Article pubs.acs.org/JCTC
New Coarse-Grained Model and Its Implementation in Simulations of Graphene Assemblies Jun-Jun Shang, Qing-Sheng Yang,* and Xia Liu* Department of Engineering Mechanics, Beijing University of Technology, Beijing 100124, China S Supporting Information *
ABSTRACT: Graphene is a one-atom thick layer of carbon atoms arranged in a hexagonal pattern, which makes it the strongest material in the world. The Tersoff potential is a suitable potential for simulating the mechanical behavior of the complex covalently bonded system of graphene. In this paper, we describe a new coarse-grained (CG) potential, TersoffCG, which is based on the function form of the Tersoff potential. The TersoffCG applies to a CG model of graphene that uses the same hexagonal pattern as the atomistic model. The parameters of the TersoffCG potential are determined using structural feature and potential-energy fitting between the CG model and the atomic model. The modeling process of graphene is highly simplified using the present CG model as it avoids the necessity to define bonds/angles/dihedrals connectivity. What is more, the present CG model provides a new perspective of coarse-graining scheme for crystal structures of nanomaterials. The structural changes and mechanical properties of multilayer graphene were calculated using the new potential. Furthermore, a CG model of a graphene aerogel was built in a specific form of assembly. The chemical bonding in the joints of graphene-aerogel forms automatically during the energy relaxation process. The compressive and recover test of the graphene aerogel was reproduced to study its high elasticity. Our computational examples show that the TersoffCG potential can be used for simulations of graphene and its assemblies, which have many applications in areas of environmental protection, aerospace engineering, and others.
1. INTRODUCTION Graphene has a unique 2D hexagonal lattice that consists of covalently bonded carbon atoms via sp2-orbitals.1−3 The unusual properties of graphene due to its specific nanostructure have attracted interest by researchers worldwide.4 For instance, great progress has been made in the synthesis of large-scale graphene sheets5,6 and graphene assemblies, such as graphene fiber,7 graphene paper,8 and graphene aerogel.9−12 Therefore, there has been intense interest in developing coarse-grained (CG) potentials that would help promote the molecular dynamics (MD) simulations of complex graphene assemblies. This paper describes, in detail, a new CG potential (TersoffCG) based on the classic Tersoff potential. The TersoffCG potential can predict the covalent bonding in the assembly structure for a given local environment, and it can compute the mechanical properties of the system. This paper reviews some earlier CG potentials and MD simulations on graphene, followed by the description of the TersoffCG potential developed in this work. In addition, CGMD simulations on multilayer graphene and graphene aerogel were carried out, which confirm the accuracy of the new potential. General simulation methods to study the mechanical properties of graphene include the continuum method13−15 and the molecular dynamics (MD) method.16−18 In contrast to © 2017 American Chemical Society
the continuum method, the MD method is very good at predicting the mechanical response of graphene on a microscale, by calculating and summing evolutions of all atoms. Several types of potentials have been used for MD simulations of graphene, such as the Morse potential,19 the Tersoff potential,20,21 the Brenner potential,22,23 the REBO (reactive empirical bond order) potential,24 and the AIREBO (adaptive intermolecular reactive empirical bond order) potential.25 These potentials have different features. In the Morse potential, the bond-angle-bending term has little effect on the prediction of fracture of materials. The Tersoff potential is an empirical many-body potential that can describe chemical bonding between carbon atoms. The Brenner potential was developed for hydrocarbons based on the Tersoff potential. The REBO potential, also called the Tersoff-Brenner potential, is the improvement and extension of the Brenner potential. The AIREBO potential was introduced for hydrocarbons with intermolecular interactions based on the REBO potential. These many-body potentials allow for the formation and dissociation of covalent chemical bonds during simulation, and they can provide an accurate description of covalent bonding processes in nonelectrostatic systems. Received: January 18, 2017 Published: July 6, 2017 3706
