Tersoff-Brenner Potential Research Articles

Development of Virtual Internal Bond method based material model for Carbon fiber and its application to Carbon fiber reinforced epoxy system

The present work contributes to the current state of art in fracture modeling of carbon fiber through development of virtual internal bond (VIB) model based on Tersoff-Brenner potential for carbon fiber. The underlying VIB method is multiscale in nature and the stress–strain relation which while retaining the microscopic attributes handles crack initiation and propagation without addition criterion. A variant of VIB method viz. multi-dimensional virtual internal bond (MVIB) method is employed to model epoxy resin, wherein the stiffness involved in the interaction potential is penalized based on the strain levels to capture damage. Following calibration based on elastic properties and tensile strength, the potential of the current work is demonstrated through simulation of damage in two-dimensional assemblage of carbon and epoxy under the action of bi-axial loading characterized via bi-axiality ratio.

Bending a graphene cantilever by a diamagnetic force

The application of a magnetic field perpendicular to the surface of a graphene cantilever generates a bending force owing to the strong anisotropy of the magnetic susceptibility. We calculate the mechanically stable equilibrium shape of a graphene cantilever in the presence of a magnetic field by minimizing the magnetic and bending energies, which are calculated using the tight-binding model and the Tersoff–Brenner potential, respectively. Furthermore, the introduction of a continuous model enables the size-dependence of the displacement by bending to be considered.

Assessment, improvement, and comparison of different computational tools used for the simulation of heat transport in nanostructures

In this work we compare different implementations of two interatomic potential models, one the empirical Tersoff–Brenner and the other the semi-empirical tight-binding, to be used in the thermal transport study of silicon nanosystems. The calculations are based on molecular dynamics simulations. In the case of Tersoff–Brenner potential, two free software packages were used, while for tight-binding potential, an in-house code was developed. Both approaches require an enormous amount of computing effort, so the use of acceleration tools for adequate performance is crucial. We present a detailed study of each computational tool used: efficiency, advantages and disadvantages, and the results of application to the calculation of thermal conductance of structured silicon nanocrystals subjected to a temperature gradient.

Comparing different multibody reactive potentials for the elastic properties and nonlinear mechanics of the carbon nanostructures

A multiscale model in conjunction with the finite element method is employed to investigate the accuracy of the different reactive empirical bond order potentials involving material and geometric nonlinearities. The four different sets of empirical parameters for the Tersoff-Brenner potential are considered in the present study and among these four sets of empirical parameters, one set is newly proposed herein and three are taken from the available reference sources. The atomistic-continuum coupling is established using kinematics of quadratic type Cauchy-Born rule for establishing the energy density of the unit cell. The finite element method is used to solve the governing equations at the continuum scale. The revised set of empirical parameters is found better for predicting the elastic properties and bond length than the formerly proposed three sets of empirical parameters when compared to those reported using quantum mechanics calculations. The results of the newly proposed potential are compared with DFT based continuum models for a wide range of numerical problems. The proposed potential parameters are tested for a variety of problems such as the energy of the carbon nanotubes with different radii, torsion of the carbon nanotubes, free vibrations of the graphene sheets and carbon nanotubes and numerical simulations of the nanoindentation experiments for circular graphene sheets.

Elastic Properties and Nonlinear Elasticity of the Noncarbon Hexagonal Lattice Nanomaterials Based on the Multiscale Modeling

Abstract This study presents the elastic properties and nonlinear elasticity of the two-dimensional noncarbon nanomaterials of hexagonal lattice structures having molecular structure XY. Four nitride-based and two phosphide-based two-dimensional nanomaterials, having graphene-like hexagonal lattice structure, are considered in the present study. The four empirical parameters associated with the attractive and repulsive terms of the Tersoff–Brenner potential are calibrated for noncarbon nanomaterials and tested for elastic properties, nonlinear constitutive behavior, bending modulus, bending and torsional energy. The mathematical identities for the tangent constitutive matrix in terms of the interatomic potential function are derived through an atomistic–continuum coupled multiscale framework of the extended version of Cauchy–Born rule. The results obtained using newly calibrated empirical parameters for cohesive energy, bond length, elastic properties, and bending rigidity are compared with those reported in the literature through experimental investigations and quantum mechanical calculations. The continuum approximation is attained through the finite element method. Multiscale evaluations for elastic properties and nonlinear stretching of the nanosheets under in-plane loads are also compared with those obtained from atomistic simulations.

