The COMPASS force field: Validation for carbon nanoribbons

Physica E 118 (2020) 113937

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Physica E: Low-dimensional Systems and Nanostructures journal homepage: www.elsevier.com/locate/physe

The COMPASS force field: Validation for carbon nanoribbons A.V. Savin a,b , M.A. Mazo a ,∗ a b

N.N. Semenov Federal Research Center for Chemical Physics, Russian Academy of Science (FRCCP RAS), Moscow, 119991, Russia Plekhanov Russian University of Economics, Moscow, 117997, Russia

ARTICLE

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Keywords: COMPASS force field Graphene Graphane Fluorographene Out-of-plane vibrations Dispersion curves

ABSTRACT The COMPASS force field has been successfully applied in a large number of materials simulations, including the analysis of structural, electrical, thermal, and mechanical properties of carbon nanoparticles. This force field has been parameterized using quantum mechanical data and is based on hundreds of molecules as a training set, but such analysis for graphene sheets was not carried out. The objective of the present study is the verification of how good the COMPASS force field parameters can accurately describe the frequency spectrum of atomic vibrations of graphene, graphane and fluorographene sheets. We showed that the COMPASS force field allows us to describe with good accuracy the frequency spectrum of atomic vibrations of graphane and fluorographene sheets, whose honeycomb hexagonal lattice is formed by sp3 hybridization. On the other hand, the force field does not describe very well the frequency spectrum of graphene sheet, whose planar hexagonal lattice is formed by sp2 banding. In that case the frequency spectrum of out-of-plane vibrations differs greatly from the experimental data — bending stiffness of a graphene sheet is strongly over estimated. We present the correction of parameters of out-of-plane and torsional potentials of the force field, which allows achieving the coincidence of vibration frequency with experimental data. After such corrections, the COMPASS force field can be used to describe the dynamics of flat graphene sheets and carbon nanotubes.

1. Introduction The obtaining of the monolayer graphene membrane with the unique physical properties [1] caused the unprecedented rise in research of single and multilayer graphene sheets, graphene nanoribbons and nanoscrolls, other graphene-based and functional graphene nanostructures [2–8]. The possibility of using such structures in electronics [9,10], optics [11], and in many other industries has been much discussed recently [5,7,12–17]. The remarkable properties of graphene make it one of the key components in creating nanomaterials for energy storage [10,18–21], polymer nanocomposites [2,22] and for medicine [23,24]. Experimental studies of these nanostructures are difficult because of their small size. So particular attention is paid to their computer simulation, where quantum mechanics (QM) equations, molecular dynamics (MD) or molecular mechanics (MM) equations are used. QM simulation allows the consideration only small-sized molecular structures, which enables justifying empirical interatomic potentials in MM/MD modeling. However, the accuracy of MD simulations depends on the parametrization of the empirical potentials that describe the atomic interactions. Also, both the experimental data and results of QM calculation are used for the determination of these parameters. For MM/MD

modeling, carbon nanoforms a variety of carbon interatomic potentials have been used. Examples of the carbon potentials are reactive empirical bond-order potential REBO/AIREBO [25–28], reactive force field ReaxFF [29], long-range carbon bond order potential LCBOP [30], analytical bond order potential (BOP) [31], environment dependent interatomic potential (EDIP) [32], the modified embedded atom method (MEAM) potential [33], the DREIDING force field [34], standard atomic potentials for large polymeric macromolecules [35–38]. It is assumed that the force field parameters should reproduce the value of the mechanical moduli and vibration spectrum of nanoparticles, but the standard parameter sets of the above-mentioned force fields have been received taking into account only in-plane frequency spectrum or/and in-plane molecular mechanics. Taking into account out-of-plane vibrations made it possible to significantly improve the matching of the bending rigidity modulus and dispersion curves calculated in MD with the available experimental data and QM simulation for force fields Morse [39,40], MEAM [41] and DREIDING [42]. Recently, a comparison was made of the effectiveness of the popular interatomic potentials for carbon with respect to the elastic properties of graphene, where Young’s modulus, Poisson’s ratio and bending rigidity, were compared with the experimental and ab initio data [43].

∗ Corresponding author. E-mail addresses: [email protected] (A.V. Savin), [email protected] (M.A. Mazo).

https://doi.org/10.1016/j.physe.2019.113937 Received 1 March 2019; Received in revised form 19 December 2019; Accepted 25 December 2019 Available online 6 January 2020 1386-9477/© 2020 Elsevier B.V. All rights reserved.

Physica E: Low-dimensional Systems and Nanostructures 118 (2020) 113937

A.V. Savin and M.A. Mazo

Fig. 1. Structures of nanoribbons (a) graphene (C12 H2 )11 C10 H12 (of size 29.9 × 13.4 Å2 ) and (b) graphane (C12 H14 )11 C10 H22 (of size 29.1 × 11.7 Å2 ).

