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School of Metallurgical and Materials Engineering, College of Engineering, University of Tehran, Tehran, Iran

Email: movahedirad@ut.ac.ir

Copyright © 2014 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Received 7 March 2014; revised 5 April 2014; accepted 1 May 2014

ABSTRACT

Grain boundary energy is very important in determining properties of ultra fine grain and nano structure materials. Molecular dynamics were used to simulate grain boundary energy at different misorientations for Al, Cu and Ni elements. Obtained results indicated well compatibility with theoretic predictions. It was obtained that higher cohesive energy results in higher grain boundary energy and depth of CSLs. In this manner, Ni had the highest and Al had the lowest cohesive energy and grain boundary energy. Also, a linear correlation was obtained between GBE of elements, which was related to relative cohesive energy.

Keywords:Molecular Dynamic, Cohesive Energy, Grain Boundary Energy, Misorientation Angle

1. Introduction

Grain boundary (GB) is one of the planar defects in microstructure of crystalline materials which can be considered as one of the most important parameters which affect properties of ultra fine grain (UFG) and nano structure (NS) materials because of the increased area fraction of GBs. Grain boundary energy (GBE) can be so critical in the field of stability of grains and deformation mechanisms of UFG and NS materials.

 

There are many reports in literature that have studied different properties of GBs experimentally from different aspects [1] [2] . However, nowadays computational simulation provides an opportunity to improve our understanding about crystallographic defects more easily in addition to experimental attempts. In this regards and in the field of GBE, Uehara et al. [3] carried out atomistic simulation and studied GBE both before and after relaxation and stability of GBs in Al. Holm et al. [4] showed strong linear correlation between GBEs of different materials by the means of molecular dynamic (MD) simulation.

Dependence of the GBE on the composition ratio in the bcc iron-chromium alloy was investigated by classical molecular dynamics simulation [5] . In another investigation, a phase-field model was used to investigate the simultaneous effects of grain boundary energy anisotropy and the presence of second-phase particles on grain growth in polycrystalline materials [6] . Also, some researchers have simulated the evolution of grain boundaries at elevated temperatures [3] [7] .

As mentioned above, it can be seen that computational simulations have been successfully employed for GB studies in recent years. Nevertheless, to the best of authors’ knowledge, little work has been done until yet to study the GB energy as a function of misorientation angle for different FCC metals, comprehensively. So, the aim of this paper is to use MD simulation with LAMMPS, as the advanced computer program for modeling defects in materials, to investigate GBE in Al, Cu and Ni elements. LAMMPS is a classical molecular dynamic simulation code designed to run efficiently on parallel computers.

2. Calculation Method

2.1. Embedded Atom Method (EAM)

In this method, each individual atom is considered as an embedded particle in a host aggregate composed of other atoms and it is assumed that the embedding energy of each atom is a function of the electron density of the host before the atom is added [8] .

It can express an equation for the total energy of the system which contains N atoms to solve practical problems in metals with many-body bonding properties [9] [10] . The total energy of the system that is composed of N atoms is given by the following equation:

(1)

where E_i is the energy of atom i and E_total is the total energy.

The EAM potential is made up of two components. First, embedding energy as a function of local electron density and the second, pair potential between atoms i and j.

(2)

where F is the embedding energy which is a function of the atomic electron density, φ is a pair potential interaction, and α and β are the element types of atoms i and j. ρ_α is the electron density at site i. The multi-body nature of the EAM potential is a result of the embedding energy term [10] .

2.2. Calculation of Cohesive Energy in Solids

Cohesive energy in solids is defined as the required amount of energy in order to break the atoms into isolated atomic species. The cohesive energy is calculated by using following equation:

(3)

where i indicates the atoms which constitute the solid.

2.3. Definition of the Grain Boundary Energy

In order to calculate the grain boundary energy, following equation is defined:

(4)

where E₀ is the bulk energy, S_gb is grain boundary area, and the summation of all atoms in the domain V gives the grain boundary energy (E_gb). Domain V is chosen around the GB, excluding the surfaces of the model. According to the Equation (4), in the bulk region the value of E_gb is zero. It is because of the fact that energy of the atom in the bulk is equal to E₀. However, in the boundary area the value of (E_i − E₀) would be non-zero.

