Using spiral chain models for study of nanoscroll structures

PHYSICAL REVIEW B 97, 165436 (2018)

Using spiral chain models for study of nanoscroll structures Alexander V. Savin,1,2 Ruslan A. Sakovich,1 and Mikhail A. Mazo1 1

Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow 119991, Russia 2 Plekhanov Russian University of Economics, Stremyanny per. 36, Moscow 117997, Russia (Received 3 January 2018; revised manuscript received 11 April 2018; published 26 April 2018)

Molecular nanoribbons with different chemical structures can form scrolled packings possessing outstanding properties and application perspectives due to their morphology. Here, we propose a simplified two-dimensional model of the molecular chain that allows us to describe the molecular nanoribbon’s scrolled packings of various structures as a spiral packaging chain. The model allows us to obtain the possible stationary states of single-layer nanoribbon scrolls of graphene, graphane, fluorographene, fluorographane (graphene hydrogenated on one side and fluorinated on the other side), graphone C4 H (graphene partially hydrogenated on one side), and fluorographone C4 F. The obtained states and the states of the scrolls found through all-atomic models coincide with good accuracy. We show the stability of scrolled packings and calculate the dependence of energy, the number of coils, and the inner and outer radius of the scrolled packing on the nanoribbon length. It is shown that a scrolled packing is the most energetically favorable conformation for nanoribbons of graphene, graphane, fluorographene, and fluorographane at large lengths. A double-scrolled packing when the nanoribbon is symmetrically rolled into a scroll from opposite ends is more advantageous for longer length nanoribbons of graphone and fluorographone. We show the possibility of the existence of scrolled packings for nanoribbons of fluorographene and the existence of two different types of scrolls for nanoribbons of fluorographane, which correspond to the left and right Archimedean spirals of the chain model. The simplicity of the proposed model allows us to consider the dynamics of molecular nanoribbon scrolls of sufficiently large lengths and at sufficiently large time intervals. DOI: 10.1103/PhysRevB.97.165436 I. INTRODUCTION

Due to its unique electrical and mechanical properties, graphene in various conformations is of great interest [1–5]. Graphene is a two-dimensional structure with a peak rigidity and tensile strength, but easily flexing in space. Secondary structures of graphene (folds, scrolls) can be attributed to a special class of carbon nanomaterials, the stability of which is provided by weak nonvalent (van der Waals) interactions of carbon atoms. In 1960 it was found that the use of graphite lubricant from flat pieces of graphite causes the forming of microscopic scrolls that play the role of roller bearings and provide a low value of the coefficient of friction [6]. A graphene nanoribbon, folding into a scroll, forms a new quasi-one-dimensional structure, which has a cross section in the form of a truncated Archimedean spiral. The geometric shape of the scroll is determined by the energy balance between the increasing van der Waals energy of the contacting areas of the graphene sheet and the energy lost due to the nanoribbon’s bending. There are several experimental technologies for obtaining scrolls of graphene nanoribbons and studying their structure and properties [7–12]. The properties of the scrolled packing of carbon nanoribbons were studied in a series of theoretical research projects. Electrical, optical, and mechanical properties of short nanoribbons scrolls were modeled from first principles [13–15]. The mechanical properties of longer nanoribbon scrolls and different scenarios of their self-assembly were described using the molecular dynamics method in numerous 2469-9950/2018/97(16)/165436(12)

articles [16–26]. The mechanical properties of long nanoribbon scrolls were described in the context of the continuum model of an elastic coiled rod [16,21,27,28], in which the flexural energy of the rod is compensated by the energy from the interaction of the contacting surfaces. All-atom models have always been used for modeling the dynamics of folds and scrolls of nanoribbons. Such models require considerable computer resources and do not allow us to consider the dynamics of long nanoribbons over significant time intervals. Therefore, only scrolls having two or three coils are usually considered. The complexity of the all-atomic models also makes it difficult to carry out a full analysis of possible stationary packings of long nanoribbons. To overcome these difficulties, we proposed a two-dimensional (2D) model of the molecular chain [29,30] that allows us to describe with high accuracy the possible stationary states of graphene nanoribbon scrolled packings. Here, we propose a more simplified model of a 2D chain that allows us to describe the stationary states of scrolled packings of molecular nanoribbons with different chemical structures. Using the model, we obtained possible stationary states of single-layer nanoribbon scrolls of graphene, graphane, fluorographene, fluorographane, graphone C4 H, and fluorographone C4 F. The model can be used both for the description of scrolled packings of molecular nanoribbons with the same surfaces (nanoribbons of boron nitride, silicene, phosphorene, carbon nitride, etc.), and for the nanoribbons which sides have different chemical modification (nanoribbons of graphone and their analogs).

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PHYSICAL REVIEW B 97, 165436 (2018)

The remainder of the paper is organized as follows. In Sec. II the chain model of the molecular nanoribbon is introduced and the parameters of the model are fitted to some results in the frame of the full-atomic model. In Sec. III the chain model is applied to analyze the stationary state of nanoribbon scrolls. We consider the scroll structures of graphane, fluorographene, fluorographane, and graphone nanoribbons in Secs. IV–VII, respectively. Some conclusions are offered in Sec. VIII.

H C

z

II. CHAIN MODEL OF MOLECULAR NANORIBBON

For simplicity, let us consider molecular nanoribbons with straights edges—a rectangular narrow strip, cut from a singlelayered molecular plane. The simplest example of such molecular plane is a graphene sheet (isolated monolayer of carbon atoms of crystalline graphite) and its various chemical modifications: graphane (a graphene sheet fully hydrogenated on both sides), fluorographene (fluorinated graphene), fluorographane (a graphene sheet hydrogenated on one side and fluorinated on the other side), graphone C4 H (a graphene sheet hydrogenated on one side with a density of 0.25), and fluorographone C4 F; see Fig. 1, panels (a), (b), (c), (d), (e), and (f).

(a)

n−1 n n+1

y

V (r)

x

(b)

(c)

U (θ) FIG. 2. View (a) from the top and (b) from the side of the graphane nanoribbon (C8 H10 )∞ with a zigzag structure (the nanoribbon lies in the plane xz); (c) a mechanical model of a chain of particles in the plane xy in which the particle defines the position of the corresponding transverse line of nanoribbon atoms (n is the line number). The potential V (r) describes the longitudinal rigidity, and the angular potential U (θ), the flexural rigidity of the chain.

