Molecular Dynamics Simulation in Modeling Carbon Nanotubes
R. Ansari and B. Motevalli
3. Algorithms of Performing an MD code
1
2/
,
( ) 1
1 ( )
2 1
cos( ) cos( )
cc e
cc cc jk cc
Br
S r R
jk cc cc
cc jk jkl jl
l j k
i cc
jki jki ij i j
ij jki
jki i ij ij
V r D S
f e G f r
x S
G f x x
f G
x r r
(2.25)
with the exerted forces are utilized to update the new positions and velocities. To obtain the microscopic properties of a system, the molecular dynamics simulation must be conducted in an ensemble [53, 54]. The common ensemble used to model carbon nanotubes is the canonical ensemble or constant-NVT. This ensemble represents a system that is in contact with a heat bath; therefore, the temperature of the system is held constant through the simulation. Subsequently, a point of importance in applying algorithms to solve the equations of motion of a system of particles, in the canonical ensemble, is the treatment of the temperature, which must be held constant in the process. Different thermostat techniques are utilized to hold the temperature of the system constant [55-58]. Since the temperature of the system of particles is computed from the total kinetic energy of the system, most thermostat techniques are based on scaling the velocities of the particles. In this book chapter, the Nose- Hoover heat bath technique is illustrated as an example [56, 57, 58]. In this technique, the velocities of the particles involved in the simulation are scaled by introducing a dynamical friction coefficient defined as follows
2
0 1
( ) 1 (3 1)
N
i i B
i
t m r N k T
Q
(3.3)Where,
is the dynamical frictional coefficient, Q is a parameterize parameter, and T0 is the reference temperature. As revealed from Eq. (3.3), the derivative of the dynamical friction coefficient is in relation with the difference between the current kinetic energy and the reference one. Subsequently, this parameter is introduced in the equations of motion asi i
( )
imr E t mr
(3.4)It can be seen from Eq (3.3) that if the current kinetic energy is higher than the reference kinetic energy corresponding to constant temperature T0,
becomes positive; therefore, the frictional force imposed in Eq. (3.4) tends to decelerate the motion of the atoms. Obviously, when the current kinetic energy is low,
becomes negative, and the corresponding frictional force tends to accelerate the motion of the atoms. To solve the equation of motion with a Nose-Hoover thermostat technique, a typical algorithm is applied to Eqs. (3.3) and (3.4) as below
2
( ) ( ) ( ) ( ) ( ) ( )
i i i
2
i ir t t r t t r t t r t t r t
(3.5a)
( ) ( ) ( ) ( ) ( )
2 2
i i i i
t t
r t r t r t t r t
(3.5b)2
0 1
( ) ( ) ( ) 3( 1)
2 2
N
i i B
i
t t
t t m r t N k T
Q
(3.5c)2
0 1
( ) ( ) ( ) 3( 1)
2 2 2
N
i i B
i
t t t
t t t m r t N k T
Q
(3.5d)( ) 2 ( ) ( )
2 ( ) 2 2
i i i
t t
r t t r t r t t
t t t
(3.5e)To demonstrate how to implement the above algorithm in simulation of a carbon nanotube, an example is illustrated in this section.
3.2. Implementation of the Nose-Hoover Algorithm in Simulation of Carbon Nanotubes
In this section, the way of handling a simulation code utilizing the Nose-Hoover algorithms of Eq. (3.5) is illustrated in more detail. Obviously, the general framework can be utilized in using other thermostat techniques. As an example, consider a carbon nanotube of (n, m), which is under axial loading. Under this loading, the Young modulus property of the CNT can be obtained, and its deformation can be studied. To perform this simulation, the following steps must be taken:
Step1. Generate the initial positions of the atoms of the carbon nanotube (n, m), and save them in the vectors X, Y, and Z. This is utilized by the concept of geometry of a carbon nanotube defined by (n, m) [59]. Utilize these positions to compute the initial acceleration vectors ax, ay, and az based on the Brenner potential energy function of Eqs.
(2.17)-(2.25). It must be mentioned that each component of these vectors corresponds to a particular atom. Moreover, distinguish the boundary atoms (e.g., two rings of atoms from both ends). Note that to simulate the axial loading, one end of the nanotube is held permanent, while the other end is displaced incrementally after each relaxation step.
Step2. Generate the initial velocity of the atoms by a random distribution and save them in the vectors Vx, Vy, and VZ. Moreover, compute the initial kinetic energy based on these velocities.
Step3. Determine the time step t, the increment axial displacement
L, an initial value for the dynamical friction coefficient
, and the reference temperature T0 (e.g.,1
t ft
,
L0.001L, 0
, and T0 300). Furthermore, define the parameter Q. A particular parameterization of Q is as
2
(3 1) B 0
Q N k T
(3.6)
where
is in the dimension of time (e.g. 10 t
).Step4. Determine the number of steps for initial relaxation K0 and relaxation after imposing each increment displacement K.
Step5. Let the nanotube be relaxed initially by K0 steps by the subroutine, which is based on the Nose-Hoover algorithm of Eqs. (3.5) as follows,
Subroutine for Nose-Hoover relaxation:
Step1. Compute the dynamical friction coefficient
in time t2t from Eq (3.5c).Step2. Compute the positions and velocities of the N atoms in tt and 2 tt
, from Eqs. (3.5a) and (3.5b), respectively. Moreover, compute the sum square of the velocities
in
2
t t
inside the loop of this step.
