MD calculation mechanical properties of fluorographene

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UDC 539. 3
MD CALCULATION MECHANICAL PROPERTIES OF FLUOROGRAPHENE
M. A. Mazo1, E.B. Gusarova1, T.P. Tovstik2, N.K. Balabaev3 1 Semenov Institute of Chemical Physics, RAS, Moscow, Russia
2Institute for Problems in Mechanical Engineering, RAS, Saint Petersburg, Russia 3Institute of Mathematical Problems of Biology, RAS, Pushchino, Russia mikhail. mazo 1 @gmail. com, tovstik_t@mail. ru
PACS 46. 25. Cc, 68. 35. Rh
We present the results for the elastic properties of a single layered carbon monofluoride or fluorographen (FG). The calculations were performed by molecular dynamics (MD) simulation using a force field with both bonded and non-bonded interatomic contributions, and the periodic boundary conditions in two dimensions, representing an infinite & quot-nanoplate"-. Simulations were fulfilled both for three basic conformations of FG [1] and for the FG with number counts of structural defects. The elastic modulus was calculated from the curves of force versus displacement obtained at slow rates of deformation. Bending stiffness was estimated independently from the nonlinear deformation under compression. The atomistic results are explained in terms of a continuum model for the thin plates.
Keywords: fluorographene. 1. Introduction
In the last years the great attention is paid to the investigations of properties of one layered membranes of graphen and similar materials. It is connected with their unique electric and heart-conducting properties. The last time the special interest to FG appears after the experimental and theoretical investigation of its electronic properties [2−5], which show that FG is very perspective material for using in microelectronic. In details the investigations of partial cases of FG was fulfilled in the work [2], were it was shown experimentally that FG has the relatively large Young'-s modulus ^ 100 N/m and the essential bending stiffness ^ 15 N/m. The way of the graphen membrane producing is studied in the work [6].
The quantum chemical calculation for two plane crystallic structures of FG was firstly fulfilled in 1993 in the work [7], where it was shown that the & quot-Chair"- configuration is energetically more favorable than the & quot-Boat"- configuration. Then by using the more occurred algorithm this calculations were fulfilled also in the works [1], [8−10]. In the work [9] the structure and electronic properties of graphen in derivatives involving Group IA or Group VIIA elements [9] is studied. The detailed computational study of structure of FG in three-dimensional stacked arrangements were fulfilled [1]. The structural and electronic properties of four crystallic structures of graphane and of graphene fluoride is investigated in [8]. In this work the Young'-s modulus and the Poisson'-s ratios are found, and the theoretical estimations of the membrane stiffness'-s occurs 2.5 times larger than the experimental ones.
The MD simulation in contrary to the quantum chemical calculation allows us to study the membranes FG with large dimensions and to calculated all their mechanical characteristics and also to study the influence of temperature and the structural defect properties. It is essential that in the frame of the molecular-dynamic experiments it is possible to analyze the more complex systems, for example the systems containing FG structures or composite materials, but for the
molecular simulation it is necessary to know reliable potentials (force field). Earlier to simulation the perfluoroalkanes from gas-phase and liquids. Some systems of force parameters are presented [11−13], but it is not clear that is it possible to use them for the MD simulation. In the work [1] Artyukhov and Chernozatonskii it was firstly derived the molecular mechanics force field for FG (CF)n that performs exceptionally well at reproducing our quantum chemical results and fits into a very general OPLS/AA molecular mechanics framework.
To verify the presented potentials and also to estimate the mechanical properties of various plane crystallic and defective structures of FG were fulfilled the MD experiments with the FG membranes by using the various potentials. In this work the potential OPLS-AA [11] is used. The unstressed membranes FG with temperatures 1,100, 300 and 500 K are studied and the results of their one-dimension deformations of temperature 300 K are analyzed.
2. FG Structures and force field
We consider three basic conformations of single-layered monofluoride crystals (Fig. 1. a, b, c) and the FG with the number random structural defects (adatoms (F) are positioned above or down the center of FG membrane).
Fig. 1. Different structures of FG. a) & quot-Chair"-, b) & quot-Boat"-, c) & quot-Zigzag"-, d) structure & quot-Chair"- with many random defects- the different colors represent respectively C atoms — gray, F atoms — black
The total potential energy of in-plane lattice FG comprises the bond stretching, the angle bending, the torsion energy, the electrostatic, and the Van-der-Waals potential energy
E bond (rij)
}_l, bond (_ 0 2
2ij (l v 1 ij) ,
Etorsioa (di, k^ = 1 [Fi (1 + cos^. fc/) + Fa (l _ COS 29ijki) +
+^3(1 + cos 3dijki) + F4(1 — cos 4dijki)], E n™b™d^® = J2E q (nj) + J2E VdW (),