FIG. 1. Structures of nanoribbons: (a) graphene (C12 H2 )11 C10 H12 ˚ 2 ), (b) graphane (C12 H14 )11 C10 H22 , (c) fluoro(of size 29.85×13.44 A graphene (C12 F14 )11 C10 F22 , (d) fluorographane (C12 H8 F6 )11 C10 H15 F5 , (e) graphone (C12 H3 )11 C18 , (f) fluorographone (C12 F3 )11 C18 (gray spheres, carbon; white, hydrogen; yellow, fluorine atoms).

As is known, graphene and its modifications are elastically isotropic materials, the longitudinal and flexural rigidity of which is weakly dependent on chirality of the structure [31–33]. Therefore, for definiteness, we will consider nanoribbons with the zigzag structure shown in Fig. 1 and Fig. 2(a). Suppose that in the ground state the nanoribbon lies in the plane xz of the three-dimensional space along the axis x; see Fig. 2(a). Such nanoribbon is a periodic structure with a constant step. Translational cells of this structure form atoms located along lines parallel to the z axis (along transverse lines). We consider such motions of the nanoribbon, when its atoms located on lines parallel to the z axis move as a rigid whole in the xy plane, keeping its coordinates along the z axis. Then these atomic lines can be considered as effective particles moving in the xy plane, and the movement of the nanoribbon

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TABLE I. Values of the parameters of the two-dimensional chain model for the nanoribbon of graphene CC, graphane HCCH, fluorographene FCCF and fluorographane HCCF, graphone C4 H, and fluorographone C4 F. Nanoribbon

M (mp )

˚ a (A)

K (N/m)

 (eV)

θ0 (deg)

0 (eV)

˚ r0 (A)

k

l

24 26 62 44 24.5 32.5

1.228 1.261 1.285 1.285 1.250 1.260

910.0 607.2 606.6 607.0 800.0 800.0

7.00 3.38 4.00 3.70 5.00 6.00

180 180 180 177 171 173

0.01970 0.00984 0.00715 0.00325 0.01430 0.01900

3.68 5.01 5.71 5.06 4.30 4.60

8 16 18 16 20 24

5 6 16 3 5 7

CC HCCH FCCF HCCF C4 H C4 F

is reduced to the motion of a chain of these particles; see Figs. 2(b), 2(c). Thus, the longitudinal and flexural movements of the nanoribbon can be described as the motion of a chain of pointwise particles in the xy plane. The simplest model of a molecular chain in two-dimensional space is shown in Fig. 2(c). The Hamiltonian of the chain has the following form: H =

N  1 n=1

+

2

   N−1 M x˙n2 + y˙n2 + V (Rn )

N−1  n=2

n=1

U (θn ) +

N−5 

N 

W (rnm ),

(1)

n=1 m=n+5

where N is the number of particles in the chain, M is the particle mass, and the vector un = (xn ,yn ) defines the position of the nth particle. The potential V (R) = K(R − a)2 /2

(2)

is responsible for the longitudinal rigidness of the chain; K is the interaction rigidness, a is the equilibrium bond length (chain step), and Rn = |un+1 − un | is the distance between neighbor particles n and n + 1. The potential U (θ ) = [1 − cos(θ − θ0 )] ≈ (θ − θ0 )2 /2

(3)

describes the flexural chain mobility; θ is the the valence angle formed by two neighbor bonds, θ0 is the equilibrium angle, and the parameter  > 0 specifies the flexural rigidity of the chain. For the nth valent angle cos(θn ) = −(un−1 ,un )/|un−1 | · |un |. The Lennard-Jones (l,k) potential W (r) = ε[l(r0 /r)k − k(r0 /r)l ]/(k − l)

(4)

describes a weak nonvalent interaction between remote chain particles; ε is the bond energy, r0 is the equilibrium bond length, k > l (degree of repulsion k is always greater than the degree of attraction l), and rnm = |un − um | is the distance between particles n and m. The step of the chain a is found as half of the period of the ground state of a flat zigzag nanoribbon. The parameter K, which determines the longitudinal rigidness, and the parameter , which determines the flexural rigidness of the chain, can be obtained from the analysis of the dispersion curves of the nanoribbon. The linear chain in 2D space has two dispersion curves for the longitudinal and transversal oscillations of the chain. The values of the chain stiffness parameters K and  were chosen so that the shape of the dispersion curves of the

chain best corresponded to the shape of the dispersion curves of the longitudinal and bending vibrations of the nanoribbons. The scheme for constructing and comparing the dispersion curves for a graphene nanoribbon is described in detail in Refs. [29,30]. The parameters of the Lennard-Jones potential (4) can either be directly calculated as sums of the nonvalent interactions of one atom with the transverse line of nanoribbon atoms, or estimated from the analysis of the structure of the nanoribbon scroll obtained using a full-atomic model. The scheme for constructing the reduced potential for the chain model of graphene nanoribbons is described in detail in Ref. [29]. Values of the parameters of the chain model for the nanoribbons of graphene CC, graphane HCCH, fluorographene FCCF, fluorographane HCCF, graphone C4 H, and fluorographone C4 F are presented in Table I. While constructing the model, wide nanoribbons are considered, so the chemical modification of its edges can be ignored, and all parameters of the nanoribbon should be normalized on its width. In this case, the particle mass M for a graphene nanoribbon will be equal to the mass of two carbon atoms (M = 2MC = 24mp ), for a nanoribbon of graphane M = 2(MC + MH ) = 26mp , of fluorographene M = 2(MC + MF ) = 62mp , of fluorographane M = 2MC + MH + MF = 44mp , of graphone M = 2MC + 0.5MH = 24.5mp , and of fluorographone M = 2MC + 0.5MF = 32.5mp , where mp is the proton mass. The longitudinal and flexural rigidness parameters of the chain K and  for graphene are obtained in [29,30]. For values of K = 910 N/m and  = 7 eV, the dispersion curves of the chain most exactly coincide with the dispersion curves of a flat graphene nanoribbon corresponding to its longitudinal and flexural vibrations. The values of K and  for other nanostructures can also be obtained from the analysis of the dispersion curves of nanoribbons obtained using a full-atomic model. The force fields COMPASS [34] and CFF91 were used for the analysis of the structure and dynamics of nanoribbons of graphane, fluorographene, and fluorographane. Van der Waals interactions of atoms in the COMPASS force field are described by the Lennard-Jones potential (4) with parameters l = 6, k = 9. We consider nanoribbon of graphene and using this potential to calculate the interaction energy of two bound carbon atoms located on one transverse line with all carbon atoms located on the other transverse line of the nanoribbon. The calculations show that the dependence of this interaction energy of two different transverse lines of atoms on the distance between the lines is well described by the Lennard-Jones potential (4) with parameters l = 5, k = 8; see

