Elsevier

Carbon

Volume 126, January 2018, Pages 165-175
Carbon

Interatomic potential suitable for the modeling of penta-graphene: Molecular statics/molecular dynamics studies

https://doi.org/10.1016/j.carbon.2017.10.002Get rights and content

Abstract

We test the potentials available for elemental carbon, with the scope to choose the potential suitable for the modeling of penta-graphene, the latest two dimensional carbon allotrope. By using molecular statics and molecular dynamics simulations we show that there is only one potential – namely the Tersoff-type potential proposed by Erhart and Albe in 2005 – which is able to correctly describe all the important features of penta-graphene. We show that this potential gives structural, mechanical and energetic parameters which are in accordance with the previously reported ab initio results.

Introduction

The synthesis of carbon nanotubes in 1992 [1], [2] and the discovery of graphene in 2004 [3], [4] have opened a completely new area of search for new carbon allotropes. Based on theoretical considerations many new carbon forms have been proposed since then, and as a result, the set of known carbon forms has been enriched by several low-dimensional [5], [6], [7], [8], [9], [10], [11] and bulk [12], [13], [14], [15], [16] structures. Despite the fact that many of these structures have not been synthesized yet, this new trend has shown the importance of theoretical studies – especially ab initio quantum mechanical calculations – for the search for new materials of designed properties, often unique.

The fact that most of the carbon allotropes have a structure consisting of hexagons as its main feature, has motivated the search for a new carbon allotrope which would have a structure completely composed of pentagons. As a result a new two-dimensional carbon allotrope, called penta-graphene, was proposed by Zhang et al. in 2014 [9], as an, exfoliated equivalent” of the T12-carbon [12], the existence of which was also predicted based on the ab initio calculations.

Penta-graphene (PG) can be seen as a layer of pentagons constructed from a mixture of sp2- and sp3-bonded carbon atoms. Zhang et al. reported that due to its unusual atomic structure – closely resembling the well-known Cairo pentagonal tiling – PG possesses many unique properties. It was found to be mechanically, dynamically and thermally stable, and able to withstand temperatures as high as T=1000 K. It was also reported that PG displays ultra-high mechanical strength, with the strain at maximum stress being as high as 21%. The stiffness of PG was also found to be very high, with the corresponding (in-plane) Young's modulus being as high as E=263 GPa nm, which is more than two-thirds of that of graphene (E=345 GPa nm). PG was also found to exhibit auxetic behavior, that is the anomalous property of becoming wider rather than thinner when stretched. Zhang et al. also reported that PG exhibits an intrinsic (and indirect) band gap, as large as Eg=3.25 eV. This feature renders PG as a better material – than graphene – for two-dimensional transistors, which require the presence of a large band gap to obtain a good switch-off.

Since its discovery PG has been studied intensively by several groups [11], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. The stability of PG and its experimental reachability was questioned by Ewels et al. in Ref. [17]. Based on the ab initio calculations they concluded that PG should be difficult to isolate, also pointing out that PG should rapidly restructure toward graphene in the presence of even a few catalytic impurities. A similar observation about the potential instability of PG was made by Cranford [18], who studied finite, hydrogen-terminated sheets of PG. By using the MD simulation he concluded that bond breaking should be observed even at relatively small deformations (ca. 5%), leading to transformations of pentagons into hexagons and heptagons. According to his work, a similar effect should be observed at elevated temperatures (ca. 600 K), resulting in transformation of PG into (defective) graphene.

Sun et al. [11] presented the results of extensive studies on the mechanical properties of PG. By combining DFT calculations with the fourth order continuum elasticity theory they calculated a complete set of (fifteen) anisotropic nonlinear elastic constants of monolayer PG, showing that the applied continuum formalism (originally proposed by Wei et al. for graphene [30]) is able to accurately describe the non-linear elastic behavior of PG in a wide range of strains, even as large as 30%. Sun et al. also investigated the fracture of PG. They concluded that due to longer bond length and lower charge density the sp3 bonds are more vulnerable to failure than the sp2 bonds. Sun et al. also studied the mechanism of PG deformation, demonstrating that the negative Poisson's ratio of PG originate from de-wrinkling of different atomic planes.