DOI: 10.1021/acs.jctc.7b00051 J. Chem. Theory Comput. 2017, 13, 3706−3714
Article
Journal of Chemical Theory and Computation Table 1. CGMD Methods Used for Problems Involving Graphene
hybrid force fields with Morse bond potential44
MARTINI force field29
hybrid force fields with harmonic bond potential34,35,42
theoretical basis
the reproduction of partitioning free energies between polar and apolar phases of a large number of chemical compounds
strain-energy conservation between an atomistic system and a reduced order model
strain-energy conservation between the CG model and well-known elastic mechanical properties of graphene
expression
Vbond(R ) = 1/2Kbond(R − R bond)2
Vb(d) = D0[1 − e−α(d − d0)]2
Vangle(θ) = 1/2K angle{cos(θ) − cos(θ0)}2
⎧1/2k (0)(r − r )2 if r < r ⎪ T 0 1 ϕT(r ) = H(rbreak − r )⎨ ⎪1/2k (1)(r − r )2 if r ≥ r ⎩ T 1 1
Vid(θ) = K id(θ − θid)2
ϕB(φ) = 1/2kB(φ − φ0)2
Vd(ϕ) = kϕ[1 − cos(2ϕ)]
ULJ(r ) = 4εij[(σij/r )12 − (σij/r )6 ]
ϕweak (r ) = 4ε((σij/r )12 − (σij/r )6 )
Vnb(r ) = 4εLJ [(σij/r )12 − (σij/r )6 ]
mapping approach
Each bead represents 4 atoms of the graphene atomistic lattice for most simulations.
application
sandwiched graphene-membrane superstructures30
The equilibrium bead distance can be 10 or 50 Å for Each bead represents 4 atoms of the CNTs, and each CG particle can represent a 25 Å × 25 graphene atomistic lattice. Å planar section of a graphene sheet. CNT array35 nanoindentation of a graphene monolayer and the uniaxial tensile test CNT networks36,37 of a graphene paper44 39 CNT buckpaper multilayer graphene assemblies45,46 40,41 CNT fiber multilayer graphene-based assemblies47,48 self-folding of graphene sheets42
method
self-assembly for nonionic surfactants on graphene nanostructures31 self-assembly nanostructures of CTAB on nanoscale graphene32 high density lipoprotein behavior on a few layer graphene33
Va(θ) = kθ(θ − θ0)2
structure and conformational behavior of twisted ultralong multilayer graphene ribbons43
al.42 simulated the self-folding of mono- and multilayer graphene sheets. In addition, the structure and conformational behavior of twisted ultralong multilayer graphene ribbons were also investigated by Cranford et al.43 Ruiz et al.44 developed a CGMD method to reproduce quantitatively graphene’s mechanical response in the elastic and fracture regimes. The group used a strain-energy conservation approach, where the force-field parameters of the model are tuned using well-known (elastic) mechanical properties of graphene. The potential energy of the system consists of bond energy, angle energy, dihedral energy, and nonbonded interaction energy. The bond energy is presented in the form of a Morse potential. These approaches, which are perfectly suitable in the context of their studies, reveal the behavior of nanomaterials, and they reduce the degrees of freedom to simplify the calculation effectively. To summarize, all these CG methods are of the “bead−spring” method, in which a single graphene layer or a single CNT is represented by a continuous network of beads and springs. The main CGMD methods used to solve problems involving graphene are summarized in Table 1. In previous research, most of the CG models of graphene retain the hexagonal lattice pattern.44 In this coarse-graining method, the degree of freedom is reduced by three-quarters, but the deformation and failure mechanism can still be described well. However, the potential energy of the system is generally divided into contributions from bonds, angles, dihedrals, and nonbonded interactions, based on the function terms of the potentials. This makes the modeling of graphene assemblies very difficult because more information than the atomic coordination is needed, such as bond connection, bond angle connection, dihedral connection, and others. Therefore, it is desirable to develop a simple CG potential that calculates the connection automatically.