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W (eV)

0.01

1

2

3

0

4 5

−0.01

6 −0.02 3

4

r (˚ A)

5

6

7

8

9

FIG. 3. Paired potentials of the nonvalent interaction of the chain nodes W (r) for the nanoribbons of graphene CC, graphane HCCH, fluorographene FCCF, fluorographane HCCF, graphone C4 H, and fluorographone C4 F (curves 1, 2, 3, 4, 5, and 6). Dotted curves give the dependencies calculated using a full-atomic model of a wide nanoribbon.

Fig. 3. Similar calculations of the transverse lines of atoms’ interaction energy can also be made for graphane nanoribbons (here it will be necessary to calculate the interaction energy of a group of bound atoms H-C-C-H of one transverse line with all atoms of another transverse line). The calculations show that the energy of the interaction of transverse lines for graphane can also be described with good accuracy by a potential (4) with parameters l = 6, k = 16. For nanoribbons of fluorographene, fluorographane, graphone, and fluorographone parameters of the interaction potential (4) can be more conveniently estimated from the analysis of stationary states of nanoribbons scrolls obtained with help of the full-atomic models. The analysis shows that the fluorographene nanoribbon has the weakest nonvalent interaction (this is due to Coulomb repulsion of fluorine atoms). III. STATIONARY STATES OF NANORIBBON SCROLLS

To find the stationary state of nanoribbon scrolls, it is necessary to solve the minimum problem Etotal → min ,

(5)

i.e., to minimize the potential energy of a full-atom model of the nanoribbon along all coordinates of its atoms, starting from the initial scroll-like configuration. Using a chain model to find a scroll structure, it is necessary to solve the minimum problem E=

N−1  n=1

V (Rn ) +

N−1  n=2

U (θn ) +

N−3 

N 

W (rnm )

n=1 m=n+3

= EV + EU + EW → min : {un }N n=1 ,

(6)

where N is the the number of particles in the chain [nanoribbon length L = (N − 1)a]. The first sum specifies the energy of the longitudinal deformation of the chain EV , the second sum, the energy of the flexural deformation EU , and the third sum EW , the energy of nonbonded interactions between chain particles.

FIG. 4. Stable stationary conformations of graphane nanoribbon with length L = (N − 1)a = 25.09 nm (number of chain particles N = 200): (a) double-folded (the specific energy of the structure E/N = −0.02582 eV), (b) triple-folded (−0.02496 eV), (c) doublefolded with one-side winding (−0.02884 eV), (d) double-folded with two-side winding (rolled-collapsed, −0.02975 eV), and (e) rolled (−0.03359 eV). Not shown here is the straight stable configuration (flat nanoribbon, energy −0.0060 eV).

Problem (6) was solved numerically using the conjugate gradient method. To verify stability of the obtained stationary configuration {un0 }N n=1 we found the eigenvalues of the matrix of second derivatives of dimension 2N × 2N :  N,N  ∂E  B= . (7) ∂un ∂um {ul0 }Nl=1 n=1,m=1

The stationary configuration of the chain will be stable only if all eigenvalues of the symmetric matrix B are nonnegative: λi  0, i = 1,2, . . . ,2N . Note that for the stable configuration the first three eigenvalues are always zero: λ1 = λ2 = λ3 = 0. These eigenvalues correspond to the motion of chain in the plane as a rigid body (shift in two coordinates and rotation). The remaining positive eigenvalues λi > 0 correspond to the natural oscillations of the structure with frequencies ωi = √ λ3+i /M,i = 1, . . . ,2N − 3. The structure of the stationary state of the chain is determined by its initial configuration used in solving the minimum problem (6). Varying the initial configuration, it is possible to obtain various stable chain packages. The linear configuration of the chain (flat nanoribbon) is stable if the angle θ0 = 180◦ . The presence of nonvalent interactions of chain particles leads to the existence on the plane of other, more energy-efficient stationary packings of the chain; see Fig. 4. A typical view of the scrolled packing nanoribbon and the corresponding two-dimensional spiral packing is shown in Fig. 5. As can be seen from the figure, the spiral packing of the chain practically coincides with the cross section of the nanoribbon scroll. The geometry of a scroll (spiral) is given by the number of its coils Nc and its internal and external radii R1 and R2 . Scrolled packing of a nanoribbon corresponds to the chain arrangement in the shape of an Archimedean spiral with an inner cavity. The center of the spiral is conveniently defined as

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FIG. 5. Scrolled packing (a) of graphane nanoribbons ˚ 2 and its (b) two-dimensional (C12 H14 )99 C10 H22 of size 252 × 13.7 A chain model (number of chain links N = 200, number of coils ˚ scroll outer radius Nc = 2.67, scroll inner radius R1 = 11.7 A, ˚ R2 = 19.0 A).

the center of its mass u0 = (x0 ,y0 ) =

N 1  0 u , N n=1 n

where un0 = (xn0 ,yn0 ) is the two-dimensional radius vectors of the nth chain node of the stationary spiral. In the polar coordinate system it can be written as xn0 = x0 + rn cos(φn ), yn0 = y0 + rn sin(φn ), where the radius rn = |un0 − u0 | and the angle φn increases monotonically with increasing node number n = 1,2, . . . ,N. The spiral can be characterized by the number of coils Nc = (φn − φ1 )/2π. It is also convenient to define the integer number of coils nc = [Nc ] + 1, where [x] is the integer part of x. Let us define the inner radius of the scroll by its first coil: R1 =

n1 1  rn , n1 n=1

where n1 is the number of chain nodes involved in formation of the spiral first coil (maximal value of index n wherein φn < φ1 + 2π ). The outer radius of the scroll can be defined by its last coil as R2 =