Eq (r%3) =
QiQj 0r.
E vdW (r y)= ?ij

(Ril 9 (Rii V rv J ~ V Tij J
(1)
(2)
where the potentials (1)-(2) parameters are taken from [11] with corrections [1].
For the numerical solution of the Newtonian equations of motion, the fast Verlet algorithm [14] with an integration step of 1 fs is used. The temperature in the system is maintained by a
6
& quot-collision"- thermostat [15,16] with the parameters A = 5.1 ps 1 and m0 = 1 a.m. The pressure is set and maintained by a Berendsen barostat [17].
3. Results and discussion 3.1. Analysis of the energy
We calculate the contribution all of this potential energies and total potential energy of the crystals structures & quot-Chair"-, & quot-Boat"- and & quot-Zigzag"-. The & quot-Chair"- configuration is energetically more favourable. In Tables 1, 2 we present the relative energies with respect to & quot-Chair"- of temperature 0 K. In Table 1 through Ebond, Eangle, Etorsion, Eq, EvdW, Epot the bond stretching, the angle bending, the torsion energy, the electrostatic, the Van-der-Waals energy, and the total potential energy are denoted respectively.
In Table 2 we compare the presented energy with the energies obtained by the previous authors. We can see that for the & quot-Boat"- structure our energy is close to the quantum chemical approach. But for the & quot-Zigzag"- structure our energy is mach larger.
Table 1. Calculated relative energies (kcal/mol)
A?(& quot-Boat"--"-Chair"-) A?(& quot-Zigzag"--"-Chair"-)
j^jbond 0. 4253 0. 0369
j^angle 0. 6683 1. 0605
j^torsion -0. 5681 0. 3317
Eq 0. 0879 0. 195
j^vdW 0. 8983 0. 0724
Epot 1. 5118 1. 6965
Table 2. Analysis of the relative energies Epot (kcal/mol)
A?(& quot-B"--"-C"-) A?(& quot-Z"--"-C"-)
Presented study 1. 512 1. 697
O. Leenaerts, quantum chemical calculation 1. 73 0. 83
V. Artyukhov, quantum chemical calculation 1. 71 0. 82
V. Artyukhov, GROMACS 1. 691 0. 855
3.2. Equilibrium structure
The geometrical parameters in isolated layers for three configurations are presented in Tables 3−5. The crystal structure parameters a, b and the distances between the atoms (in X) and angles (in deg) are given.
3.3. Deformation and Temperature extension
We design the FG membrane as an orthotropic material with the various stiffness coefficients in directions X and Y. For calculation of these stiffness and of the corresponding
Mazo M. A., Gusarova E. B., Tovstik T. P., Balabaev N. K. Table 3. & quot-Chair"-, 0 K
a b a/{h/Ti) d d C F '-Pccc '-PcGF a CGC F
Presented study 4. 579 2. 644 1. 000 1. 590 1. 393 112.5 106.2 64. 2
V. Artyukhov 4. 521 2. 611 1. 000 1. 584 1. 382 110.9 108.0 61. 9
O. Leenaerts 4. 503 2. 600 1. 000 1. 579 1. 371 110.8 108.1 —
Table 4. & quot-Boat"-, 0 K
a b a/{h/Ti) d d G F '-¦Pccc '-PcGF a CGC F
Presented study 4. 642 2. 629 1. 019 1. 592 1. 635 1. 392 111.3 115.5 104.2 104.4 -113.3 67. 2
V. Artyukhov 4. 617 2. 585 1. 031 1. 574 1. 667 1. 377 114.0 104.0 107.4 -115.9 66. 9
O. Leenaerts 4. 603 2. 574 1. 032 — - - - -
Table 5. & quot-Zigzag"-, 0 K
a b a/{b/Ti) d ucc d C F '-Pccc P& gt-CCF a CCC F
Presented study 4. 257 2. 668 0. 921 1. 594 1. 588 1. 393 113.6 115.7 103.8 101.8 -68.1 158. 0
V. Artyukhov 4. 200 2. 635 0. 920 1. 567 1. 602 1. 382 115.6 103.6 106.2 -65.8 156. 6
O. Leenaerts 4. 183 2. 625 0. 920 — - - - -
Poisson'-s ratios we use the relations [18]
^13

E3 V23
033,
1 v12
en = - er 11 — - er 22
21, 1 e22 = -?r& lt-T 11 + TTi722 p
2 & amp-3
_ V31 U32 1
e33 — -c^*722 l^*733'-
2 ^3
1

-& lt-733,
613
e23
e12
G13 1
G23 1
^13,
^23,
(3)
G
-^12 •
12
Here E'-j, Gij, and ^ are the Young modules, the shear modules and Poisson'-s ratios respectively, and the following restrictions are valid
E1 U12 = E2 U21,
E1 V13 = E3 V31,
E223 = E3 U32 •
(4)
For estimation of the FG stiffness we fulfill the extension-compression simulation of FG membranes. The velocities of deformation are equal to 0. 002 and 0. 005 X/ps (or 0.2 and 0.5 m/s). Two problems are solved, namely the problem of extension-compression with 1D stress state in the direction X and in the direction Y. As in all 2D plate theories we put a33 = 0. Under there assumptions we get
en e22
& lt-7ll
E1'-
C22 E2
& amp-22 = - ^21
en