The influence of doping and functionalization on the properties of PG has been also investigated. Berdiyorov et al. [19] found that electronic properties of PG can be fine-tuned, by substituting C atoms with Si, B and N. They showed that the band gap size can be significantly reduced, even to 0.2 eV. According to them the strongest reduction of the band gap is obtained for Si substitutions on the top (or bottom) plane of PG. They also showed that surface termination with fluorine and hydroxyl groups results in an increase in the band gap. The functionalization of PG sheets was also investigated by Li et al. [20], who found that hydrogenation and fluorination can effectively tune the electronic and mechanical properties of PG, changing the Poisson's ratio from negative to positive, and reducing the Young's modulus.

The thermal conductivity of PG has been also studied. Using classical equilibrium molecular dynamics (MD), Xu et al. [21] found that the thermal conductivity of PG at the room temperature is about 170 W/(m K), which is much lower than that of graphene, which is 2000–4000 W/(m K) [31]. They also identified the main mechanism of thermal conduction. By analyzing phonon frequencies and phonon mean free paths they found out that the acoustic phonons make a contribution of about 90% to the thermal conductivity, also showing that phonons with mean free paths larger than 100 nm make a contribution over 50%. They have also demonstrated that the remarkably lower thermal conductivity of PG (compared with graphene) results from the lower phonon group velocities and fewer collective phonon excitations.

The influence of functionalization on the thermal properties of PG has been also studied. Using DFT calculations combined with an iterative solution of the phonon Boltzmann transport equation, Wu et al. [22] found that hydrogenation of PG leads to large (76% increase) improvement in thermal conductivity.

Other spatial forms of PG have been also investigated. Yu and Zhang studied the electronic properties of layered PG [23]. They showed that there is no direct-to-indirect band gap transition in PG by varying strain, layer number, and stacking misalignment. Owing to its characteristics, few-layer PG was recognized by Yu and Zhang [23] as a very promising material for optoelectronic and photovoltaic applications. Recently, Rajbanshi et al. [24] by using DFT studied penta-graphene nanoribbons (PGNRs), concluding that PGNRs are thermodynamically meta-stable with respect to graphene nanoribbons. They also found that on application of uniaxial strain the band gap of PGNRs decreases continuously, yielding a strain-tunable optoelectronic material.

Another potential application of PG has been recently highlighted by Xiao et al. [25]. By using DFT calculations they found that PG provides very high ion storage capacity and fast ion diffusivity, and therefore is a promising anode material for the Li/Na-ion batteries.

The above presented analysis of the current state of knowledge shows how significant the future role of PG may be. At the same time, it illustrates the increasing importance of atomistic modeling techniques. So far, the DFT methods have been mainly used in studies on PG and – to the best of our knowledge – there are only three papers, which have reported the application of empirical potentials to model PG [18], [21], [26].

In the first article Cranford [18] has employed the reactive force field [32] (REAX) to study the mechanical properties and the chemical stability of PG, while in the second Ebrahimi [26] has used the reactive empirical bond order (REBO) potential [33] to study the effect of hydrogen coverage on the buckling of PG. It is important to note that no preliminary validation of the chosen description method has been carried out in both of the aforementioned works.

Contrary to this, Xu et al. [21] performed such validation by testing four different interatomic potentials: the original Tersoff potential [34], [35], the optimized Tersoff potential [36], [37], the REBO potential [33] and the environment-dependent interatomic potential [38] (EDIP). By comparing the calculated structural and mechanical parameters with the results of the ab initio calculations they concluded that among the tested interaction models the original Tersoff potential most closely reproduces the properties of PG. Therefore, it was used in Ref. [21] to study the transport phenomena in PG. However, it must be noted that the Poisson's ratio of PG calculated with the use of the original Tersoff potential was found in Ref. [21] to be equal to 0.174, which strongly (more than 2.5-times) differs from the ab initio result of Zhang et al. [9], which is 0.068. Such a significant difference suggests that the original Tersoff potential may not be the best choice when modeling PG at the empirical level.

It is well known that the credibility of the results obtained from any atomistic simulation depends first and foremost on the quality of the model employed to describe interatomic interactions. The last three decades have brought significant advances in the development of empirical potentials. This is especially true in the case of carbon. Due to the significance of this element for the nanotechnology revolution, many new potentials have been proposed for carbon [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], in order to describe its various – and often very different – forms. Since all potentials for carbon have been parameterized without accounting for the properties of PG – which was unknown when the potentials were developed – before starting modeling PG at the empirical level it is important to first test how well, or whether at all, the existing potentials reproduce the properties of this new – and very exotic – form of carbon.