1 n2

N 

rn ,

n=N−n2 +1

where n2 is the number of chain nodes involved in the formation of the spiral last coil (N − n2 + 1 is the minimal value of n wherein φn > φN − 2π ). The twisting rigidity of the spiral √ is characterized by the lowest natural frequency ω1 = λ4 /M. This frequency corresponds to the periodic twisting/untwisting oscillations of the spiral. In the approximation of a continuous elastic rod this oscillation motion has been studied in [16,27]. IV. SCROLLED PACKING OF GRAPHANE NANORIBBONS

The structure of the scrolled packing of graphene nanoribbons was considered in Refs. [29,30]. The simpler model

proposed here leads to the same results, so we consider the scroll packings of the other five types of nanoribbons. Let us first describe the possible stationary structures of graphane nanoribbons. To do this, we consider the dependence of the number of coils Nc , inner R1 and outer radius R2 , and the smallest natural frequency ω1 on the number of chain nodes N [on the chain length L = (N − 1)a]; see Fig. 6. A typical form of packages for nanoribbons of different lengths is shown in Fig. 7. The single-coil configuration (the number of coils nc = 1) of the scroll (a) can exist only for ˚ (for 41  N  62). nanoribbons of length L ∈ [50.4,76.6] A The double-coil configuration (nc = 2) of the scroll (b) is ˚ (49  N  150), (c) the stable at lengths L ∈ [60.5,187.9] A ˚ (134  N  270), three-coil (nc = 3) at L ∈ [167.7,339.2] A ˚ (250  N  (d) the four-coil (nc = 4) at L ∈ [314.0,503.1] A 400), and (e) structures with five or more coils (nc  5) are ˚ (N  400). stable for L  503.1 A At certain lengths, there may be two stable configurations of the scroll packing; see Figs. 7(c) and 7(d). This bistability is due to the nonvalent interaction of the nanoribbon ends. In one configuration, the ends interact more strongly (are closer to each other), in the other, more weakly (the ends are more distant from each other). Such bistability also exists for scrolls of graphene nanoribbons [30]. Therefore, the dependence of Nc , R1 , R2 , ω1 on the length (on the number of particles in the chain N ) is divided into branches corresponding to the configurations of the scrolled packing with the same number of coils nc ; see Fig. 6. As can be seen from Fig. 6 increasing the nanoribbon length leads to a monotonic increase of the coils of its scrolled packing according to the power law Nc ≈ 0.105N 0.61 for N → ∞. With increasing length of the power law the radius ˚ and the outer scroll radius of inner cavity R1 ≈ 5N 0.16 A 0.49 ˚ A also increase. The specific energy of the R2 ≈ 1.45N spiral (scroll) E/N decreases monotonically with increasing number of particles (the longer the nanoribbon, the greater the energy gain from its assembly into a scroll): E/N  E∞ for N → ∞, where E∞ = −0.067 eV is specific energy of crystal from the parallel straight chains. The eigenmode of the lowest positive frequency ω1 is the twisting-untwisting mode when all nodes of the model chain move along the Archimedes spiral. The increase in the scroll length leads to a decrease in the frequency of this oscillation ω1 ≈ 135/N cm−1 for N → ∞; see Fig. 6(d). By a similar law, the smallest natural frequency of the scrolled packing of a graphene nanoribbon decreases [16,27,30]. To verify the results obtained via the chain model, stationary scrolled packings of a graphane nanoribbon of various lengths were also found using a full-atomic model. The graphane ˚ corresponding to a chain of nanoribbon of width D = 13.7 A N links can be described by the formula (C12 H14 )N/2−1 C10 H22 ; see Figs. 1(b) and 2(b). As can be seen from Fig. 6, the fullatomic model leads to the same values of the number of coils Nc and the radii of the scroll R1 , R2 , as the two-dimensional chain model. Thus, the chain model for graphane nanoribbons makes it possible to find their scrolled packings with good accuracy. For each fixed value of the chain length, we can numerically find all the main steady state packings of the chain. Except for

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9 8 7 6 5 4 3 2 1

PHYSICAL REVIEW B 97, 165436 (2018)

(a)

1

nc =5 nc =4 nc =3 nc =2 nc =1

E/N (eV)

2

3

10

10

(b)

0

−0.03

2

−0.06

3

R1 , R2 (˚ A)

2

3

10

50

10

(c)

5

FIG. 7. Scrolled packing of graphane nanoribbons for chain with number particles: (a) N = 62 (number of the scroll coils Nc = 0.97, ˚ specific inner and outer radius of the scroll R1 = R2 = 11.98 A, ˚ energy E/N = 0.0042 eV); (b) N = 62 (Nc = 1.33, R1 = 9.33 A, ˚ E/N = 0.0050 eV); (c) N = 270 (Nc = 2.99, R1 = R2 = 10.29 A, ˚ E/N = −0.0400 eV); (d) N = 270 (Nc = ˚ R2 = 22.82 A, 13.31 A, ˚ R2 = 22.10 A, ˚ E/N = −0.0402 eV); (e) N = 3.23, R1 = 11.68 A, ˚ R2 = 50.74 A, ˚ E/N = −0.0606 1400 (Nc = 8.6, R1 = 14.64 A, eV).

30 4

10

ω0 (cm−1)

2

3

10

10

(d)

0

10

6 −1

10

2

10

N

3

10

FIG. 6. Dependencies of (a) number of scroll coils Nc of graphane nanoribbon, (b) the specific energy of the structure E/N for the scroll (curves 2), (c) inner R1 and outer radius R2 of the scroll (curves 4 and 5), (d) the lowest frequency of the scroll’s natural oscillations ω1 (curve 6) on the number of chain particles N . Markers give values obtained using a full-atomic model of a nanoribbon ˚ Dashed lines give (C12 H14 )N/2−1 C10 H22 with width D = 13.7 A. ˚ R2 = power-law dependencies Nc = 0.105N 0.61 , R1 = 5N 0.16 A, ˚ ω1 = 135/N cm−1 , and dashed right line (curve 3) 01.45N 0.49 A, corresponds to specific energy E∞ = −0.067 eV for a crystal of parallel straight chains.