A condensed-phase optimized ab-initio COMPASS force field has been developed recently [44,45] and has been successfully applied in a large number of soft materials simulations with carbon nanostructures (see [46–56] and references in them). The force field has been parameterized using hundreds of molecules as a training set including molecules with sp3 - and sp2 -hybridized carbon atoms, but the parametrization and validation using planner monomolecular lattice structures like graphene sheets and its modifications have not been yet performed. The purpose of this article is to check how well the force field of the COMPASS reproduces a vibration range of three nanoribbons: graphene, graphane and fluorographene. For this we calculated the frequency spectrum and dispersion curves of these nanoribbons using the COMPASS force field, and compared them with the results of abinitio data. It was found that the force field could give an accurate reproduction of experimental spectrum lattice of sp3 hybridization, such as graphene or fluorographene. However, the obtained frequency spectrum of graphene turned out to be noticeably shifted to the highfrequency region with respect to the known theoretical and experimental data for out-of-plane vibrations of carbon. We made a correction of parameters of torsional potentials, which allowed us to rectify the deficiency of the COMPASS force field. 2. A model of a graphene and graphane nanoribbons Molecular nanoribbon is a narrow, straight-edged strip, cut from a single-layered molecular plane. The simplest example of such a molecular plane is a graphene sheet (isolated monolayer of carbon atoms of crystalline graphite) and its various chemical modifications: graphane (fully hydrogenated on both sides graphene sheet) and fluorographene (fluorinated graphene). As is known, graphene and its modifications are elastically isotropic materials, the longitudinal and flexural rigidity which is poorly dependent on the chirality of the structure. Therefore, for definiteness, we will consider nanoribbons with the zigzag structure shown in Figs. 1 and 2. Let us consider a rectangular ribbon cut from a flat sheet of graphene [Fig. 1(a)] and graphane [Fig. 1(b)] in the zigzag directions. The nanoribbons structure can be obtained by longitudinal translation of a transverse unit cell consisting of 𝑁𝑒 = 𝑁C + 𝑁H atoms (see Fig. 2).

Fig. 2. Schematic view of the structures of zigzag (a) graphene (C8 H2 )∞ and (b) graphane (C8 H10 )∞ nanoribbons. The indices 𝑛 and 𝑘 enumerate the unit cells in the nanoribbon and the atoms in the cell. Dotted lines indicate the cell borders. The gray color region is the elementary cell of the nanoribbon. Nanoribbons lie in the 𝑥𝑦-plane.

2

Physica E: Low-dimensional Systems and Nanostructures 118 (2020) 113937

A.V. Savin and M.A. Mazo

Table 1 Values of the parameters for valence bond potential 𝑉1 (𝑏) = 𝑘𝑏,2 (𝑏 − 𝑏0 )2 + 𝑘𝑏,3 (𝑏 − 𝑏0 )3 + 𝑘𝑏,4 (𝑏 − 𝑏0 )4 , angle potential 𝑉2 (𝜃) = 𝑘𝜃,2 (𝜃 − 𝜃0 )2 + 𝑘𝜃,3 (𝜃 − 𝜃0 )3 + 𝑘𝜃,4 (𝜃 − 𝜃0 )4 and torsional potential 𝑉𝑡 (𝜙) = 𝑘𝜙,1 (1 − cos 𝜙) + 𝑘𝜙,2 (1 − cos 2𝜙) + 𝑘𝜙,3 (1 − cos 3𝜙). 𝑏0 (Å)

𝑘𝑏,2 (kcal/mol Å2 )

𝑘𝑏,3 (kcal/mol Å3 )

𝑘𝑏,4 (kcal/mol Å4 )

CC atoms) CH (sp2 atoms) CC (sp3 atoms) CH (sp3 atoms) CF (sp3 atoms)

1.4170 1.0982 1.530 1.101 1.390

470.8361 372.8251 299.670 345.000 403.032

−627.6179 −803.4526 −501.77 −691.89 0

1327.6345 894.3173 679.81 844.60 0

Angle

𝜃0 (deg)

𝑘𝜃,2 (kcal/mol rad2 )

𝑘𝜃,3 (kcal/mol rad3 )

𝑘𝜃,4 (kcal/mol rad4 )

CCC (sp2 atoms) CCH (sp2 atoms) CCC (sp3 atoms) CCH (sp3 atoms) HCH (sp3 atoms) CCF (sp3 atoms) FCF (sp3 atoms)

118.90 117.94 112.67 110.77 107.66 109.20 109.10

61.0226 35.1558 39.5160 41.4530 39.6410 68.3715 71.9700

−34.9931 −12.4682 −7.4430 −10.6040 −12.9210 0 0

0 0 −9.5583 5.1290 −2.4318 0 0

Torsion

𝑘𝜙,1 (kcal/mol)

𝑘𝜙,2 (kcal/mol)

𝑘𝜙,3 (kcal/mol)

CCCC (sp2 atoms) CCCH (sp2 atoms) HCCH (sp2 atoms) CCCC (sp3 atoms) CCCH (sp3 atoms) CCCF (sp3 atoms) HCCH (sp3 atoms) FCCF (sp3 atoms)

8.3667 0 0 0 0 0 −0.1432 0

1.2000 3.9661 2.3500 0.0514 0.0316 0 0.0617 0

0 0 0 −0.1430 −0.1681 0.1500 −0.1530 −0.1

Bond (sp2

Table 2 Values of parameters for cross-coupling bond/bond potential 𝑉 (𝑏, 𝑏′ ) = 𝑘𝑏𝑏′ (𝑏 − 𝑏0 )(𝑏′ − 𝑏′0 ), bond/angle potential 𝑉 (𝑏, 𝜃) = 𝑘𝑏𝜃 (𝑏 − 𝑏0 )(𝜃 − 𝜃0 ) angle/angle potential, 𝑉 (𝜃, 𝜃 ′ ) = 𝑘𝜃𝜃′ (𝜃 − 𝜃0 )(𝜃 ′ − 𝜃0′ ) and angle/angle/torsion potential 𝑉 (𝜃, 𝜃 ′ , 𝜙) = 𝑘𝜃𝜃′ 𝜙 (𝜃 − 𝜃0 )(𝜃 ′ − 𝜃0′ ) cos 𝜙.

