Empirical Potentials

Molecular Dynamics Simulation in Modeling Carbon Nanotubes

R. Ansari and B. Motevalli

2. Empirical Potentials

2.1. General Structure of Potential Function

Although quantum mechanics models are the most precise ones in modeling materials at atomistic scales, they can only handle very small nanoscale systems of about a few hundred atoms. Due to this fact, a more efficient way of describing the energetic of nanostructures is utilizing interatomic potential energy functions. These potentials are only a function of relative positions of the atoms, and through them, the total interaction energy of the whole atomic system is evaluated. Obviously, the more accurate these potentials are and the better they describe the bonding and energetic of the atomistic structure, the more reliable are the results obtained and the nearer is the model to reality. Indeed, there is not a specific potential function that is the best one for modeling all systems of particles. The appropriate potential function that can model all the bonding and dynamics of the nanostructure to the best possibility may vary with respect to the case of study and the system of particles under investigation. Therefore, the first step for implementing a successful MD code is to choose an appropriate potential energy function for the case study.

For a system of particles, a general structure of the total potential energy can be expressed by a potential function, which is in terms of the position vectors of the constituent particles, as follows:

1 2 1 2 3

( , ,..., _N) ( )_i ( , )_{i j} ( , , ) ..._{i j k}

i i j i i j i k j

V r r r v r v r r v r r r

  















(2.1)

Where r_i is the position vector of particle i, and v_n is the n-body interatomic potential function. The first term in Eq. (2.1) considers the effect of an external field; the second term includes the pair-wise interaction between particles i and j; and the third term computes for the three-body potential containing particles i, j, and k. Higher order interactions can be included in the potential energy function of Eq. (2.1), but in many cases of study, these terms have an insignificant effect. Additionally, these terms increase the computational time of simulation extensively.

2.2.1. Pair-Wise Potentials

There are number of potential functions to model the interaction between pairs of atoms or molecules efficiently. These potential energy functions are usually composed of two terms of attraction and repulsion. In long distances, the attraction term is dominant and the pair of particles tends to each other, while in short distances the repulsion term prevails and the pair of particles tends to repel. Actually, the pair-wise potentials that are applied for the case of carbon nanotubes are for the evaluation of the vdW interaction potential energies between the walls of multiwalled carbon nanotubes. Here, the two most popular pair-wise potentials, which are utilized extensively in the literature, are introduced. The first one is known as the Lennard-Jones (LJ) potential, which is used in the simulation and modeling of various nanoscale system of particles. This function is given as follows:

12 6

( ) 4

LJ ij

ij ij

V r

r r

 



^{    }

         

(2.2)

The first term in Eq. (2.2) stands for the repulsion effect, while the second one is the attraction partition. Note that the pair-wise interaction potentials depend only on the magnitude of the separation distance between particles i and j (i.e. r_ij  r_i r_j ). The constant parameters in Eq. (2.2) are the collision diameter  and the dislocation energy . These parameters depend upon the system of particles to be simulated. Subsequently, the corresponding LJ interaction force between a pair of particles is computed as

12 6

13 7

( ) 24 2 ^ij

i ij i LJ

ij ij ij

F r V r

r r r

 



^{ }

      ^(2.3)

Another commonly used potential function is the Morse pair-wise interaction potential, which is given by



^{2 ( ) ( )}



( )

^rij

2

^rij

M ij

V r   e

^{ }

 e

^{ } (2.4)

Similarly, the first term stands for the repulsion part, while the second term is the attraction one. The corresponding interaction force due to the Morse potential is obtained from



^{2 ( ) ( )}



( ) 2 ^{rij rij ij}

M ij

ij

F r e e r

r

   



^{ }

  (2.5)

Ref. [30] have applied different pair-wise potential functions for the systems of double-walled carbon nanotubes oscillators and have studied the behavior of such systems under these potentials. Based on their results, they concluded that applying different potentials may lead to a total different conclusion on the behaviors of carbon nanotube oscillators. Thus, in respect to the type of the system to be modeled, different pair-wise potentials may cause quite different results.

2.2.4. Many Body Potential Functions

As mentioned, the pair-wise potentials introduced in the previous subsection compute the vdW potential energy between two distinct carbon nanotubes. To accurately evaluate the interatomic potential energy within the atomistic structure of a carbon nanotube, an appropriate potential energy function must be applied. Herein, the most popular potential energy functions that are appropriate to model atomistic interaction within the structure of a carbon nanotube are presented.

