Atomistic Simulation Method

0.3truein Molecular statics and molecular dynamics are widely used in performing atomistic simulations. Molecular statics is used in determining the equilibrium positions of atoms in a crystal, by minimizing the total energy of the crystal at 0 K. The interatomic potential which prescribes the interactions between atoms is a necessary input for the simulation. An accurate and relatively simple model of the energetics is required so that a sufficiently large set of atoms can be considered in the computational crystal. Pair potentials were generally used for modeling grain boundaries in earlier works [e.g. 4, 5, 6]. The main limitation of the pair potential models is that they fail to take into account the metallic bonds, i.e. coordinate-dependent or many body interactions. In contrast, EAM interatomic potentials include in an implicit way the many-body effects, and have been proven to be more reliable in representing atomic interactions in metallic systems [7, 8]. EAM accounts for many-body effects, yet does not require significantly more computational effort than the simple pair potential models. The basic equations of the EAM are:
 

 

 
where E_tot is the total energy of an assembly of atoms, E_i is the internal energy associated with atom i, is the electron density at atom i due to the rest of the atoms in the system, is the energy to embed atom i into the electron density , is the two-body pair potential between atom i and atom j separated by the distance r_ij, and f(r_ij) is the contribution to the electron density at atom i due to atom j at the distance r_ij from atom i.

In order to apply the EAM, functions F, f and must be given. Oh and Johnson [9] have presented an analytical EAM model for close-packed metals. In their model, these functions are given by:
 

 

 
where
 

 

 

 
Here r_e is the equilibrium interatomic separation distance, r_c is the cutoff distance, f_c(r) and are cutoff functions to ensure f(r_c)=f'(r_c)=0 and , which provides a smooth cutoff at the cutoff distance r_c. , , , f_e, , a, b, and n are EAM parameters to be determined by fitting above EAM functions to materials properties (cohesive energy, elastic constants, and vacancy formation energy etc.). For aluminum these constants are: , , , , a=-4.8144, b=0.47685 and n=0.39948. The parameter f_e need not be specified since only ratios of occur in equation (4).

Figure 1 shows the f(r) and of aluminum as a function of the separation distance r between two atoms. Both of the functions smoothly reach to zero at cutoff distance (). Also shown in Figure 1 is the total energy of an atom as a function of its nearest neighbor distance r in a perfect crystal calculated from equation (2). The second and higher order neighbor distances and the number of neighbors at each r is determined according to the FCC arrangement of aluminum crystal. Neighbors beyond the cutoff distance (r_c) are not included in the model. It can be seen from Figure 1 that the equilibrium structure of a perfect Al crystal is obtained at r=2.86 (or lattice parameter 4.05 , which exactly matches the measured lattice parameter). As seen from the figure an interatomic distance of r in either direction of r_e (r<r_e or r>r_e) would result in a higher energy state. In order to obtain the equilibrium structure and compute the energy, the EAM model described above was incorporated in a DYNAMO program developed at the Sandia National Laboratory Livermore in which the conjugate gradient method is used to minimize the energy.