For numerical simulation, we used COMPASS force-field functional form [44]: ∑ 𝐸𝑡𝑜𝑡𝑎𝑙 = [𝑘𝑏,2 (𝑏 − 𝑏0 )2 + 𝑘𝑏,3 (𝑏 − 𝑏0 )3 + 𝑘𝑏,4 (𝑏 − 𝑏0 )4 ] 𝑏

𝑘𝑏𝑏′ (kcal/mol Å2 )

Bond/bond sp2 )

CC/CC (first neighboring, CC/CH (first neighboring, sp2 ) CC/CC (second neighboring, sp2 ) CC/CH (second neighboring, sp2 ) CH/CH (second neighboring, sp2 ) CC/CH (first neighboring, sp3 ) CH/CH (first neighboring, sp3 ) CC/CC (first neighboring, sp3 )

68.2856 1.0795 53.0 −6.2741 −1.7077 3.3872 5.3316 0

Bond/angle

𝑘𝑏𝜃 (kcal/mol Å rad)

C–C/C–C–C (sp2 atoms) C–C/C–C–H (sp2 atoms) C–H/C–C–H (sp2 atoms) C–C/C–C–C (sp3 atoms) C–C/C–C–H (sp3 atoms) C–H/C–C–H (sp3 atoms) C–H/H–C–H (sp3 atoms)

28.8708 20.0033 24.2183 8.016 20.754 11.421 18.103

Angle/angle

𝑘𝜃𝜃′ (kcal/mol rad2 ) 2

C–C–C/C–C–C (sp atoms) C–C–C/C–C–H (sp2 atoms) H–C–C/C–C–H (sp2 atoms) C–C–C/C–C–C (sp3 atoms) C–C–C/C–C–H (sp3 atoms) H–C–C/C–C–H (sp3 atoms)

0 0 0 −0.1729 −1.3199 −0.4825

Angle/angle/torsional

𝑘𝜃𝜃′ 𝜙 (kcal/mol rad2 )

C–C–C/C–C–C/C–C–C–C (sp2 ) C–C–C/C–C–H/C–C–C–H (sp2 ) H–C–C/C–C–H/H–C–C–H (sp2 ) C–C–C/C–C–C/C–C–C–C (sp3 ) C–C–C/C–C–H/C–C–C–H (sp3 ) H–C–C/C–C–H/H–C–C–H (sp3 )

0 −4.8141 0.3598 −22.045 −16.164 −12.564

+

∑ [𝑘𝜃,2 (𝜃 − 𝜃0 )2 + 𝑘𝜃,3 (𝜃 − 𝜃0 )3 + 𝑘𝜃,4 (𝜃 − 𝜃0 )4 ]

+

∑ [𝑘𝜙,1 (1 − cos 𝜙) + 𝑘𝜙,2 (1 − cos 2𝜙)

𝜃

𝜙

+𝑘𝜙,3 (1 − cos 3𝜙)] ∑ ∑ + 𝑘𝜒 𝜒 2 + 𝑘𝑏𝑏′ (𝑏 − 𝑏0 )(𝑏′ − 𝑏′0 ) 𝑏𝑏′

𝜒

+

∑

𝑘𝑏𝜃 (𝑏 − 𝑏0 )(𝜃 − 𝜃0 ) +

∑ 𝜃,𝜃 ′

𝑏,𝜃

𝑘𝜃𝜃 ′ (𝜃 − 𝜃0 )(𝜃 ′ − 𝜃0′ )

+

∑ (𝑏 − 𝑏0 )[𝑘𝑏𝜙,1 cos 𝜙 + 𝑘𝑏𝜙,2 cos 2𝜙 + 𝑘𝑏𝜙,3 cos 3𝜙]

+

∑ (𝜃 − 𝜃0 )[𝑘𝜃𝜙,1 cos 𝜙 + 𝑘𝜃𝜙,2 cos 2𝜙 + 𝑘𝜃𝜙,3 cos 3𝜙]

𝑏,𝜙

𝜃,𝜙

+

∑

𝜃,𝜃 ′ ,𝜙

+

∑ 𝑖,𝑗

𝑘𝜃𝜃′ 𝜙 (𝜃 − 𝜃0 )(𝜃 ′ − 𝜃0′ ) cos 𝜙

𝑞𝑖 𝑞𝑗 ∕𝑟𝑖𝑗 +

∑

𝜖𝑖𝑗 [2(𝑟0𝑖𝑗 ∕𝑟𝑖𝑗 )9 − 3(𝑟0𝑖𝑗 ∕𝑟𝑖𝑗 )6 ].

(1)

𝑖,𝑗

The functions can be divided into two categories which are namely: (1) valence terms including diagonal and off-diagonal cross-coupling terms and (2) nonbonded interaction terms. The valence terms represent internal coordinates of bond 𝑏, angle 𝜃, torsion angle 𝜙, and out-ofplane angle 𝜒, and the cross-coupling terms include combinations of two or three internal coordinates. The nonbonded interactions include an LJ-9-6 potential for the van der Waals (vdW) term and a Coulombic potential for an electrostatic interaction. Graphene has a flat form due to the sp2 hybridization of all valence bonds [Fig. 1(a)]. Graphane (fully hydrogenated graphene) as a twodimensional crystal was predicted in theoretical works [57,58] and was confirmed by experimental data in [59]. Graphane is an analogue of graphene with the unique properties [60–63]. Graphane nanoribbon is a strip with a constant width cut from a two-sized hydrogenated graphene sheet [see Figs. 1(b), 2(b)]. Several conformations of the graphane sheet exist, and we consider the most energy-efficient configuration armchair, where hydrogen atoms are connected from different sides to all adjacent carbon atoms. The substitution of hydrogen atoms for fluorine atoms leads to the formation of the fluorographene sheet

The number of the carbon atoms in the unit cell is always multiple of two, and the number of hydrogen atoms 𝑁H = 2 for graphene nanoribbon and 𝑁H = 𝑁C + 2 for graphane. Further, we use the following notation (see Fig. 2): each atom is numbered with at twocomponent index 𝛼 = (𝑛, 𝑘), where 𝑛 = 0, ±1, ±2, … defines the unit cell number and 𝑘 = 1, … , 𝑁𝑒 number of atoms in the unit cell. 3