Fracture Toughnesses and Crack Growth Angles of Single-Layer Graphyne Sheets

Recently, Shang et al. (Angew Chem Int Ed 57(3):774–778, 2018) have developed a method to synthesize ultrathin (around 1.9 nm) graphyne nanosheets. We reported here the mixed-mode I–II fracture toughnesses and crack growth angles of single-layer graphyne sheets using molecular dynamics (MD) simulations and the finite element (FE) method based on the boundary layer model, respectively. The various carbon–carbon bonds of graphyne sheets in the FE method are equated with the nonlinear Timoshenko beams based on the Tersoff–Brenner potential, where all the parameters of the nonlinear beams are completely determined based on the continuum modeling. All the results from the present FE method are reasonable in comparison with those from our MD simulations using the REBO potential. The present results show that both the critical stress intensity factors (SIFs) and the crack growth angle strongly depend on the chirality and loading angle $$\varphi $$ ( $$\varphi =90^{\circ }$$ and $$\varphi =0^{\circ }$$ representing pure mode I and pure mode II, respectively). Meanwhile, the fracture properties of single-layer cyclicgraphene and supergraphene sheets are also studied in order to compare with those of the graphyne sheets. The critical equivalent SIFs are derived as $$1.55<K_{{\text {eq-cy}}}$$ (cyclic) $$<1.95$$ nN A $$^{-3/2}$$ , $$1.64<K_{{\text {eq-gy}}}$$ (graphyne) $$<2.64$$ nN A $$^{-3/2}$$ and $$0.61<K_{{\text {eq-su}}}$$ (super) $$<2.04$$ nN A $$^{-3/2}$$ in the corresponding zigzag and armchair sheets using the MD simulations, while the SIFs are $$0.32<K_{{\text {eq-cy}}}$$ (cyclic) $$<0.48$$ nN A $$^{-3/2}$$ , $$1.96<K_{{\text {eq-gy}}}$$ (graphyne) $$<2.49$$ nN A $$^{-3/2}$$ and $$1.42<K_{{\text {eq-su}}}$$ (super) $$<2.95$$ nN A $$^{-3/2}$$ using the FE method. These findings should be of great help for understanding the fracture properties of carbon allotropes and designing the carbon-based nanodevices.

A derivative-free upscaled theory for analysis of defects

Notwithstanding its recent focus on microstructure-driven non-classical aspects, a breakthrough model in continuum mechanics that can evolve the macroscopic deformation of a solid body undergoing fracture whilst systematically incorporating the microstructural information remains elusive. In addressing this issue, we presently obtain, based on the molecular level information, a derivative-free balance law pertaining to a higher scale of interest and useful in a continuum or discrete setting. Derived using a probabilistic projection technique, the law exploits certain microstructural information in a weakly unique manner. The projection generalizes the notion of directional derivative and, depending on the application, may be interpreted as a discrete Cauchy–Born map with the structure of the classical deformation gradient emerging in the infinitesimal limit. As an illustration, we use the Tersoff–Brenner potential and obtain a discrete macroscopic model for studying the deformation of a single-walled carbon nanotube (SWCNT). The macroscopic (or continuum) model shows the effect of chirality – a molecular phenomenon – in its deformation profile. We also demonstrate the deformation of a fractured SWCNT, which is a first-of-its-kind simulation, and predict crack branching phenomena in agreement with molecular dynamics simulations. As another example, we have included simulation results for fractured SWCNT bundle with a view to establishing our claim regarding the efficacy of the proposed method.

Effect of initial strain and material nonlinearity on the nonlinear static and dynamic response of graphene sheets

Computationally efficient multiscale modelling based on Cauchy–Born rule in conjunction with finite element method is employed to study static and dynamic characteristics of graphene sheets, with/without considering initial strain, involving Green–Lagrange geometric and material nonlinearities. The strain energy density function at continuum level is established by coupling the deformation at continuum level to that at atomic level through Cauchy–Born rule. The atomic interactions between carbon atoms are modelled through Tersoff–Brenner potential. The governing equation of motion obtained using Hamilton's principle is solved through standard Newton–Raphson method for nonlinear static response and Newmark's time integration technique to obtain nonlinear transient response characteristics. Effect of initial strain on the linear free vibration frequencies, nonlinear static and dynamic response characteristics is investigated in detail. The present multiscale modelling based results are found to be in good agreement with those obtained through molecular mechanics simulation. Two different types of boundary constraints generally used in MM simulation are explored in detail and few interesting findings are brought out. The effect of initial strain is found to be greater in linear response when compared to that in nonlinear response.