spiral packing, the chain can form still two-layer and multilayered folds and also combinations of folds with windings; see Fig. 4. The dependence of the specific energy E/N of these conformations on the number of chain particles N is shown in

Fig. 8(a). As the figure shows, for graphane flat nanoribbons are the most energetically favorable for N  72 (for length ˚ A double fold of the nanoribbons is the most enerL  89.5 A). ˚ getically favorable for 72 < N < 108 (89.5 < L < 134.9 A), and the scrolled packing becomes more profitable for N  108 ˚ (L  134.9 A). For dense conformations of the fold type only the end part of the chain participates in bending deformation, which does not change with increasing chain length; see Table II. Therefore the specific energy of flexural deformation EU /N decreases as N −1 for N → ∞, and the energy gain from nonbonded interactions EW /N converges to the final value of multilayered flat nanoribbons. Owing to it the specific energy of such conformations will tend to finite value: E/N  Em for N → ∞, where Em is specific energy of system of m parallel straight molecular chains (E1 = −0.0062, E2 = −0.0358, E3 = −0.0465, E4 = −0.0519, ..., E∞ = −0.067 eV). In scrolled packings, all flexural deformations are evenly distributed along all chain, and the number of roll coils monotonically grows with increase in chain length. In this case, the specific energy of scrolled packing monotonically tends to the greatest possible value E∞ , which corresponds to the energy of the infinite crystal from parallel straight

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E/N (eV)

0.02

(a)

5

0

E1

2

TABLE II. Dependencies of the specific energies EU /N , EW /N on the number of chain particles N for (a) double-folded, (b) triplefolded, (c) double-folded with one-side winding, (d) double-folded with two-side winding, and (e) rolled stationary conformations of the chain for graphane and fluorographene nanoribbons. Graphane

−0.02

E2

3

−0.04

4

E3

6 1

−0.06

E∞ 2

E/N (eV)

0.01

E4

3

10 5

10

(b)

0

E1

2

N

EU /N

100 200 400 800

0.01461 0.00730 0.00365 0.00183

100 200 400 800

0.03236 0.01624 0.00812 0.00406

100 200 400 800

0.02838 0.01420 0.00710 0.00355

100 200 400 800

0.01663 0.00831 0.00416

100 200 400 800

0.01815 0.01182 0.00688 0.00447

−0.01

E2

3

−0.02

4 1

−0.03

6

E∞ 2

10

N

E3 E4

3

10

FIG. 8. Dependencies of the specific energy E/N of scrolled, straight configuration, double-folded, triple-folded, double-folded with one-side winding and double-folded with two-side winding packings of the chain (curves 1, 2, 3, 4, 5, and 6 respectively) on the number of chain particles N for (a) graphane and (b) fluorographene nanoribbons. Dashed lines correspond to values of energy E1 > E2 > E3 > E4 > E∞ , where Em is specific energy of a crystal from the m parallel straight chains (m-layer plane nanoribbon).

molecular chains. All this allows concluding that for larger lengths scrolled packings of graphane nanoribbons will be the most energetically favorable. V. SCROLLED PACKING OF FLUOROGRAPHENE NANORIBBONS

The fluorographene nanoribbon differs from the graphane nanoribbon in such a way that its carbon atoms are joined by fluorine atoms instead of hydrogen atoms. The fluorine atom is much larger than the hydrogen atom, and the C-F valence bond is more strongly polarized than the C-H bond (in the COMPASS force field [34] it is assumed that on the C-F bond atoms there are charges qC = 0.25e, qF = −0.25e, and on the C-H bond atoms, charges qC = −0.053e, qH = 0.053e, where e is the electron charge). Coulomb repulsion of negatively charged fluorine atoms leads to a significant weakening of the interaction between the contacting sections of the nanoribbon. The analysis of the full-atom model of the nanoribbon shows that the contacting regions of the nanoribbon continue to be attracted to each other, and the nanoribbons can form stable scrolled structures; see Fig. 9.

Fluorographene EW /N (a) −0.02912 −0.03316 −0.03519 −0.03620 (b) −0.03388 −0.04127 −0.04491 −0.04673 (c) −0.03765 −0.04311 −0.04583 −0.04719 (d) −0.04645 −0.05038 −0.05235 (e) −0.03158 −0.04546 −0.05409 −0.06044

EU /N

EW /N (eV)

0.01125 0.00562 0.00281 0.00141

−0.01177 −0.01601 −0.01813 −0.01919

0.02078 0.01221 0.00611 0.00253

−0.00741 −0.01731 −0.02163 −0.02423

0.02572 0.01255 0.00629 0.00316

−0.01583 −0.02061 −0.02329 −0.02462

0.01444 0.00711 0.00356

−0.02182 −0.02519 −0.02698

0.01228 0.00728 0.00489 0.00306

−0.01125 −0.01981 −0.02633 −0.03026

The large size of the fluorine atoms and the strong polarization of the valence bonds C-F make it difficult to directly calculate the nonvalent interaction potential W (r) for the chain model. The parameters k and l of the potential (4) can be estimated from the asymptotics of the interaction energy of the nanoribbon sections as they approach and become distant from each other. The parameters r0 and ε here were chosen so that the structure of the scrolls obtained using a full-atomic

FIG. 9. Scrolled (C12 F14 )99 C10 F22 of N = 200) with (a) ˚ outer R1 = 18.2 A, E = 42.52 eV) and E = 42.45 eV).

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packing of fluorographene nanoribbons ˚ 2 (number of particles size 256.3 × 14.2 A number of coils Nc = 1.94 (inner radius ˚ energy radius of the scroll R2 = 23.5 A, ˚ R2 = 22.1 A, ˚ (b) Nc = 2.16 (R1 = 15.7 A,

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(a) 1

Nc

6 5

nc =5 nc =4 nc =3 nc =2

4 3

E/N (eV)

2 1 2

3

10

10

0.01

(b)

0

−0.01 2

−0.02

R1 , R2 (˚ A)

−0.03 60 50

3 2

3

10

10

5

(c)