Physica E: Low-dimensional Systems and Nanostructures 118 (2020) 113937

A.V. Savin and M.A. Mazo

Table 3 Values of parameters for cross-coupling bond/torsion potential 𝑉 (𝑏, 𝜙) = (𝑏 − 𝑏0 )[𝑘𝑏𝜙,1 cos 𝜙 + 𝑘𝑏𝜙,2 cos 2𝜙 + 𝑘𝑏𝜙,3 cos 3𝜙]. and cross-coupling angle/torsion potential 𝑉 (𝜃, 𝜙) = (𝜃 − 𝜃0 )[𝑘𝜃𝜙,1 cos 𝜙 + 𝑘𝜃𝜙,2 cos 2𝜙 + 𝑘𝜃𝜙,3 cos 3𝜙]. Bond/torsion

𝑘𝑏𝜙,1 (kcal/mol Å)

𝑘𝑏𝜙,2 (kcal/mol Å)

𝑘𝑏𝜙,3 (kcal/mol Å)

C–C/C–C–C–C (sp2 atoms, central bond) C–C/C–C–C–H (sp2 atoms, central bond) C–H/H–C–C–H (sp2 atoms, central bond) C–C/C–C–C–C (sp2 atoms, terminal bond) C–C/C–C–C–H (sp2 atoms, terminal bond) C–H/C–C–C–H (sp2 atoms, terminal bond) C–H/H–C–C–H (sp2 atoms, terminal bond) C–C/C–C–C–C (sp3 atoms, central bond) C–C/C–C–C–H (sp3 atoms, central bond) C–C/H–C–C–H (sp3 atoms, central bond) C–C/C–C–C–C (sp3 atoms, terminal bond) C–C/C–C–C–H (sp3 atoms, terminal bond) C–H/C–C–C–H (sp3 atoms, terminal bond) C–H/H–C–C–H (sp3 atoms, terminal bond)

27.5989 0 0 −0.1185 0 0 0 −17.787 −14.879 −14.261 −0.0732 0.2486 0.0814 0.2130

−2.3120 −1.1521 4.8228 6.3204 −6.8958 −0.4669 −0.6890 −7.1877 −3.6581 −0.5322 0 0.2422 0.0591 0.3120

0 0 0 0 0 0 0 0 −0.3138 −0.4864 0 −0.0925 0.2219 0.0777

Angle/torsion

𝑘𝜃𝜙,1 (kcal/mol rad)

𝑘𝜃𝜙,2 (kcal/mol rad)

𝑘𝜃𝜙,3 (kcal/mol rad)

C–C–C/C–C–C–C (sp2 atoms) C–C–C/C–C–C–H (sp2 atoms) C–C–H/C–C–C–H (sp2 atoms) H–C–C/H–C–C–H (sp2 atoms) C–C–C/C–C–C–C (sp3 atoms) C–C–C/C–C–C–H (sp3 atoms) C–C–H/C–C–C–H (sp3 atoms) H–C–C/H–C–C–H (sp3 atoms)

1.9767 0 0 0 0.3886 −0.2454 0.3113 −0.8085

1.0239 2.5014 2.7147 2.4501 −0.3139 0 0.4516 0.5569

0 0 0 0 0.1389 −0.1136 −0.1988 −0.2466

Table 4 Values of parameters for LJ-9-6 potential 𝑉 (𝑟) = 𝜖[2(𝑟0 ∕𝑟)9 −3(𝑟0 ∕𝑟)6 ] and charge valence bond increments 𝑞1 , 𝑞2 . Atom atom

𝜖 (kcal/mol)

𝑟0 (Å)

C C (sp2 atoms) C H (sp2 atoms) H H (sp2 atoms) C C (sp3 atoms) H H (sp3 atoms) F F (sp3 atoms) C H (sp3 atoms) C F (sp3 atoms)

0.0680 0.0271 0.0230 0.0400 0.0230 0.0598 0.0215 0.0422

3.9150 3.5741 2.878 3.854 2.878 3.200 3.526 3.600

Atom atom

𝑞1 (e)

𝑞2 (e)

0 −0.1268 0 −0.053 +0.25

0 +0.1268 0 +0.053 −0.25

C C C C C

C (sp2 atoms) H (sp2 atoms) C (sp3 atoms) H (sp3 atoms) F (sp3 atoms)

where 𝑛 – number of the unit cell, 𝐮𝑛 – 3𝑁𝑒 -dimensional vector defining coordinates of atoms in the unit cell, 𝐌 – the diagonal matrix of cell atomic masses, 𝑃 (𝐮𝑛−1 , 𝐮𝑛 , 𝐮𝑛+1 ) – of the cell atom’s interaction with each other and with atoms from neighboring cells. In the ground state each nanoribbon unit cell is obtained from the previous one by shifting by the step 𝑎: 𝐮0𝑛 = 𝐮0 + 𝑎𝑛𝐞𝑥 , where the unit 𝑁𝑒 vector 𝐞𝑥 = {(1, 0, 0)𝑘 }𝑘=1 (nanoribbon lies along the 𝑥 axes). To find the ground state (the period 𝑎 and the coordinate vector 𝐮0 ) we should solve the minimum problem: 𝑃 (𝐮0 − 𝑎𝐞𝑥 , 𝐮0 , 𝐮0 + 𝑎𝐞𝑥 ) → min ∶ 𝐮0 , 𝑎.