Large deformation static and dynamic response of carbon nanotubes by mixed atomistic and continuum models

Multiscale modelling based finite element method is employed to investigate the nonlinear static and dynamic response of single walled carbon nanotubes and graphene sheets. The study is carried out using a four noded Kirchhoff rectangular field consistent finite element formulated in cylindrical coordinate system considering geometric and material nonlinearities. The atomic interactions are modelled through the Tersoff–Brenner potential. To avoid membrane locking, the circumferential strain is interpolated consistently using smoothed interpolation functions derived through the least square minimization. The accuracy and efficiency of the present approach are investigated by comparing the multiscale modelling based results with those obtained through molecular mechanics simulation. The quantitative contributions of material nonlinearity and different types of geometric nonlinearities (Green–Lagrange and von–Karman type nonlinearity) on the nonlinear response of carbon nanotubes and graphene sheets subjected to different kinds of load are brought out.

Mechanical behavior enhancement of defective graphene sheet employing boron nitride coating via atomistic study

The most stable form of boron nitride polymorph naming hexagonal boron nitride sheet has recently been widely concerned like graphite due to its interesting features such as electrical insulation and high thermal conductivity. In this study, the molecular dynamic simulations are implemented to investigate the mechanical properties of single-layer graphene sheets under tensile and compressive loadings in the absence and presence of boron–nitride coating layers. In this introduced hybrid nanostructure, the benefit of combining both individual interesting features of graphene and boron–nitride sheets such as exceptional mechanical and electrical properties can be simultaneously achieved for future potential application in nano devices. The influences of chiral indices, boundary conditions and presence of mono-atomic vacancy defects as well as coating dimension on the mechanical behavior of the resulted hybrid structure are reported. The interatomic forces between the atoms are modeled by employing the AIREBO and Tersoff–Brenner potentials for carbon–carbon and boron–nitrogen atoms in each layer, respectively. Furthermore, the van der Waal interlayer forces of carbon–boron and carbon–nitrogen are estimated by the Lennard–Jones potential field. Besides the potential improvement in electrical and physical properties of the nanostructure, it is demonstrated that the buckling load capacity of the fully coated graphene sheet with 3% concentration of mono-atomic vacancy defects noticeably enhances by amounts of 24.1%.

Free vibration analysis of single-walled boron nitride nanotubes based on a computational mechanics framework

Free vibration behaviors of single-walled boron nitride nanotubes are investigated using a computational mechanics approach. Tersoff-Brenner potential is used to reflect atomic interaction between boron and nitrogen atoms. The higher-order Cauchy-Born rule is employed to establish the constitutive relationship for single-walled boron nitride nanotubes on the basis of higher-order gradient continuum theory. It bridges the gaps between the nanoscale lattice structures with a continuum body. A mesh-free modeling framework is constructed, using the moving Kriging interpolation which automatically satisfies the higher-order continuity, to implement numerical simulation in order to match the higher-order constitutive model. In comparison with conventional atomistic simulation methods, the established atomistic-continuum multi-scale approach possesses advantages in tackling atomic structures with high-accuracy and high-efficiency. Free vibration characteristics of single-walled boron nitride nanotubes with different boundary conditions, tube chiralities, lengths and radii are examined in case studies. In this research, it is pointed out that a critical radius exists for the evaluation of fundamental vibration frequencies of boron nitride nanotubes; opposite trends can be observed prior to and beyond the critical radius. Simulation results are presented and discussed.

The chirality-dependent fracture properties of single-layer graphene sheets: Molecular dynamics simulations and finite element method

The chirality-dependent mixed-mode I-II fracture toughness and crack growth angles of single-layer graphene sheets are determined using molecular dynamics (MD) simulations and the finite element (FE) method based on the boundary layer model, respectively. The carbon–carbon bond in the FE method is equivalent to a nonlinear Timoshenko beam based on the Tersoff–Brenner potential. All the results of the present FE method agree well with those of our MD simulations performed using the REBO potential. The chiral crack angles of α = 0° (zigzag), 15°, 30° (or 90°, armchair), and 45° at different loading angles from 0° ≤ φ ≤ 90° (φ = 90° for mode I and φ = 0° for mode II) are studied. The present results show that both critical stress intensity factors (SIFs) and crack growth angles strongly depend on the chiral angle α, the dimensions [in two-dimensional (2D) or three-dimensional (3D) states], as well as the temperature, for a given loading angle φ. The critical equivalent SIFs change from 2.52 to 4.07 nN Å−3/2 in the 2D state and from 2.46 to 5.06 nN Å−3/2 in the 3D state at different loading angles. The SIFs are around one order of magnitude smaller than those of ordinary steel, which indicates that chiral graphene is remarkably brittle in contrast to its ultrahigh strength. These findings should be of great help in understanding the chirality-dependent fracture properties of graphene sheets and designing graphene-based nanodevices.