Step3. In the loop of Step2, update the positions of all atoms to the current position vectors else for the boundary ones.
Step4. Compute the kinetic energy in
2 t t
, then compute
in time t
t fromEq. (3.5d). Moreover, obtain the current acceleration vectors based on the updated position vectors.
Step5. Compute the velocities of the atoms in time t
t from Eq. (3.5e). Note that the acceleration in t
t is actually the current acceleration, which is updated in preceding step. Simultaneously, compute the sum square of the velocities, and update the velocities to the current velocities of the atoms.Step6. Compute the current kinetic energy based on the updated velocities. Then computations begin again from Step2, until the loop of relaxation is finished.
Step 5. Displace the axial coordinates of one end of the boundary atoms in amount of
L. Then, let the nanotube be relaxed for K steps by the subroutine presented above.Step 6. Compute the resultant axial force on the boundary atom, and save it in a vector. Additionally, compute the potential energy and the total mechanical energy, and save them also.
Step 7. Continue the simulation until the final axial strain is obtained.
Note that at the end of the simulation, the corresponding axial force to each axial strain is obtained. Therefore, the stress-strain curve can be depicted in order to investigate the properties of the CNTs. A similar procedure can be applied in order to impose torsional and bending loading. Fig 3. 1. represents the deformation of a (10, 10) carbon nanotube under torsional loading as an example of the illustrated algorithm in this section. The tube whose length is 9.72 nm contains 1,600 atoms (80 rings). Three rings from each end are held constant, while one of them is rotated with increment twisting angle of 1°. The time step is 1 fs, and the simulation is conducted at the constant temperature of 300 K. According to the conducted simulation, this instant armchair carbon nanotube buckles when it attains the twisting angle of 72°. Since this book chapter aims to note the important issues and the way of implementation of an MD simulation code for carbon nanotubes, presenting numerical results for investigation of the properties of these structures are out of its scope. One can conduct the procedure presented in this section in order to investigate these properties on his own. However, an extensive study on carbon nanotubes properties under different loadings is available in the literature, based on MD simulations (e.g., Refs. [1-27]). Fig 3.2. demonstrates the torsional buckling of a double-walled carbon nanotube [5]. To simulate multiwalled carbon nanotubes, a potential function such as Lennard-Jones potential must be utilized to model the vdW interaction force between the walls of the tubes, while each tube is treated by a suitable interatomic potential like the Brenner potential, individually. Therefore, in the presented procedure, only in the steps of computing the accelerations for each atom, the acceleration caused by the vdW interaction force must be added also.
To conduct a successive MD code, some important factors must be considered. These factors are listed in summary below
1) Choosing an appropriate potential energy function to model accurately the bonding in the system under investigation.
2) Choosing an appropriate thermostatic technique and algorithm scheme that can simulate the procedure correctly.
3) Choosing appropriate time steps, number of relaxation steps, and amount of displacement to the boundary atoms in each step. Obviously, smaller steps and displacements and higher number of relaxation steps result in a more accurate simulation, but, on the other hand, the time expense increases. Thus, in choosing these parameters, a balance must be taken between accuracy and the speed of the simulation.
a) b)
Figure 3.1. Deformation of a (10,10) carbon nanotube under torsional loading, a) at 100 twisting angle, b) at 120 twisting angle.
Figure 3.2. Deformation of a DWCNT under torsional loading, taken from Ref. [5].
Mylvaganam and Zhang [11] have discussed the above factors in their research article.
They utilized two different potential functions of Tersoff and Tersoff-Brenner to model the binding between the atoms. Moreover, they applied different thermostat techniques and algorithm schemes in order to find the best for modeling of carbon nanotube structures. Fig 3.
3. displays the results taken from these different schemes [11]. As revealed from this Figure, in early stages of simulation, different schemes have almost similar results, but in higher stages, they follow quite different paths. These Figures demonstrate how critical the choice of the scheme utilized in MD simulation is when large deformations are examined. It is observed from Fig 3. 3b. that only two schemes could correctly predict the necking of the CNT and the formation of chain atoms in large strains, while the nanotube is suddenly separated into two parts after a certain strain in other schemes.
In the end, the procedure presented in this book chapter can be used in order to investigate different computational experiments on carbon nanotubes via MD simulations, such as:
1) Axial tension and compression of single-walled and multiwalled CNTs. The elastic properties, axial buckling, and large deformations can be investigated [1, 2, 4, 6-12, 14, 16, 17, 19-21, 23, 25, 26].
2) Imposing torsion and bending moment and studying buckling [1, 5, 12, 13, 22, 25].
3) Imposing different external pressure and examining the deformation of CNT [12].
4) Imposing the combination of foregoing loadings simultaneously.
5) Imposing the above-mentioned loadings to CNTs containing defects [3] and CNTs filled by other molecules such as fullerene.
6) Imposing axial tensile and compressive loads to bundle of CNTs [18].
7) Investigation of the thermal effects on the properties of CNTs [12, 15, 24], and so on.
Figure 3. 3. The stress–strain curves of (a) a (10,10) armchair SWNT and (b) a (17,0) zigzag SWNT using Tersoff and Tersoff–Brenner potentials Ref. [11].
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