2.4. System Characteristics

For having a sufficiently large system, the area of every region is 200 unit cells in x direction, 200 unit cells in y direction, and 1 unit cell in z direction and total number of atoms is 160,000. Also, periodic boundary condition is applied in three directions.

Symmetrical tilt GBs were created by arranging atoms in upper and lower groups in x-direction and rotating them in θ/2 and −θ/2, respectively, where θ was given as the misorientation angle. Simulation was done every 5˚ from 0˚ till 90˚ and also for five specific angles, 22.6˚, 28.1˚, 36.9˚, 53.1˚, and 61.9˚ related to the Coincidence-Site Lattice (CSL) boundaries. In addition, since some atoms may be too close to each other due to the rotation in the GBs, a threshold distance was chosen to delete one of the overlapped atoms. So, Delete_atoms command was used in freely available LAMMPS code to delete created atoms in the GBs.

In order to understand the effect of types of materials on the GB energy, Al, Cu, and Ni elements with FCC crystal structure were selected. In this study, Mishin Al EAM [11] , Mishin Ni EAM [11] , and Mishin Cu EAM1 [12] , and are parameterized to represent Al, Cu, and Ni, respectively. In addition, so as to avoid activating any thermal phenomena, all simulations were carried out at 0.0 K.

2.5. GB Relaxation

For having reliable results, it is essential to reach an equilibrium state. It is done via performing minimize command by iteratively adjusting atom coordinates in two stages. The initial relaxation was applied to the system in order to release it from internal stresses. Then, additional relaxation was programmed to remove hydrostatic pressure of atoms gradually and achieving full relaxation state. In this stage, the GB energies were calculated and the configurations were obtained for all atomistic specimens.

3. Results and Discussion

The variation of the potential energy with time step of the investigated elements is shown inngcc Figure 1 for selected misorientation of 53.1˚. Similar results were obtained for other misorientations. As can be seen in this figure, the potential energy reaches to a steady state shortly after start. This behavior shows that atoms experience a relaxation state in which they are displaced from their initial positions and try to find more energetically favorable positions. It should be stated here that all potential energies in this paper were calculated after reaching to steady state.

To have a better understanding about the movement of atoms during relaxation state, atomic structure of the material near the grain boundaries at different misorientations is shown in Figure 2 for both before and after relaxation of atoms. It can be observed from Figure 2 that not only the atoms which are on the grain boundaries, but also the atoms adjacent to the grain boundaries also experience some relaxation. These can be related to steps 1 and 2 of relaxation, respectively, as described in Part 2.5. Another important feature in this figure is that atoms tend to decrease empty region at the grain boundaries by relaxation. This behavior seems reasonable since the grain boundary energy is a function of empty region at the boundary and thus systems have a tendency to change atomic order to decrease energy by decreasing empty regions. It is also evident that dislocations exist at some specific intervals along LAGBs (Figure 2(a)).

To calculate grain boundary energy according to Equation (4), firstly, it was necessary to calculate cohesive energies of the elements. The calculated cohesive energies of Al, Ni and Cu are given in Table1 These values are in agreement with previous reports for these elements [4] and were used for further calculations.

The calculated grain boundary energy as a function of misorientation angle is shown in Figure 3 for Al, Cu and Ni. The rotation misorientations were measured against axis, which is a four bold symmetrical axis, and thus the calculations were made between 0˚ - 90˚. It can be seen that at all misorientations, grain boundary energy of Ni is higher than Cu and Cu is higher than Al. By considering the fact that GBE calculation is based on both the number and energy of broken atomic bonds, it can be concluded that GBE is dependent directly on the cohesive energy. Consequently, under similar conditions where two elements with similar crystallographic structure have equal number of broken bonds, GBE of the element with higher cohesive energy would be higher. Accordingly, the GBE of the investigated elements can be rationalized according to their cohesive energy data, given in Table1 As a result, Ni has the highest GBE because its cohesive energy is the highest and Al has the lowest GBE because its cohesive energy is the lowest one in the investigated elements.