The problem (3) is solved by the conjugate gradient method. The numerical solution shows that the ground state calculated by the COMPASS force field agrees well with the experimental values (the equilibrium values of bond lengths, valence angles and torsional angles are the same). In order to verify the coincidence of the nanoribbon frequency spectrum with the experimental data. To accomplish this, we found the dispersal curve of the nanoribbon.

structure (fully fluorinated graphene). In comparison with graphene, graphane and fluorographene are not two-dimensional crystal structures, but corrugated sheets with sp3 hybridized carbon atoms and with atoms of hydrogen or fluorine connected to the different sides of the sheet. For graphene nanoribbon simulation the potentials (1) with the parameters of an atom with sp2 hybridization should be used, and for graphane and fluorographene nanoribbon simulation — potentials with the parameters of an atom with sp3 hybridization. The quired parameters for interatomic interaction potentials are shown in Table 1. Out-of-plane angle potential 𝑉3 (𝜒) = 𝑘𝜒 𝜒 2 appears in the Hamiltonian only for graphene nanoribbons with planar sp2 valence bands: for C– C–C–C atom group 𝜅𝜒 = 7.1794 kcal/mol, for C–C–C–H atoms 𝜅𝜒 = 4.8912 kcal/mol. Parameters for cross-coupling interaction potentials are shown in Tables 2 and 3. Parameters for nonvalence interactions can be found in Table 4. Parameter values for C and H atoms are taken from [44], for fluorine atom, the parameter values — from force field CFF91. Taking into consideration the noncovalent interactions of atoms only from neighboring unit cells, the Hamiltonian of a nanoribbon can be written in the following form 𝐻=

+∞ [ ] ∑ 1 (𝐌𝐮̇ 𝑛 , 𝐮̇ 𝑛 ) + 𝑃 (𝐮𝑛−1 , 𝐮𝑛 , 𝐮𝑛+1 ) , 2 𝑛=−∞

(3)

3. Dispersion equation Let us introduce a 3𝑁𝑒 -dimensional vector 𝑁

𝑒 𝐯𝑛 = {𝐮(𝑛,𝑘) − 𝐮0(𝑛,𝑘) }𝑘=1

describing the displacement of atoms in the 𝑛th cell from their equilibrium positions. Then the Hamiltonian (2) of a nanoribbon can be written in the following form: 𝐻=

+∞ [ ] ∑ 1 (𝐌𝐯̇ 𝑛 , 𝐯̇ 𝑛 ) + 𝑈 (𝐯𝑛−1 , 𝐯𝑛 , 𝐯𝑛+1 ) , 2 𝑛=−∞

(4)

where interaction potential 𝑈 (𝐯1 , 𝐯2 , 𝐯3 ) = 𝑃 (𝐮0 − 𝑎𝐞𝑥 + 𝐯1 , 𝐮0 + 𝐯2 , 𝐮0 + 𝑎𝐞𝑥 + 𝐯3 ). The Hamiltonian (4) corresponds to the equations of motion −𝐌𝐯̈ 𝑛 = 𝑈𝐯1 (𝐯𝑛 , 𝐯𝑛+1 , 𝐯𝑛+2 ) + 𝑈𝐯2 (𝐯𝑛−1 , 𝐯𝑛 , 𝐯𝑛+1 ) + 𝑈𝐯3 (𝐯𝑛−2 , 𝐯𝑛−1 , 𝐯𝑛 ),

(2)

were vector 𝑈𝐯𝑖 = 𝜕𝑈 (𝐯1 , 𝐯2 , 𝐯3 )∕𝜕𝐯𝑖 , 𝑖 = 1, 2, 3. 4

(5)

Physica E: Low-dimensional Systems and Nanostructures 118 (2020) 113937

A.V. Savin and M.A. Mazo

Fig. 3. Structure of 3𝑁𝑒 dispersion curves for zigzag graphene nanoribbon (C64 H2 )∞ for (a) in-plane and (b) out-of-plain vibrations. The thick red curves correspond to oscillations localized at the nanoribbon edges. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

For small displacements, system (5) can be written as a system of linear equations −𝐌𝐯̈ 𝑛 = 𝐁1 𝐯𝑛 + 𝐁2 𝐯𝑛+1 + 𝐁∗2 𝐯𝑛−1 + 𝐁3 𝐯𝑛+2 + 𝐁∗3 𝐯𝑛−2 ,

(6)

where the matrices 𝐁1 = 𝑈𝐯1 ,𝐯1 + 𝑈𝐯2 ,𝐯2 + 𝑈𝐯3 ,𝐯3 , 𝐁2 = 𝑈𝐯1 ,𝐯2 + 𝑈𝐯2 ,𝐯3 , 𝐁3 = 𝑈𝐯1 ,𝐯3 and the matrices of partial derivatives are 𝑈𝐯𝑖 ,𝐯𝑗 =

𝜕2 𝑈 (𝟎, 𝟎, 𝟎), 𝑖, 𝑗 = 1, 2, 3. 𝜕𝐯𝑖 𝜕𝐯𝑗

The solution of the systems of linear equations (6) can be written in the standard form of the wave 𝐯𝑛 = 𝐴𝐰 exp(𝑖𝑞𝑛 − 𝑖𝜔𝑡),

(7)

Fig. 4. Structure of 3𝑁𝑒 dispersion curves for zigzag graphane nanoribbon (C64 H66 )∞ . The thick red curves correspond to oscillations localized at the nanoribbon edges. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

where 𝐴 – amplitude, 𝐰 – eigenvector, 𝜔 is the phonon frequency with the dimensionless wave number 𝑞 ∈ [0, 𝜋]. Substituting Eq. (7) into Eq. (6), we obtain the eigenvalue problem 𝜔2 𝐌𝐰 = 𝐂(𝑞)𝐰,