Great progress has been made on the synthesis of large-scale graphene assemblies. The size and arrangement of graphene assemblies have an important effect on their macroscale mechanical properties. Therefore, it is very desirable to improve the understanding of the relationship between the macroscopic mechanical behavior and the microstructural evolution of the graphene assemblies. However, full-atomic MD simulations of large-scale graphene materials are time-consuming. Therefore, the coarse-grained MD (CGMD) method with a higher computational efficiency is required. In the CGMD method, a CG model substitutes the atomic structure of the material, and CG potentials are used to describe the interactions inside the CG model. A suitable CG model has the potential to be very effective because it allows simulating mesoscale physical processes while retaining the molecular details of the system. The CGMD method has been used extensively to study the mechanical properties of graphene and its allotrope, carbon nanotube (CNT). Zhigilei et al.26 presented a mesoscopic force field (MFF) to describe the internal interactions with CNTs based on the results of MD simulations on an atom scale. Volkov et al.27 developed the mesoscopic tubular potential for the van der Waals interaction between straight single-walled CNTs of arbitrary length and orientation. Using both the MFF and mesoscopic tubular potential, Volkov et al.28 investigated the factors responsible for self-organization of CNTs into mechanically stable and reproducible network structures, which are so characteristic of CNT films and mats. Marrink et al.29 proposed the MARTINI force field for biomolecular simulations, and the method was used to investigate many problems involving graphene.30−33 Buehler34 developed a CGMD simulation technique to study self-assembly processes, including self-folding and bundle formation, as well as the response of bundles of CNTs to severe mechanical stimulation under compression, bending, and tension. This approach has been implemented successfully to explore the structural evolution and mechanical properties of large-area CNT arrays,35 CNT networks,36,37 and Bucky papers,38,39 as well as CNT fibers.40,41 Considering the strain energy conservation between an atomistic system and the reduced order model, Cranford et
2. METHOD The Tersoff potential21 depends on the bond order upon the local environment. Therefore, it is feasible to calculate the structure and energetics of complex covalently bonded systems. Similar to harmonic CG bond/angle/dihedral potentials, which 3707
DOI: 10.1021/acs.jctc.7b00051 J. Chem. Theory Comput. 2017, 13, 3706−3714
Article
Journal of Chemical Theory and Computation
the pairs of atoms i and j (two body), multiplied with a term (bij) that depends on the angle between i, j and a third atom k (three body), as shown in Figure 2.
are commonly used in simulations of graphene, carbon nanotube, and polymers, etc., the Tersoff potential has simple formula form and less number of parameters, making it possible to carry out the atomistic-to-CG mapping. Besides, the Tersoff potential has been widely used for simulating interatomic interactions between atoms of a hybrid structure such as carbon in graphene and carbon nanotubes. Despite the fact that ReaxFF and AIREBO potentials are successful in describing the realistic mechanical behavior of graphene, it still faces some challenges while modeling the C−C bond in hybrid structures. Therefore, we concluded that a CG potential based on the Tersoff potential would be more practical. For the TersoffCG potential developed here, the parameters were determined based on the structure feature of the CG model for graphene and potential energy fitting between the CG model and the original atomic model. The CG model for graphene and the parameter determination process are described below. Considering the structural characteristic of graphene, the CG model is composed of a hexagonal lattice of beads. Each bead represents four atoms in the atomic lattice (Figure 1). Hence,
Figure 2. Atoms described with the Tersoff potential.
The functions f R and fA are expressed as (3)
fA (r ) = −B exp( −λ 2r )
(4)
Since one bead in the CG model represents four atoms, f R and fA of the CG model should be four times as large as the fullatom model. Therefore, the values for A and B in the TersoffCG potential are obtained by multiplying the original values by four. Based on the atomistic-to-CG mapping scheme, the distance between the adjacent CG beads is twice that of the atomic distance. In order to keep the indexes of the expressions of repulsive and the attractive interactions consistent throughout the atomistic-to-CG mapping, the values of parameters λ1 and λ2 in the TersoffCG potential are half of their original values. The cutoff function f C is
Figure 1. CG mapping scheme of graphene and features of the TersoffCG model.
⎧1 r