One famous potential function is the Tersoff‘s potential, which was primarily suggested to model the bonding in Si [47, 48]. Actually, this potential function is a pair-wise potential, which takes the local atomistic environment into account. The total potential energy is given by

Tr( )

ij i j i

E V r







^(2.6)

where,

V

^Tr

( ) r

_ij is the Tersoff potential energy function between atom i and its nearest neighbor atoms j, which is computed by

( ) ( ) ( ) ( )

Tr

ij c ij R ij ij A ij

V r  f r V r B V r  ^(2.7)

The functions V_R and V_A represents the repulsive and attractive pair-wise interactions, and are given by

( ) ( )

R ij

A ij

r

R ij CC

r

A ij CC

V r A e V r B e











 ^(2.8)

The local atomic environment is imposed through the B_ij coefficient in the potential energy function of Eq. (2.7). This coefficient is evaluated as follows

 

3 3

1 2

( )

,

2 2

2

2 2

(1 ) ,

( ) ( )

( ) 1

cos

A ij ik

n n n

ij

r r

C ij ijk

k i j

ijk

ijk

B

f r e g

c c

g d d h



 

 

 



  

 



 



  

 



^(2.9)

Table 2.1. Parameters in the Tersoff potential

ACC

BCC



A





n c d h 1

RCC 2

RCC

1.3936 10 3eV 3.4674 10 2eV

3.4879 Ǻ^-1 2.2119 Ǻ^-1

1.5724 10 

^7

7.2751 10 

^1

3.8049 10 

4 4.3484

0.57058

 1.8 Ǻ

2.1 Ǻ

The constant parameters in Eq. (2.9) are given in Table 2. 1 for the case of carbon-carbon interaction.



_ijk is the angle between bonds i-j and i-k, andf_C is a cutoff function, which considers only the nearest neighbors of atom i in the calculation of its potential energy. This function is given by

1

1

1 2

2 1

2

1,

( )

1 1

( ) cos ,

2 2 0,

ij cc

ij cc

c ij cc ij cc

cc cc

ij cc

r R r R

f r R r R

R R

r R



 

   

      

 



(2.10)

Based on Tersoff potential, Brenner [49] proposed a potential that overcomes some deficiencies of the Tersoff potential and can model more precisely the bonding between the carbon atoms. This potential is as follows

( ) ( ) ( ) ( )

Br

ij c ij R ij ij A ij

V r  f r V r B V r  ^(2.11)

Similarly, V_R and V_A stand for the repulsive and attractive terms of the potential function of Eq. (2.11), respectively. These functions are expressed as

 

 

2

2/

( ) ,

1

( ) ,

1

e

cc cc ij cc

e

cc cc ij cc

S r R

cc R ij

cc

S r R

cc cc A ij

cc

V r D e

S

V r D S e

S





 

 

 

 



(2.12)

Likewise, f_C is the cut-off function and its expression is the same as Eq. (2.10). Moreover, the coefficient

B

_ij

, which includes the local atomistic structure, is given by

2

ij ji

ij

B B

B 



(2.13)

,

1 ( ) ( )

cc

ij c ik ijk

k i j

B f r G









 

   

  

^(2.14)

 

2 2

0 0

0 2 2 2

0 0

( ) 1

1 cos

ijk

ijk

c c

G a

d d

 

 

 

  

   

 

(2.15)

The parameters in Eqs. (2,12) to (2.15) are given in Table 2.2. The Brenner potential has been used thoroughly in modeling of carbon nanotubes, and successful results have been obtained for various types of problems such as axial, torsional, and bending loading of these structures.

Note that the principles of handling an MD code are similar for other types of potential functions also. Therefore, an MD code based on a specific potential energy function can be easily adapted to the other types of potentials. Since the Brenner potential function is applied successfully in modeling of carbon nanotubes, this potential is used in this book chapter in the loops of the MD code. Other potential functions can be also found in the literature (e.g., see Refs. [50, 51]).

The computation of the total force exerted on each individual atom is the most expensive and important part of a molecular dynamics simulation. The expression of this force is

obtained by taking the gradient of the potential energy with respect to the position of an instant atom i. The computation of this gradient for the special case of Brenner potential energy function is presented as follows

1

ˆ ˆ ˆ

^Br

( )

i i ij

i j i

i i i

F E i j k V r

x y z

_{ }

    

             

^(2.16)

Table 2.2. Parameters in the Brenner first generation potential function

Parameter 1^st parameterization 2^nd parameterization e

RCC ^{1.315 Ǻ 1.39 Ǻ}

DCC 6.325 eV 6 eV

βCC 1.5 (1/ Ǻ) 2.1 (1/ Ǻ)

SCC 1.29 1.22

δCC 0.80469 0.5

a0 0.011304 0.00020813

c0 19 330

d0 2.5 3.5

1

RCC ^{1.7 Ǻ 1.7 Ǻ}

1

RCC ^{2 Ǻ 2 Ǻ}

Eq. (2.16) can be written in three scalar equations for each component of the interatomic force. As an example, the computation of the x component is presented; obviously, other components are obtained, similarly. The x component is obtained as

,

( ) ( )

x

Br Br

ij jk

i

j i i j i k i j i

V r V r

F  x   x

 

  

 

  

^(2.17)

The second term on the right hand side of Eq. (2.17) is the contribution, which is associated with

B

_jk. Utilizing Eqs. (2.11) to (2.15), one can obtain

( ) ( )

( ) ( ) .