(8) orthogonal to the plane. Two third of the branches correspond to the atom vibrations in the 𝑥𝑦 plane of the nanoribbon (in-plane vibrations), whereas only one third corresponds to the vibrations orthogonal to the plane (out-of-plane vibrations), when the atoms are shifted along the axes 𝑧. The nanoribbon frequency spectrum consists of two intervals [0, 1911] and [3086.6, 3099.2] cm−1 (the second high-frequency interval corresponds to the edge vibrations of the valence bonds C–H). Discarding frequencies of the edge vibrations from Fig. 3, we can conclude that the graphene sheet spectrum consists of the one frequency interval [0, 𝜔𝑚 ] with the maximum frequency 𝜔𝑚 = 1911 cm−1 . The spectrum of out-of-plane vibrations also consists of the one frequency interval [0, 𝜔𝑜 ] with the maximum frequency 𝜔𝑜 = 1420 cm−1 . The frequency spectrum of the graphene sheet has been studied theoretically and experimentally in [64–67]. The maximum frequency of in-plane vibrations is 𝜔𝑚 = 1600 cm−1 , the maximum frequency of out-of-plane vibrations is 𝜔𝑜 = 868 cm−1 . Thus, the graphene frequency spectrum calculated by the COMPASS force field does not coincide with experimental valuations. The frequency spectrum of in-plane vibrations is 1.2 times more and the frequency spectrum of out-of-plane vibrations is 1.6 times more than experimental valuations, i.e. the bending stiffness of the nanoribbon is 2.5 times more.

where Hermitian matrix 𝐂(𝑞) = 𝐁1 + 𝐁2 𝑒𝑖𝑞 + 𝐁∗2 𝑒−𝑖𝑞 + 𝐁3 𝑒2𝑖𝑞 + 𝐁∗3 𝑒−2𝑖𝑞 . Using the substitution 𝐰 = 𝐌−1∕2 𝐞, problem (8) can be rewritten in the form 𝜔2 𝐞 = 𝐌−1∕2 𝐂(𝑞)𝐌−1∕2 𝐞,

(9)

where 𝐞 is the normalized eigenvector, (𝐞, 𝐞) = 1. Thus, for obtaining the dispersion curves 𝜔𝑗 (𝑞), it is necessary to find the eigenvectors of the Hermitian matrix (9) of a size 3𝑁𝑒 × 3𝑁𝑒 for each fixed wave number 0 ≤ 𝑞 ≤ 𝜋. As a result, we obtain 3𝑁𝑒 3𝑁 branches of the dispersion curve {𝜔𝑗 (𝑞)}𝑗=1𝑒 . 4. Graphene frequency spectrum Let us consider a wide nanoribbon (C64 H2 )∞ (the nanoribbon width 𝐷 = 68.79 Å, the period 𝑎 = 2.412 Å, the number of atoms in unit cell 𝑁𝑒 = 66) to find the frequency spectrum of the graphene sheet. Fig. 3 shows the 3𝑁𝑒 dispersion curves of the nanoribbon. The plane structure of the nanoribbon allows dividing its vibrations into two classes: inplane vibrations, when the atoms always stayed in the plane of the nanoribbon and out-of-plane vibrations when the atoms are shifted 5

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all the frequencies will shift down. The frequency spectrum of graphene sheet (CD)∞ also consists of three intervals [0, 620], [813, 1503], [2130, 2200] cm−1 (the high-frequency interval most strongly shifts down). The stronger increase in the weight of the connected atoms, the change of hydrogen atoms for fluorine atoms, causes the stronger frequency shift and the closure of the gaps. Fig. 5 shows the dispersion curves for fluorographene nanoribbon (C64 F66 )∞ . The figure clearly shows that the nanoribbon frequency spectrum of linear vibrations consists of two intervals: [0, 1183.6] and [1252.0, 1589.3] cm−1 . The analysis of the dispersion curves suggests that the frequency spectrum of linear vibrations of the endless fluorographene sheet (CF)∞ consists of two continuous intervals [0, 1184] and [1252, 1589] cm−1 with the narrow gap between them. Such a structure of the frequency spectrum is also in good agreement with the results of quantum chemical calculations [68]. Quantum calculations suggest the frequency spectrum intervals [0, 1039] and [1106, 1312] cm−1 . The structure of the frequency spectrum can be obtained also from the analysis of the frequency spectrum density of atomic thermal vibrations of the finite length nanoribbon. Let us consider the graphane nanoribbon (C12 H14 )99 C10 H22 with size 25.09 × 1.17 nm2 , which consisted of 𝑁 = 100 unit cells. The dynamics of the thermalized nanoribbon is described by the system of Langevin equations 𝜕𝐻 − 𝛤 𝑀𝑛,𝑘 + 𝛯𝑛,𝑘 , 𝜕𝐮𝑛,𝑘 𝑛 = 1, 2, … , 𝑁, 𝑘 = 1, 2, … , 𝑁𝑒 ,

(10)

𝑀𝑛,𝑘 𝐮̈ 𝑛,𝑘 = −

where 𝐻 – Hamiltonian of the nanoribbon, created by the COMPASS force field, 𝐮𝑛,𝑘 = (𝑢𝑛,𝑘,1 , 𝑢𝑛,𝑘,2 , 𝑢𝑛,𝑘,3 ) – 3D coordinate vector of atom (𝑛, 𝑘), 𝑀𝑛,𝑘 – mass of this atom, 𝛤 = 1∕𝑡𝑟 – damping coefficient (relaxation time 𝑡𝑟 = 0.4 ps). Normally distributed random forces 𝛯𝑛,𝑘 = (𝜉𝑛,𝑘,1 , 𝜉𝑛,𝑘,2 , 𝜉𝑛,𝑘,3 ) normalized by conditions

Fig. 5. Structure of 3𝑁𝑒 dispersion curves for zigzag graphene fluoride nanoribbon (C64 F66 )∞ . The thick red curves correspond to oscillations localized at the nanoribbon edges. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