Large stretchability and failure mechanism of graphene kirigami under tension.

From the macro- to the nanoscale, kirigami structures show novel and tunable properties by tailoring the original two-dimensional sheets. In this study, the large stretchability and failure behavior in graphene nanoribbon kirigami (GNR-k) are obtained using the finite element (FE) method and molecular dynamics (MD) simulations. The carbon-carbon bond in the FE method is equivalent to a nonlinear Timoshenko beam based on the Tersoff-Brenner potential. All the results from the present FE method are in reasonable agreement with those from our MD simulations using the REBO potential. These results from the two methods show that the maximum ultimate strain of GNR-k (around 100%) is around 4 times higher than that of a pristine graphene nanoribbon (GNR), whereas the minimum ultimate stress of GNR-k is around one order of magnitude lower than that of the GNR. In particular, the large stretchability of GNR-k is indirectly proven to be mainly derived from the out-of-plane bending deformation by measuring the nonlinear mechanical properties of paper kirigami. Our results provide physical insights into the origins of the large stretchability of GNR-k and make GNR-k applicable to flexible nanodevices.

Improved tensile and buckling behavior of defected carbon nanotubes utilizing boron nitride coating – A molecular dynamic study

Synthesizing inorganic nanostructures such as boron nitride nanotubes (BNNTs) have led to immense studies due to their many interesting functional features such as piezoelectricity, high temperature resistance to oxygen, electrical insulation, high thermal conductivity and very long lengths as physical features. In order to utilize the superior properties of pristine and defected carbon nanotubes (CNTs), a hybrid nanotube is proposed in this study by forming BNNTs surface coating on the CNTs. The benefits of such coating on the tensile and buckling behavior of single-walled CNTs (SWCNTs) are illustrated through molecular dynamics (MD) simulations of the resulted nanostructures during the deformation. The AIREBO and Tersoff-Brenner potentials are employed to model the interatomic forces between the carbon and boron nitride atoms, respectively. The effects of chiral indices, aspect ratio, presence of mono-vacancy defects and coating dimension on coated/non-coated CNTs are examined. It is demonstrated that the coated defective CNTs exhibit remarkably enhanced ultimate strength, buckling load capacity and Young's modulus. The proposed coating not only enhances the mechanical properties of the resulted nanostructure, but also conceals it from few external factors impacting the behavior of the CNT such as humidity and high temperature.

Mechanical properties of pristine and nanoporous graphene

We present molecular dynamics simulations of monolayer graphene under uniaxial tensile loading. The Morse, bending angle, torsion and Lennard-Jones potential functions are adopted within the mdFOAM library in the OpenFOAM software, to describe the molecular interactions in graphene. A well-validated graphene model using these set of potentials is not yet available. In this work, we investigate the accuracy of the mechanical properties of graphene when derived using these simpler potentials, compared to the more commonly used complex potentials such as the Tersoff-Brenner and AIREBO potentials. The computational speed up of our approach, which scales O(1.5N), where N is the number of carbon atoms, enabled us to vary a larger number of system parameters, including graphene sheet orientation, size, temperature and concentration of nanopores. The resultant effect on the elastic modulus, fracture stress and fracture strain is investigated. Our simulations show that graphene is anisotropic, and its mechanical properties are dependant on the sheet size. An increase in system temperature results in a significant reduction in the fracture stress and strain. Simulations of nanoporous graphene were created by distributing vacancy defects, both randomly and uniformly, across the lattice. We find that the fracture stress decreases substantially with increasing defect density. The elastic modulus was found to be constant up to around 5% vacancy defects, and decreases for higher defect densities.

Multiscale Modeling of Dynamic Characteristics of Carbon Nanotube Reinforced Nanocomposites

A unique atomic structure of carbon nanotube unveils outstanding properties. This makes it potentially highly valued reinforcing material to strengthen composite materials. The methodology is established in this research paper to investigate the static and dynamic characteristics of the nanocomposites. Repol polypropylene H110MA is used as a matrix material along with the different percentages of single-walled carbon nanotubes (SWCNTs). A concept of representative volume element (RVE) is considered to study the various properties of the nanocomposite material. The carbon–carbon bond of nanotube is modeled using Tersoff–Brenner potential and represented by space frame element. The matrix material properties are tested in the laboratory which are further used to model it and represented by three-dimensional continuum elements. The interaction between nanotube and polymer matrix is modeled using “Lennard–Jones 6-12” potential represented by nonlinear spring elements. The effect of reinforcement, chirality, % volume of SWCNT, atomic vacancy defect and Stone–Wales defect on the properties of nanocomposite are investigated. To see the effect of reinforcement, the eigenvalues of the RVE are extracted for different boundary conditions. The viscoplastic behavior of the matrix material is considered and the rate-dependent characteristics of the nanocomposite are studied. The damping property of the nanocomposite material is also investigated based on the phase lag between stress and strain field by applying harmonic strain at different frequencies.