40 30

4

20 10 2

10

N

3

10

FIG. 10. Dependencies of (a) number of scroll coils Nc of fluorographene nanoribbons, (b) the specific energy of the structure E/N for the scroll (curve 2), (c) inner R1 and outer radius R2 of the scroll (curves 4 and 5) on the number of chain particles N . Markers give values obtained using a full-atomic model of a nanoribbon ˚ Dashed lines give (C12 F14 )N/2−1 C10 F22 with width D = 14.2 A. power-law dependencies Nc = 0.083N 0.61 (curve 1), R1 = 7N 0.18 ˚ R2 = 1.8N 0.48 A ˚ (curves 4 and 5), and dashed right line (curve A, 3) corresponds to specific energy E∞ = −0.035 eV of a crystal of parallel straight chains.

model would best coincide with the structure of the spiral chain packages obtained using the chain model. The best match is achieved when using the values of the parameters presented in Table I. The dependence of the number of coils Nc and inner R1 and outer radius R2 of the scroll on the number of chain nodes N for the fluorographene nanoribbon is given in Fig. 10. There are only two or more coils in scrolled packings for fluorographene nanoribbons (number of coils Nc > 1). The double-coil configuration (nc = 2) can exist only for ˚ (for 68  N  230). nanoribbons of length L ∈ [86,294] A The three-coil configuration (nc = 3) of the scroll is stable at ˚ (200  N  400), four-coil (nc = 4) lengths L ∈ [256,513] A ˚ at L ∈ [513,641] A (400  N  600), five-coil (nc = 5) at

˚ (600  N  840), and structures with five L ∈ [641,1078] A ˚ (N  840). or more coils (nc  5) are stable for L  1078 A At certain lengths, there can be simultaneously two stable configurations of the scrolled packing of the nanoribbon (see Fig. 9). This bistability as well as for graphane nanoribbons is due to the noncovalent interaction of the nanoribbon ends. Because of this, the dependencies Nc , R1 , and R2 on N are divided into branches corresponding to the configurations of a scrolled packing with the same number of coils nc ; see Fig. 10. As can be seen from Fig. 10 increasing the nanoribbon length leads to a monotonic increase in the number of coils of its scrolled packing according to the power law Nc ≈ 0.083N 0.61 for N → ∞. With increasing length, the radius of the inner ˚ cavity also grows according to the power law R1 ≈ 7N 0.18 A 0.48 ˚ A. The specific and the outer radius of the scroll R2 ≈ 1.8N energy of the spiral (scroll) E/N decreases monotonically with the number of particles (the longer the nanoribbon, the greater the energy gain from its assembly into a scroll): E/N  E∞ for N → ∞, where E∞ = −0.035 eV is specific energy of a crystal of parallel straight chains. Stationary scrolled packing of fluorographene nanoribbons of various lengths was also found using an all-atomic model to verify the results obtained via the chain model. The fluo˚ corresponding rographene nanoribbon of width D = 14.2 A to a chain of N links can be described by the formula (C12 F14 )N/2−1 C10 F22 ; see Fig. 1(c). As can be seen from Fig. 8 a full-atomic model gives a good coincidence of the number of coils Nc and scroll radii R1 , R2 with the values obtained via the chain model. Thus, the chain model for fluorographene nanoribbons also makes it possible to find with good accuracy their scrolled packings. For each fixed value of the length of a chain, we can numerically find all its main steady state packings. Here too, as well as for nanoribbons of graphane, except for spiral packing of a chain, exist also double-folded and multifolded structures and combinations of folds with windings. Dependencies of the specific energy E/N on the number of chain nodes N for different conformations is shown in Fig. 8(b). As can be seen from the figure, the flat form of the fluorographane nanoribbon is the most energetically favorable for N  116 ˚ A double fold of the nanorib(for length L  147.8 A). bon is the most energetically favorable for 116 < N < 142 ˚ and the scrolled packing becomes (147.8 < L < 181.2 A), ˚ more profitable for N > 142 (L > 181.2 A). Also, as for a graphane, here for dense conformations of the fold type only the end part of the chain participates in bending deformation, which does not change with increasing chain length; see Table II. Therefore the specific energy of flexural deformation EU /N decreases as N −1 for N → ∞, and the energy gain from nonbonded interactions EW /N converges to the final value of multilayered flat nanoribbons. Owing to it the specific energy of such conformations will tend to finite value: E/N  Em for N → ∞, where Em is the specific energy of the system of m parallel straight molecular chains (E1 = −0.0032, E2 = −0.0194, E3 = −0.0248, E4 = −0.0274, ..., E∞ = −0.035 eV). In scrolled packings, all flexural deformations are evenly distributed along all chains, and the number of roll coils monotonically grows with increase in chain length. In this

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Let us consider the fluorographane nanoribbon shown in Fig. 1(d). The main feature of this nanoribbon is the nonequivalence of its sides. One side of it is hydrogenated (hydrogen atoms are attached to carbon atoms), and the other is fluorinated (fluorine atoms are attached to carbon atoms). Since fluorine atoms are larger than hydrogen atoms and have a much greater electrical charge, they are more repulsive from each other. Therefore, the flat form of such nanoribbon is not a stable state. A nanoribbon always bends and forms a convex surface the outer side of which contains fluorine atoms, and the inside contains hydrogen atoms. If nanoribbon length considerably exceeds its width, in the ground state it takes the form of a circular arc; see Fig. 11(a). Formation of the arc leads to the convergence of the nanoribbon ends, which can lead to the self-assembly of the nanoribbon into a scroll structure [35]. In the chain model the nonequivalence of nanoribbon sides is reflected in the difference of the equilibrium value of the bond angle θ0 from 180◦ . For a fluorographane nanoribbon angle θ0 = 177◦ . The convex nanoribbon can be rolled into a scroll by two ways: to the scroll in which its outer side contains fluorine atoms and to the scroll in which its outer side contains hydrogen atoms; see Figs. 11(b) and 11(c). The most advantageous in energy is always the first form of the scroll, since in it the convexity of the nanoribbon coincides with its bend in the

FIG. 11. The ground state (a) of fluorographene nanoribbons ˚ number of particles (C12 H8 F6 )47 C10 H17 F5 (ribbon width D = 13.9 A, N = 96). The least (b) and the most (c) energetically favorable scrolled packings of nanoribbon (C12 H8 F6 )127 C10 H17 F5 (N = 256): packing with fluorinated inner surface (number of scroll coils Nc = ˚ R2 = 23.4 A, ˚ energy E = 146.17 2.8, scroll radii R1 = 14.4 A, eV) and packing with fluorinated outer surface (Nc = 3.43, R1 = ˚ E = 116.45 eV). The energy difference between ˚ R2 = 21.8 A, 9.1 A, conformations (b) and (c) E = 29.7 eV.