⟨𝜉𝑛,𝑘,𝑖 (𝑡1 )𝜉𝑚,𝑙,𝑗 (𝑡2 )⟩ = 2𝑀𝑛,𝑘 𝛤 𝑘𝐵 𝑇 𝛿𝑛𝑚 𝛿𝑘𝑙 𝛿𝑖𝑗 𝛿(𝑡1 − 𝑡2 ), where 𝑘𝐵 – Boltzmann’s constant, 𝑇 – thermostat temperature. The system of equations of motion (10) was integrated numerically during the time 𝑡 = 10 ps (the stationary state of the plane nanoribbon serves as a starting point). During this time, the nanoribbon and the thermostat came to balance. Then its interaction with the thermostat was switched off and the dynamics of isolated thermalized nanoribbon were considered. The density of the frequency spectrum of atomic vibrations 𝑝(𝜔) was calculated via a fast Fourier transform. The density of the frequency spectrum of atomic vibrations of the graphane and fluorographene nanoribbons at 𝑇 = 300 K is presented in Fig. 6. The figure clearly shows that the profile of density 𝑝(𝜔) coincides well with the forms of the dispersal curves. The low-frequency gap is clearly visible for the graphane nanoribbon, and the narrow gap in the frequency spectrum is clearly visible for the fluorographene nanoribbon.

5. Graphane and fluorographene frequency spectrum Let us consider a wide nanoribbon (C64 H66 )∞ (nanoribbon width 𝐷 = 68.4 Å, period 𝑎 = 2.522 Å, number atoms in unit cell 𝑁𝑒 = 130) to find the frequency spectrum of graphane sheet. Fig. 4 shows the dispersion curves of the nanoribbon. As can be seen from the figure, the nanoribbon frequency spectrum consists of three intervals: lowfrequency interval [0, 757.8], middle-frequency interval [987.5, 1543.6] and narrow high-frequency interval [2921.8, 2966.6] cm−1 . On the average, carbon atoms account for 79.5% of the vibration energy in the first frequency interval, for 48.1% in the second frequency interval and only for 8.4% in the third. It enables us to say that the lowfrequency spectrum corresponds to the vibrations, in which the valence bond C–H and the valence angles C–C–H remains nearly unchanged, the middle interval corresponds to the vibrations, in which the valence corners C–C–H begin to take part in the vibration, and the third interval corresponds to the vibrations of the hard valence bonds C–H (the number of such modes always coincide with the number of hydrogen atoms). Since we have considered as sufficiently broad nanoribbon, the analysis of its dispersal curves suggests that the frequency spectrum of linear vibrations of the endless graphene sheet (CH)∞ consists of three continuous intervals [0, 756], [987, 1544] and [2922, 2967] cm−1 . Thus, the graphene sheet has two gaps in the frequency spectrum: narrow low-frequency [756, 987] and wide high-frequency [1544, 2922] cm−1 . Such structure of the frequency spectrum is in good agreement with the results of first-principles calculations. Quantum calculations suggest the high-frequency spectrum intervals [0, 806], [950, 1350] and [2733, 2783] cm−1 [68]. If the hydrogen atoms H change to the heavier deuterium atoms D, the three-zoned structure of the frequency spectrum will remain, but

6. Corrections of the COMPASS force field for graphene sheet The simulation shows that the COMPASS force field does not describe well the frequency spectrum of the graphene sheet. The frequency spectrum of out-of-plane vibrations differs strongly from the experimental values. The potential energy of the graphene sheet depends on variations in bond length, bond angles, and dihedral angles between the planes formed by three neighboring carbon atoms and it can be written in the form ∑ ∑ ∑ ∑ ∑ 𝐸= 𝑉1 + 𝑉2 + 𝑉3 + 𝑉4 + 𝑉5 , (11) 𝛺1

𝛺2

𝛺3

𝛺4

𝛺5

where 𝛺𝑖 , with 𝑖 = 1, 2, 3, 4, 5, are the sets of configurations including up to nearest-neighbor interactions. Owing to a large redundancy, the sets only need to contain configurations of the atoms shown in Fig. 7, 6

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Fig. 6. Density of states 𝑝(𝜔) of atom vibrations in (a) graphane nanoribbon (C12 H14 )99 C10 H22 (darker color corresponds to frequency region of the gap edge modes) and (b) ∞ graphene fluoride nanoribbon (C12 F14 )99 C10 F22 for temperature 𝑇 = 300 K. The density of states is normalized by the condition ∫0 𝑝(𝜔)𝑑𝜔 = 1.

Fig. 7. Configurations containing up to 𝑖th nearest-neighbor interactions in the graphene sheet.

Potentials 𝑉𝑖 (𝐮𝛼 , 𝐮𝛽 , 𝐮𝛾 , 𝐮𝛿 ), 𝑖 = 3, 4, 5, describes the deformation energy associated with a change of the effective angle between the planes 𝐮𝛼 𝐮𝛽 𝐮𝛾 and 𝐮𝛽 𝐮𝛾 𝐮𝛼 as shown in Fig. 7. In the COMPASS force field the first potential 𝑉1 = 𝑘𝑏,2 (𝑏 − 𝑏0 )2 + 𝑘𝑏,3 (𝑏 − 𝑏0 )3 + 𝑘𝑏,4 (𝑏 − 𝑏0 )4 describes the deformations of valence bonds, the second potential 𝑉2 = 𝑘𝜃,2 (𝜃 − 𝜃0 )2 + 𝑘𝜃,3 (𝜃 − 𝜃0 )3 + 𝑘𝜃,4 (𝜃 − 𝜃0 )4 – the deformations of valence angles, the third potential 𝑉3 (𝜒) = 𝑘𝜒 𝜒 2 – out-of-plane deformations. The potentials of dihedral angles 𝑉4 and 𝑉5 are described by a single potential of the torsional angle 𝑉𝑡 (𝜙) = 𝑘𝜙,1 (1 − cos 𝜙) + 𝑘𝜙,2 (1 − cos 2𝜙) + 𝑘𝜙,3 (1 − cos 3𝜙). The specific value of the parameter 𝜅𝜒 = 5.7537 kcal/mol can be found from the frequency spectrum of small-amplitude oscillations of a sheet of graphite [35]. For the potentials of dihedral angles 𝑉4 = 𝑐4 (𝜙 − 𝜋)2 and 𝑉5 = 𝑐5 𝜙2 according to the results of Ref. [69] coefficient 𝑐4 is close to 𝑘𝜒 , whereas 𝑐5 ≪ 𝑐4 (|𝑐5 ∕𝑐4 | < 1∕20). So for the second derivative of the torsion potential we have 𝑉𝑡′′ (0) = 𝑘𝜙,1 + 4𝑘𝜙,2 = 𝑉5′′ (0) = 0, 𝑉𝑡′′ (𝜋) = −𝑘𝜙,1 + 4𝑘𝜙,2 = 𝑉4′′ (0) = 2𝑐4 = 2𝑘𝜒 ,