It must be stressed that the unusual structure of PG makes this system somehow challenging for the empirical potentials. This originates from the fact that there are two types of bonding schemes present in PG, with the entire structure being a, mixture” of sp2- and sp3-hybridized atoms. This is a reason why the basic building blocks in PG differ significantly from their counterparts seen in diamond and graphite/graphene (e.g. the carbon-carbon bond lengths characteristic for diamond and graphite/graphene are 1.54 Å and 1.42 Å, respectively, the corresponding bonds in PG have the lengths of 1.55 Å and 1.34 Å, respectively, also the valence angles typical for PG – i.e. 98.6, 112.2, 113.5 and 134.4° – differ significantly from those typical for diamond – 109.5° — and graphite/graphene – 120°). Therefore, it is not clear if even the potentials, which are known as being able to (simultaneously) describe purely sp2- and purely sp3-systems, will be able to correctly capture the characteristics of PG.

The above observations led us to start the studies the results of which are presented in this paper. We considered 14 different empirical potentials available for carbon and based on molecular statics/molecular dynamics (MS/MD) simulations tested if any of them is able to describe (at least in a satisfactory manner) the interactions in this unusual system, possessing exotic, sp2-sp3 mixed hybridization, not seen in typical carbon forms.

This paper is organized as follows. In Section 2 we describe the structure of PG and recap the results of the ab initio calculations of Zhang et al. [9]. We also briefly present the interatomic potentials that were tested in this work. In Section 3 we present the results of MS calculations of structural, energetic and mechanical properties of PG and compare the results obtained with different empirical potentials against the results of Ref. [9]. On this basis, we choose the most promising interatomic potential for further validation. Section 4 is devoted to a presentation of studies on the finite temperature stability of PG performed on the basis of MD simulation. We conclude with a summary in Section 5.

Section snippets

Structure of penta-graphene

The structure of PG (as described by Zhang et al. in Ref. [9]) is presented in Fig. 1. PG possesses P-421m symmetry (space group number 113) with a tetragonal lattice. Its unit cell has the lattice parameters a=b=3.64 Å and contains two types of carbon atoms: sp3- (later on termed as C1 atoms) and sp2-bonded (C2 atoms). Effectively, the unit cell of PG contains two C1 atoms and four C2 atoms.

PG has a layered structure, with three layers present (see Fig. 1b). The C1 atoms are lying on the

Validation procedure

In order to test the reliability of the considered potentials for each of them have calculated the structural and mechanical properties of PG at T=0 K, using the MS technique. In choosing the most reliable potential we followed a practical approach, whereby we compared the obtained results with the ab initio calculations of Zhang et al. [9], which were taken as a reference. Our test set consisted of the following structural properties: the lattice parameter a; the interlayer spacing h; two

Simulation protocol

In order to test if the chosen T05 potential is able to reproduce the finite-temperature behavior of PG we performed a series of molecular dynamics simulations. The main scope here was to test how the temperature and the stresses influence the stability of PG described with the T05 potential, and if the obtained picture is consistent with the ab initio studies of Ref. [9].

Our detailed protocol was as follows. The structure optimized for 0 K was first equilibrated at the desired temperature T,

Summary

We performed a careful validation of the interatomic potentials available for elemental carbon, with the scope to choose the potential suitable for the modeling of penta-graphene. We considered 14 different empirical potentials, i.a., three Tersoff-type potentials, two Stillinger-Weber-type potentials, three REBO models, and three parameterizations of the REAX force field.

By using molecular statics method we calculated the properties of PG at 0 K. By comparing the obtained results with the

Acknowledgements

We acknowledge the support of the Polish Ministry of Science and Higher Education (Grant IP2012 043972) and the TASK Academic Computer Centre (Gdańsk, Poland). This research was also supported in part by the PL-Grid Infrastructure (Grant POIG.02.03.00-00-096/10).

References (75)

  • C. Su et al.