^12
& lt-7ll
Ex 0~22
E2
(^22 = 0) — (^11 = 0) —
(5)
(6)
Our algorithm allow us to measure and calculate only 2D stiffness'-s, and like this [8], [19] we introduce the & quot-2D Young'-s modules& quot- E by relations
Ei = Ei • He, E2 = E2 • HE2, (7)
where hEi, h, E2 are the effective thickness'-s of the extension-compression simulation of FG membranes.
The extension-compression simulation is fulfilled for three studied crystal configurations of FG. We compare our results with the published early theoretical results. Our results concerning the Young'-s modules are correlated very accurately with the quantum chemical calculation [8] (see Table 6). But there is the essential disagreement of the Poisson'-s ratios. Also we mark the essential influence the structural defects to the Young'-s modules.
It occurs that the experimental values [2] of the Young'-s modulus E = 100 ± 30 N/m are 2−2.5 times lower than the theoretical ones. To explain this difference we investigate the membrane with the large number of defects.
From 400 pairs of atoms of a crystal lattice (CF)m structure & quot-Chair"- we choose by a random way Ni pairs of atoms and for them we change the position of fluorine atoms into opposite ones with respect to the plane of membrane. The calculations are fulfilled for Ni = 8, 20, 40 and 100 defects. The obtained elastic modules for structures with defects better agree with the experimental results.
Fig. 2. The dependence between stress and extension for structures with defects
Our results shows (see Fig. 2) that the crystallic material is linearly elastic for the comparatively large deformations (until 2%), but for structures with defects are linear only for deformation 0.1 — 0. 2%.
Also we fulfill the simulation of FG membranes with the beam stress state and calculate the bending stiffness of membranes. The magnitudes of bending constants for FG membranes are estimated by using the Euler'-s formula of the critical compressive load for a thin beam. For the periodic boundary conditions the plate loses its stability at the second bending mode and the Euler formula in the X direction gives
42 h3
Ff=D1ly-jr, D1 = E1-^1 (8)
where D1 is the bending stiffness, and hm is the effective membrane thickness at the cylindrical bending, E1 is the Young'-s modulus and lx, ly are the cell dimensions in the X and Y directions. The compression simulation gives us the critical force F? r, and by first equation of (8)
we get Di = Di (Fccr). We assume that
hoi = hEi = hi, hD2 = hE2 = h2, (9)
and calculate the membrane effective thickness h1, h2 from (7), (8)
«=
By such way we estimate the bending stiffness'-s D1, D2 and effective thicknesses h1, h2, but in the case with the large number of defects (N = 20, 40, 100) this way is unacceptable because the critical point is not definite. In these case we suppose that hi are done and equal to the previous values of h1, h2 = 2. 6, 2.7 A, and estimate values of D1, D2 by relation (10).
To estimate the numerical error we fulfill calculations for various values Ix, Iy.
Also we fulfill calculations for various temperatures 1, 100, 300 and 500 K and calculate the coefficient of temperature extension (see Table 7). Calculations with the temperature 1 K are used for the comparison with the quantum chemical calculation at the temperature 0 K.
Table 6. Modulus E (N/m), Poisson'-s ratios v, bending stiffness D (N ¦ m) and
effective thickness h (A) of FG membrane for temperatures 1 K or 1 K / 300 K
Structure direction E ?[8] V v [8] D x 1020 h
& quot-Chair"- X 231 /236 243 0. 22 / 0. 22 0. 07 100 2. 3
Y 236 / 237 243 0. 22 / 0. 22 0. 07 120 2. 5
& quot-Boat"- X 342 / 336 230 0. 12/0. 12 -0. 01 185 2. 5
Y 234 / 236 262 0. 08 / 0. 08 -0. 01 260 3. 7
& quot-Zigzag- X 167/171 117 0. 11 /0. 12 0. 05 185 3. 6
Y 251 /252 271 0. 17/0. 17 0. 11 210 3. 2
random X 200/213 — 0. 17/0. 15 — 114 2. 6
Ni = 8 Y 220 / 222 — 0. 18/0. 18 — 134 2. 7
X 118 / 119 — 0. 08/0. 10 — 66* 2. 6*
Ni = 20 Y 177/ 178 — 0. 13/0. 15 — 108* 2. 7*
X 120/117 — 0. 09 / 0. 09 — 68* 2. 6*
Ni = 40 Y 177/176 — 0. 14/0. 13 — 108* 2. 7*
X 76/80 — 0. 11/0. 10 — 43* 2. 6*
Ni = 100 Y 141 / 137 — 0. 20 / 0. 22 — 86* 2. 7*
Table 7. The coefficients of temperature extension
& quot-Chair"- & quot-Boat"- & quot-Zigzag"- random defects
OLx X 106 4.3 3.0 7.6 1.9 — 6. 5
ay x 106 4.5 5.0 4.7 4.3 — 5. 2
In addition we get a number of calculations with the various parameters in the equations (1), (2). Our results show that the better coincidence with the quantum chemical results gives the force field OPLS/AA with correction [1], but the further corrections are desirable.
Acknowledgements
The research is supported by Division of Chemistry and Materials Technologies of RAS, grant of program N2- grant 11−02−1 453- and Russian Foundation of Basic Investigations, grant 09−01−92 002-HHC-a.
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