(a) 1

Nc E/N (eV)

VI. SCROLLED PACKING OF FLUOROGRAPHANE NANORIBBONS

11 10 9 8 7 6 5 4 3 2 1 0.04

2

nc =5 nc =4 nc =3 nc =2 2

3

10

10

(b)

0

−0.04 4 3

−0.08

R1 , R2 (˚ A)

case, the specific energy of scrolled packing monotonically tends to the greatest possible value E∞ , which corresponds to the energy of the infinite crystal from parallel straight molecular chains. All this allows concluding that for larger lengths scrolled packings of fluorographane nanoribbons also will be the most energetically favorable. The analysis of possible stationary packings of graphane and fluorographene nanoribbons allows us to conclude that for nanoribbons with larger lengths and identical surfaces (for the chains with an equilibrium value of valence angle θ0 = 180◦ ) scrolled packing will be the most energetically favorable. The situation changes if surfaces of a nanoribbon have various chemical structure and equilibrium valence angle θ0 = 180◦ .

2

60 50

3

10

10

8

(c)

40

6

30 7

20 5

10 2

10

N

3

10

FIG. 12. Dependencies of (a) the number of scroll coils Nc , (b) the specific energy E/N , (c) inner R1 and outer radius R2 of the scrolled packing of fluorographene nanoribbons on the number of chain particles N . Curves 1, 3, 5, and 6 give dependencies for scroll with fluorinated outer surface; curves 2, 4, 7, and 8, for scroll with fluorinated inner surface. Markers give values obtained using the fullatomic model of nanoribbon (C12 H8 F6 )N/2−1 C10 H17 F5 of width D = ˚ Dashed lines give power-law dependencies Nc = 0.165N 0.56 13.9 A. ˚ R2 = 1.1N 0.53 A ˚ and Nc = 0.135N 0.56 (curves 1 and 2), R1 = 8.4 A, 0.23 ˚ 0.51 ˚ A, R2 = 1.35N A (curves 7 (curves 5 and 6), and R1 = 3.8N and 8).

scroll. In the chain model these forms of the scroll correspond to the chain packings in a spiral with different chirality (right and left spiral). The dependence of the number of coils Nc , the specific energy E/N, and inner R1 and outer radius R2 of the scroll on the number of particles N is given in Fig. 12. As can be seen from the figure, a scroll of a nanoribbon with a fluorinated outer surface is always more energy efficient and has a more compact shape than a scroll with a hydrogenated outer surface. The number of coils Nc ≈ 0.165N 0.56 for the first form and Nc ≈ 0.135N 0.56 for the second form of the scroll. The inner cavity radius of the first form scroll is practically independent ˚ and the outer radius R2 ≈ 1.1N 0.53 A. ˚ of its length R1 ≈ 8.4 A,

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VII. SCROLLED PACKING OF GRAPHONE NANORIBBONS

12

(a)

10

1

Nc

8 6

2

4 2

E/N (eV)

˚ R2 ≈ 1.35N 0.51 A ˚ For the second form scroll R1 ≈ 3.8N 0.23 A, for N → ∞. It is necessary to note that both forms of a scroll having different twist are stable configurations. The most energetically advantageous first form of a scroll can be formed by ˚ (N > 35), and the second, nanoribbons of length L > 45.0 A ˚ (N > 55). only by nanoribbons of length L > 70.7 A To verify the results obtained via the chain model, stationary scrolled packing of fluorographane nanoribbons of various lengths was also found using a full-atomic model. The ˚ corresponding fluorographane nanoribbon width D = 13.9 A to the chain of N links can be described by the formula (C12 H8 F6 )N/2−1 C10 H17 F5 ; see Fig. 1(d). As can be seen from Fig. 12 the 2D chain model gives a good coincidence of the number of coils Nc and the radii of the scroll R1 , R2 with the values obtained via the full-atomic model. Thus, the chain model for fluorographane nanoribbons also allows us to find their scrolled packings.

0

2

3

10

10

(b)

−0.03 3 4

−0.06 2

R1 , R2 (˚ A)

Graphone C4 H (25% one-side hydrogenated graphene sheet) is the most stable structure formed by hydrogenation on one side of a graphene sheet [36]. Attaching a hydrogen (fluorine) atom leads to a local change in the valence bonds with sp2 hybridization by sp3 , which entails the appearance of local convexity in the sheet. The addition of hydrogen atoms along a single line leads to a break in the flat sheet of graphene along this line to the formation of a dihedral angle [37,38]. One-side hydrogenation (fluorination) of graphene nanoribbon entails nonequivalence of its sides. The flat form of the nanoribbon becomes unstable; it bends and forms a convex

50

3

10

10

(c)

40 30

8

20 6

10

7 5 2

10

N

3

10

FIG. 14. Dependencies of (a) number of scroll coils Nc , (b) specific energy E/N , (c) inner R1 and outer R2 of single-scrolled (curves 1, 3, 5, and 6) and double-scrolled (curves 2, 4, 7, and 8) packings of graphone nanoribbons C4 H on the number of chain particles N . Markers give values obtained using the full-atomic model ˚ of nanoribbon (C12 H3 )N/2−1 C18 of width D = 11.4 A.

FIG. 13. Single-scrolled (a), (d) and double-scrolled (b), (c) packings of graphone nanoribbons C4 H, C4 F with number of chain particles N = 1001. For packing (a) number of coils Nc = 9.12, ˚ R2 = 38.40 A, ˚ inner and outer radii of the scroll R1 = 5.29 A, ˚ specific energy E/N = −0.05648 eV; (b) Nc = 6.21, R1 = 5.26 A, ˚ E/N = −0.05808 eV; (d) Nc = 8.68, R1 = 6.54 A, ˚ R2 = 26.41 A, ˚ E/N = −0.07268 eV; (c) Nc = 5.95, R1 = 6.24 A, ˚ R2 = 40.14 A, ˚ E/N = −0.07122 eV. R2 = 27.56 A,

surface on the outside with attached atoms. Strong bending of the nanoribbon leads to its folding into scroll structures [39,40]. In the chain model the nonequivalence of nanoribbon sides is reflected in the difference of the equilibrium value of the bond angle θ0 from 180◦ . The nonequivalence of graphone nanoribbon sides is more revealed than that of the fluorographane nanoribbons; here the angle θ = 171◦ for C4 H and θ = 173◦ for C4 F. Graphone nanoribbons are always rolled up so that their convex side is the outer side of the scroll; see Fig. 13. As can be seen from the figure, it is possible to fold the nanoribbon into a single-scrolled (a), (d) and a double-scrolled (b), (c) packing. The double-scrolled packing becomes more energy efficient for graphone C4 H ˚ and with the number of particles N > 700 (L > 874 A), ˚ for C4 F, with N > 1400 (L > 1763 A); see Fig. 14(b) and 15(b).