( ) ( )

( ) . ( )

Br

ij c ij i j

R ij ij A ij

i ij ij

R ij A ij i j ij

c ij ij A ij

ij ij ij i

V r f r x x

V r B V r

x r r

V r V r x x B

f r B V r

r r r x

  

 

   

 

     

        

(2.18)

The derivatives of each term in Eq. (2.18) are obtained as follows

2 ( )

( ) 2

1

cc cc ij cc

S r R

R ij cc cc cc

ij cc

V r D S

r S e



_  _

  

 

(2.19a)

 

( ) 2 /

2/

1

e

cc cc ij cc

S r R

A ij cc cc cc cc

ij cc

V r D S S

r S e



  

 

 

(2.19b)

1 2

ij ij ji

i i i

B B B

x x x

    

   

    

_(2.19c)

1

, ,

( ) cos( )

1 ( ) . . ( )

cos( )

cc

ij ijk ijk ik i k

cc ijk ik ik ijk

k i j k i j

i ijk i ik ik

B G f x x

G f f G

x x r r

  

  



 

 

  

    



 



      

(2.19c)

1

, ,

( ) cos( )

1 ( ) . .

cos( )

cc

ji jik jik

cc jik jk jk

k i j k i j

i jik i

B G

G f f

x x

  

 



 

 

  

    



 



    

(2.19d)

As mentioned,



_ijk is the angle between bonds i-j and i-k; therefore,

2 2 2

cos( )

2

ij ik ij ik jk

ijk

ij ik ij ik

r r r r r

r r r r

  ^  ^{ }

(2.20)

2 2 2 2 2 2

2 2

cos( )

2 2

ijk ij ik jk i j ik ij jk i k

i ij ik ij ij ik ik

r r r x x r r r x x

x r r r r r r



      

   



(2.21)

Similarly,

2 2 2

2

cos( ) 2

jik ij jk ik i j ik i k

i ij jk ij ij ik ik

r r r x x r x x

x r r r r r r



    

   



(2.22)

Before introducing the second term in Eq. (2.17), a description in how to compute the series in ^ij

i

B x





^and

ji i

B x





is presented. As mentioned, these potential functions evaluate the interaction of an atom with its nearest neighbors. Fig 2. 1 represents the local structure of atoms within a carbon nanotube.

Atoms j in all series are the first neighbors of atom i, and the bond i-j is the bond between these atoms. Since the function Gijk includes triplet atoms where the atom i is the center, the atoms k in series of Bij and ^ij

i

B x





are also the first neighbors where when j specifies one of the neighbors, k is the other neighbors of atom i. These atoms are identified by k1 in Fig 2. 1. In

function Gjik, which appears in Bji and ^ji

i

B x





, one of the first neighbors (atom j) is the center atom in the triplets of jik; therefore, in these triplets, atoms k are the first neighbors of atom j, which means they are in the second neighborhood of atom i. These atoms are represented by atoms k2 in Fig 2. 1.

Figure 2.1. Adjacent atoms of an instant atom i within a carbon nanotube structure.

Note that in the total potential energy of atoms j (the first neighbors of atom i), the coefficient Bjk (k i) includes bond i-j through

G ( 

_jkl

) |

^{l j k}_{k i j}^_^,_, , which must be taken into account in the computation of the force exerted on atom i, since this coefficient has derivative with respect to xi. The second term in Eq. (2.17) is the contribution on the atom i by the potential energy of atom j. This term is extracted as follows

 

( ) 2/

1

e

cc cc jk cc

Br

S r R

jk cc cc jk

jk

i cc i

V r D S B

x S f e x



 

  

   _(2.23)

, ,

1 ( ) ( )

cc

jk c jl jkl

l j k k i j

B f r G







 

 

   

  

(2.24)

In the expression of Bjk, k is the neighbor of j, which is not atom i (i.e., k is the second neighbor of atom i), while l is a neighbor of j, which is not atom k. Obviously, one of the neighbor atoms l can be atom i. Consequently, the terms including atoms i have derivatives with respect to xi.

Finally, the last term is obtained as follows

  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)