    Two-dimensional carbon allotrope with strong electronic anisotropy

    Phys. Rev. B

    (2013)
  • W.-J. Yin et al.

    R-graphyne: a new two-dimensional carbon allotrope with versatile dirac-like point in nanoribbons

    J. Mater. Chem. A

    (2013)
  • B.R. Sharma et al.

    pentahexoctite: a new two-dimensional allotrope of carbon

    Sci. Rep.

    (2014)
  • Z. Wang et al.

    Phagraphene: a low-energy graphene allotrope composed of 5-6-7 carbon rings with distorted dirac cones

    Nano Lett.

    (2015)
  • S. Zhang et al.

    Penta-graphene: a new carbon allotrope

    Proc. Natl. Acad. Sci.

    (2015)
  • H. Sun et al.

    Mechanical properties of monolayer penta-graphene and phagraphene: a first-principles study

    Phys. Chem. Chem. Phys.

    (2016)
  • Z. Zhao et al.

    Tetragonal allotrope of group 14 elements

    J. Am. Chem. Soc.

    (2012)
  • X. Jiang et al.

    The r3-carbon allotrope: a pathway towards glassy carbon under high pressure

    Sci. Rep.

    (2013)
  • N.U. Zhanpeisov

    Theoretical dft study on structure and chemical activity of new carbon k4 clusters

    Res. Chem. Intermed.

    (2013)
  • S. Zhang et al.

    Stable three-dimensional metallic carbon with interlocking hexagons

    Proc. Natl. Acad. Sci.

    (2013)
  • C.P. Ewels et al.

    Predicting experimentally stable allotropes: instability of penta-graphene

    Proc. Natl. Acad. Sci.

    (2015)
  • G.R. Berdiyorov et al.

    Band gap engineering in penta-graphene by substitutional doping: first-principles calculations

    J. Phys. Condens. Matter

    (2016)
  • X. Li et al.

    Tuning the electronic and mechanical properties of penta-graphene via hydrogenation and fluorination

    Phys. Chem. Chem. Phys.

    (2016)
  • W. Xu et al.

    Thermal conductivity of penta-graphene from molecular dynamics study

    J. Chem. Phys.

    (2015)
  • X. Wu et al.

    Hydrogenation of penta-graphene leads to unexpected large improvement in thermal conductivity

    Nano Lett.

    (2016)
  • Z.G. Yu et al.

    A comparative density functional study on electrical properties of layered penta-graphene

    J. Appl. Phys.

    (2015)
  • B. Xiao et al.

    Penta-graphene: a promising anode material as the li/na-ion battery with both extremely high theoretical capacity and fast charge discharge rate

    ACS Appl. Mater. Interfaces

    (2016)
  • S. Ebrahimi

    Effect of hydrogen coverage on the buckling of penta-graphene by molecular dynamics simulation

    Mol. Simul.

    (2016)
  • P. Avramov et al.

    Translation symmetry breakdown in low-dimensional lattices of pentagonal rings

    J. Phys. Chem. Lett.

    (2015)
  • T. Stauber et al.

    Tight-binding approach to penta-graphene

    Sci. Rep.

    (2016)
  • X. Wei et al.

    Nonlinear elastic behavior of graphene: Ab initio calculations to continuum description

    Phys. Rev. B

    (2009)
  • E. Pop et al.

    Thermal properties of graphene: fundamentals and applications

    MRS Bull.

    (2012)
  • K. Chenoweth et al.

    Reaxff reactive force field for molecular dynamics simulations of hydrocarbon oxidation

    J. Phys. Chem. A

    (2008)
  • D.W. Brenner et al.

    A second-generation reactive empirical bond order (rebo) potential energy expression for hydrocarbons

    J. Phys. Condens. Matter

    (2002)
  • J. Tersoff

    Modeling solid-state chemistry: interatomic potentials for multicomponent systems

    Phys. Rev. B

    (1989)
  • J. Tersoff

    Erratum: modeling solid-state chemistry: interatomic potentials for multicomponent systems

    Phys. Rev. B

    (1990)
  • L. Lindsay et al.

    Optimized tersoff and brenner empirical potential parameters for lattice dynamics and phonon thermal transport in carbon nanotubes and graphene

    Phys. Rev. B

    (2010)
  • Cited by (38)

    View all citing articles on Scopus
    View full text