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specific energy takes place with the number of chain particles ˚ for single-scrolled packing of the N = 701 (L = 882 A) nanoribbon; see Fig. 15(b). For double-scrolled this switching ˚ The increase in the entakes place at N = 1601 (L = 2016 A). ergy of the scrolled packing for large nanoribbon lengths is due to the fact that for larger scroll diameters, its further winding of the nanoribbon becomes less energetically favorable (the small curvature of the outer layers does not allow the realization of the natural curvature of the layer with nonequivalent sides). Therefore, for the length L > 100 nm (L > 202 nm) graphone nanoribbons C4 H (C4 F) will form not the scrolls having a cross-sectional shape of a flat spiral, but more complex 3D spiral structures. To verify the results obtained via the chain model, stationary scrolled packing of graphone nanoribbons C4 H (C4 F) of different lengths was also found using the full-atomic model. ˚ [D = 11.5 A] ˚ The graphone nanoribbon of width D = 11.4 A corresponding to a chain of N particles can be described by formula (C12 H3 )N/2−1 C18 [(C12 F3 )N/2−1 C18 ]; see Figs. 1(e), 1(f). As can be seen from Figs. 14 and 15 the two-dimensional chain model for graphone nanoribbons also gives a good coincidence of the number of coils Nc and the radii of the scroll R1 , R2 with the values obtained via the full-atomic model.

12

(a)

Nc

10

1

8 6

2

4

E/N (eV)

2 0

2

3

10

10

(b)

−0.03

3 4

−0.06

2

60

3

R1 , R2 (˚ A)

10

50

10

(c)

40

VIII. CONCLUSIONS

30

8 6

20 10

7 5 2

10

N

3

10

FIG. 15. Dependencies of (a) number of scroll coils Nc , (b) specific energy E/N , (c) inner R1 and outer R2 of single-scrolled (curves 1, 3, 5, and 6) and double-scrolled (curves 2, 4, 7, and 8) packings of fluorographone nanoribbon C4 F on the number of chain particles N . Markers give values obtained using the full-atomic model ˚ of nanoribbon (C12 F3 )N/2−1 C18 of width D = 11.5 A.

The dependencies of the number of coils Nc , the specific energy E/N, and inner R1 and outer radius R2 of singlescrolled and double-scrolled nanoribbon packings on their length (on the number of chain particles N ) for C4 H and C4 F are shown in Figs. 14 and 15. A feature of graphone scrolls is a decrease in the inner radius R1 with an increase in the nanoribbon length. The inner cavity decreases monotonically, and the number of coils and the outer diameter of the scroll increase monotonically with increasing nanoribbon length. For graphone C4 H specific energy E/N decreases mono˚ and increases monotonitonically at N < 361 (L < 450 A) cally at N > 361 for single-scrolled (double-scrolled) packing of the nanoribbon; see Fig. 14(b). For double-scrolled the specific energy decreases monotonically at N < 801 (L > ˚ and increases monotonically at N > 801. For fluo1000 A) rographone C4 F switching from decrease to increase in the

All-atomic modeling of long-nanoribbon dynamics requires considerable computing resources and to solve this problem we propose a simple 2D model of a molecular chain that allows us to describe folded and scrolled packages of nanoribbons. Here, we propose the most simplified 2D model of a two-dimensional chain, allowing us to describe the scroll conformations of graphene-like single-layer nanoribbons of graphene, graphane, fluorographene, fluorographane (graphene hydrogenated on one side and fluorinated on the other side), graphone C4 H (graphene partially hydrogenated on one side), and fluorographone C4 F. The Hamiltonian of the chain model (1) takes into account the longitudinal and flexural nanoribbon rigidness, as well as nonvalent interactions between the transverse layers of nanoribbon atoms. Using the model, possible stationary states of the scrolls were obtained. The dependencies of energy, the number of coils, and the inner and outer radii of the scrolled packing on the nanoribbon length were analyzed and it was shown that a scrolled packing is the most energetically favorable conformation for the nanoribbons depicted above. For the longer nanoribbons of graphone and fluorographone, a doublescrolled packing is more advantageous when the nanoribbon is symmetrically rolled into scrolls from opposite ends. The possibility of the existence of rolled packing for nanoribbons of fluorographene was shown and two different types of fluorographane rolls that in the chain model correspond to the left and right spirals of Archimedes were discovered. When constructing the model, the concrete structure of the edges of the nanoribbons is not taken into account. It is assumed that the nanoribbons are sufficiently wide and it is possible to normalize nanoribbon properties on its width. To verify the results obtained via the chain model we also used the full-atom model and found scrolled packings of graphene, graphane, fluorographene, fluorographane, graphone C4 H, and

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fluorographone C4 F nanoribbons of different lengths. The chain model allows finding the scrolled packings with good accuracy for these nanoribbons with a width of D  1 nm. The simplicity of the proposed model allows us to consider the dynamics of molecular nanoribbon scrolls of sufficiently large lengths and at sufficiently large time intervals. The model can be used for the description of scrolled packings of molecular nanoribbons with the same surfaces (nanoribbons of boron nitride, silicene, phosphorene, carbon nitride, etc.), of nanoribbons whose sides have different chemical modifi-

cation (nanoribbons of graphone and their analogs), and for describing the dynamics of multilayered nanoribbons.

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ACKNOWLEDGMENTS

This work is supported by the Russian Science Foundation under Grant No. 16-13-10302. The research was carried out using supercomputers at the Joint Supercomputer Center of the Russian Academy of Sciences (JSCC RAS).

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