	Any modification	Ref.	Work done
Albe et al.^150	—	157	Effect of temperature and strain rate on mechanical properties of BNNS
	—	169	Radial compression of BNNTs
	—	171	Bending rigidity of BNNSs
	—	183	Structural and thermal behavior of BNNTs with SW defects
	—	184	Fracture and buckling behavior of BNNTs
	—	188	Torsional vibrational performance of BNNTs
	—	189	Torsional and buckling behavior of BNNTs
	—	191	Effect of interfacial structure on mechanical properties of BNNBs
	—	194	Effect of diameter and temperature on thermal conductivity of BNNTs, tensile properties of BNNTs
	—	201	Effect of temperature and diameter on coefficients of thermal (CTE) expansion of BNNTs
	—	207	Thermal-stability and compressive behavior of BNNT and BNNT@CNT
Sevik et al.^146	—	146	Characterization of thermal transport in BN nanofillers
	Cut-off function f(r) = 1	161	Effects of vacancy and SW defects on mechanical properties of BNNSs
	Tersoff–Brenner potential	181	Effect of vacancy and SW defects on failure behavior of BNNTs under tension
	—	187	Effect of chirality on elastic properties of BNNTs under tension and torsion
	R ij = Sij	190	Size effects on mechanical properties of BNNRs
	—	197	Thermal conductivity of graphene–BNNS hybrids
	—	199	Impact of isotopes on thermal transport in BN nanostructures
Matsunaga et al.^147	—	159	Investigation of tensile properties and thermal conductivity of BNNS
	—	160	Mechanical properties of hybrid graphene and BNNS
	Inclusion of attractive potential term to deal with homo-elemental bonds	177	Effects of vacancies and functionalization on Young's modulus of BNNTs
	—	182	Mechanical properties of polycrystalline BNNSs
	—	206	Thermal transport in hybrid of graphene and h-BN
	—	220	Elastic properties of BNNTs embedded in amorphous Si–B–N
Sekkal et al.^145	Slightly modified to get better fit to the bond length and bond energy	26	Elastic and shear moduli of BN nanofillers
Liao et al.^168	Modified version of parameters from Albe et al. to, better capture of properties of nanofillers, predict better bond breaking behavior	168	Deformation behaviors of BNNTs under axial tensile strains in armchair direction
	—	167	Buckling behavior of pristine and hydrogenated BNNTs
	—	172	Effect of temperature on tensile and compressive behavior of BNNTs
UFF & Dreiding	—	163	Mechanical properties of hydrogenated BNNTs
	—	174	High-temperature thermal stability and in-plane compressive properties of a BNNT
	—	202	High-temperature thermal stability and axial compressive properties of a coaxial CNT@BNNT
	—	205	Structure stability and high-temperature distortion resistance of trilayer hybrid of graphene and BNNS
	—	204	Structural stability and elastic properties of bilayer hybrid of graphene and BNNS
	—	213	Axial compression of copper-filled BNNT
	—	225	Compressing behavior of coaxial silicon nanowires inside a BNNT
Harmonic potential or LJ potential	Harmonic potential for bonded interactions and LJ potential non-bonded interactions	170	Buckling behavior of BNNTs under axial compression
	Parameters from Liao et al. were used to obtain harmonic potential parameters for bonded interactions	179	Effects of vacancy defects on the compressive buckling of boron nitride nanotubes
ReaxFF	—	180	Point and line defects on BNNTs
	—	186	Fracture behavior of defective BNNTs
MM3 + LJ	—	209	Interactions between polymers SWBNNT
AMBER force field + LJ	—	211	Dispersion of BNNTs inside water by triton X-100 surfactant
	—	216	Liquid phase exfoliation and crumpling of BNNSs
CHARMM36 force field	—	222	Stability of BNNTs into lipid bilayers

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4. Conclusion and future aspects

In this review, the authors have summarised recent progress made in the field of atomistic modeling of BN nanofillers. Due to their exceptional properties, BN nanofillers are emerging as potential candidates for a wide range of practical applications. On the basis of a wide band gap in the range of ∼5.5 eV, higher stability against oxidation even at higher temperatures and thermal stability, BN nanofillers have emerged as superior to carbon based nanofillers for specific applications and hence their further exploration has always drawn the attention of researchers.

From the preceding review on atomistic modeling of BN nanofillers, it can be concluded that these approaches are capable of producing results that are in good agreement with experimental results. When compared to each other, each of these techniques has its own advantages along with certain limitations associated with it, as discussed in the review. In spite of the fact that DFT based simulations are more accurate, they have limitations of being computationally intensive and limit the overall system size. Although, tight binding calculations are faster they need further improvements, like in their scalability. Continuum based FEM techniques proved to be accurate for predicting the global properties, but have limitations while modeling the localised effects such as bond formation and rotation under an atomistic atmosphere. When it comes to MD, Tersoff type potentials have attracted more attention among the researchers for simulating the bonded interactions, due to accurate parameterizations that produce an accurate fit with the DFT calculations and experimental results. So far, different sets of Tersoff type potential parameters have been proposed by the researchers. Sekkal et al.,¹⁴⁵ Sevik et al.,¹⁴⁶ Matsunaga et al.,¹⁴⁷ and Albe et al.¹⁵⁰ are the prominent contributors for these parameters. Due to the empirical nature of these parameters, an overall scattered result in terms of mechanical, fracture and thermal properties has been reported in MD based simulations performed with these parameters. Some studies^21,161 on mechanical properties of BN nanofillers with minor modifications in the cut-off function have predicted better results that are in good agreement with DFT and experiments. These studied were performed using the parameters from Sevik et al.¹⁴⁶ However, no study with other sets of parameters, used in conjunction with a modified cut-off function, has been reported yet. So, a scope of further improvements, such as cut-off function optimisation, exists, as far as MD simulations of BN nanofillers are concerned.

Overall, atomistic modeling and simulation techniques are proving to be an excellent complement to the experimental techniques to explore further research directions. Properties which cannot be explored with experiments can be traceable using these techniques and hence, further improvements in these techniques will make them more versatile.

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