Fig. 8. Structure of 3𝑁𝑒 dispersion curves for zigzag graphene nanoribbon (C64 H2 )∞ for (a) in-plane and (b) out-of-plain vibrations. The thick red curves correspond to oscillations localized at the nanoribbon edges. The force field COMPASS with correction of parameters (12) was used. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

therefore coefficients 𝑘𝜙,2 = 𝑘𝜒 ∕4 = 1.4384 kcal/mol, 𝑘𝜙,1 = −4𝑘𝜙,2 = −5.7537 kcal/mol. Now let us consider the COMPASS force field with the new parameters values 𝑘𝜒 = −𝑘𝜙,1 = 5.7537, 𝑘𝜙,2 = 1.4384 kcal/mol.

including their rotated and mirrored versions. The potential 𝑉1 (𝐮𝛼 , 𝐮𝛽 )

(12)

Since all the parameters for the off-diagonal interactions bond/torsion, angle/torsion and angle/angle/torsion have become undefined after such modification, remove them away from the force field. Then we find the frequency value of the graphene sheet via that modification of the force field.

describes the deformation energy due to a direct interaction between pairs of atoms with indexes 𝛼 and 𝛽. The potential 𝑉2 (𝐮𝛼 , 𝐮𝛽 , 𝐮𝛾 ) describes the deformation energy of the angle between the valent bonds 𝐮𝛼 𝐮𝛽 and 𝐮𝛽 𝐮𝛾 . 7

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Fig. 8 shows the dispersal curves of the graphene nanoribbon (C64 H2 )∞ calculated via the modified COMPASS force field. From the figure, we can now conclude that the spectrum of the graphene sheet consists of the frequency interval of in-plane vibrations [0, 1823] and the frequency interval of out-of-plane vibrations [0, 876] cm−1 , that is in good agreement with the results described in [64–67]. Thus, modification (12) of the COMPASS force field allows taking into account the flexural mobility of graphene sheet (the initial set of parameters led to an over estimation of the bending stiffness)

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7. Conclusion The study shows that the COMPASS force field allows describing with good accuracy the frequency spectrum of atomic vibrations of graphane and fluorographene sheets, whose honeycomb hexagonal lattice is formed by sp3 valence bands. On the other hand, the force field does not describe very well the frequency spectrum of the graphene sheet, whose planar hexagonal lattice is formed by sp2 valence bands. In that case, the frequency spectrum of out-of-plane vibrations differs greatly from the experimental data (bending stiffness of a graphene sheet is strongly overestimated). The correction (12) of parameters of out-of-plane and torsional potentials of the force field was made, and that allows us to achieve the coincidence of vibration frequency with experimental data. After such corrections, the COMPASS force field can be used to describe the dynamics of flat graphene sheets and carbon nanotubes. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement A.V. Savin: Conceptualization, Methodology, Software, Investigation. M.A. Mazo: Investigation, Data curation, Validation, Writing original draft, Writing - review & editing. Acknowledgments The work was supported by the Russian Science Foundation (award No. 16-13-10302). The research was carried out using supercomputers at the Joint Supercomputer Center of the Russian Academy of Sciences (JSCC RAS). References [1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Electric field effect in atomically thin carbon films, Science 306 (2004) 666–669, http://dx.doi.org/10.1126/science.1102896. [2] V.B. Mohan, K. Lau, D. Hui, D. Bhattacharyya, Graphene-based materials and their composites: A review on production, applications and product limitations, Composites B 142 (2018) 200–220, http://dx.doi.org/10.1016/j.compositesb. 2018.01.013. [3] D. Akinwande, C.J. Brennan, J.S. Bunch, et al., A review on mechanics and mechanical properties of 2D materials-graphene and beyond, Extreme Mech. Lett. 13 (2017) 42–77. [4] V. Harik, Mechanics of Carbon Nanotubes. Fundamentals, Modelling and Safety, Academic Press, 2018, http://dx.doi.org/10.1016/C2016-0-00799-4. [5] W. Feng, P. Long, Y. Feng, Y. Li, Two-dimensional fluorinated graphene: Synthesis, structures, properties and applications, Adv. Sci. 3 (2016) 1500413, http://dx.doi.org/10.1002/advs.201500413. [6] T. Chen, R. Cheung, Mechanical properties of graphene, in: M. Aliofkhazraei, N. Ali, W.I. Milne, C.S. Ozkan, S. Mitura, J.L. Gervasoni (Eds.), Graphene Science Handbook: Mechanical and Chemical Properties, CRC Press, London, 2016, pp. 3–15, http://dx.doi.org/10.1201/b19674. [7] G.R. Bhimanapati, Z. Lin, V. Meunieret, et al., Recent advances in twodimensional materials beyond graphene, ACS Nano 9 (2015) 11509–115399, http://dx.doi.org/10.1021/acsnano.5b05556. 8

Physica E: Low-dimensional Systems and Nanostructures 118 (2020) 113937

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