Nonlinear dynamics of bi-layered graphene sheet, double-walled carbon nanotube and nanotube bundle

Due to strong van der Waals (vdW) interactions, the graphene sheets and nanotubes stick to each other and form clusters of these corresponding nanostructures, viz. bi-layered graphene sheet (BLGS), double-walled carbon nanotube (DWCNT) and nanotube bundle (NB) or ropes. This research work is concerned with the study of nonlinear dynamics of BLGS, DWCNT and NB due to nonlinear interlayer vdW forces using multiscale atomistic finite element method. The energy between two adjacent carbon atoms is represented by the multibody interatomic Tersoff–Brenner potential, whereas the nonlinear interlayer vdW forces are represented by Lennard-Jones 6–12 potential function. The equivalent nonlinear material model of carbon–carbon bond is used to model it based on its force–deflection relation. Newmark’s algorithm is used to solve the nonlinear matrix equation governing the motion of the BLGS, DWCNT and NB. An impulse and harmonic excitations are used to excite these nanostructures under cantilevered, bridged and clamped boundary conditions. The frequency responses of these nanostructures are computed, and the dominant resonant frequencies are identified. Along with the forced vibration of these structures, the eigenvalue extraction problem of armchair and zigzag NB is also considered. The natural frequencies and corresponding mode shapes are extracted for the different length and boundary conditions of the nanotube bundle.

Nanomechanics analysis of perfect and defected graphene sheets via a novel atomic-scale finite element method

Due to their accuracy and reliability, atomistic-based methods such as molecular dynamics (MD) simulations have played an essential role in the field of predictive modeling of single layered graphene sheets (SLGSs) at the nanoscale. However, their applications are limited due to the computational costs. Additionally, consistent with the discrete nature of SLGSs, conventional continuum-based methods cannot be utilized to study the mechanical characteristics of these nanostructures. To overcome these issues, a new Atomic-scale Finite Element Method (AFEM) based on the Tersoff-Brenner potential has been developed in this study. Efficiency of the proposed method is demonstrated employing several numerical examples and its applicability is carefully testified in the case of perfect and defected SLGSs. To facilitate a better comparison, the mechanical behavior obtained by this method is compared with the one determined via MD simulation in various case studies. The results reveal that the proposed method has the accuracy of MD simulations and the speed of continuum-based approaches, simultaneously.

The Elastic Property of Bulk Silicon Nanomaterials through an Atomic Simulation Method

This paper reports a systematic study on the elastic property of bulk silicon nanomaterials using the atomic finite element method. The Tersoff-Brenner potential is used to describe the interaction between silicon atoms, and the atomic finite element method is constructed in a computational scheme similar to the continuum finite element method. Young’s modulus and Poisson ratio are calculated for[100],[110], and[111] silicon nanowires that are treated as three-dimensional structures. It is found that the nanowire possesses the lowest Young’s modulus along the[100] direction, while the[110] nanowire has the highest value with the same radius. The bending deformation of[100] silicon nanowire is also modeled, and the bending stiffness is calculated.

Vibration characteristics of open- and capped-end single-walled carbon nanotubes using multi-scale analysis technique incorporating Tersoff–Brenner potential

This research work addresses questions on the vibration characteristics of single-walled carbon nanotubes (SWCNTs) using multi-scale analysis. Atomistic finite element method (AFEM) is one such multi-scale technique where sequential mode is used to transfer information between two length scales to model and simulate the nanostructures at continuum level. This method is used to investigate the vibration characteristics of SWCNTs. Open- and capped-end armchair and zigzag nanotubes are considered with clamped-free and clamped–clamped boundary conditions. The dependence of vibration characteristic of SWCNTs on their length, diameter and atomic structure is also demonstrated. The body interatomic Tersoff–Brenner (TB) potential is used to represent the energy between two carbon atoms. Based on the TB potential, a new set of force constant parameters is established for carbon nanotubes and presented in this paper. Molecular and structural mechanics analogy is used to find the equivalent geometric and elastic properties of the space frame element to represent the carbon–carbon bond. To validate the vibration results of AFEM incorporating the proposed new set of force constants, molecular dynamics simulation is also carried out on the same structure of carbon nanotube, and it is found that they are in good agreement with each other.