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Embedded Atom Method (EAM) Simulation in the MedeA® Environment
This application note provides an overview of the use of EAM based forcefield simulations in the MedeA®environment using the MedeA LAMMPS interface. The emphasis is on the background to EAM based simulation and the properties which may be obtained using the method.
Embedded Atom Method (EAM) forcefield based simulations provide a computationally efficient description of structural, mechanical, and thermal properties of metallic systems. Within the EAM description, the evaluation of energy and forces, even for large assemblies of atoms, is several orders of magnitude faster than comparable first-principles calculations and this performance scales linearly with the number of atoms. Hence EAM calculations are able to span length and time scales inaccessible to first-principles methods. This application note provides an overview of EAM forcefields 1, their use within the MedeA®environment, and a brief illustration of the properties which EAM forcefields can provide.
Embedded Atom Method Forcefields
As its name implies, the embedded atom method accounts for the behavior of an atom placed in a defined electron density. The method therefore captures a significant portion of the physical reality of metallic bonding. Related to the effective medium theory of Nørskov and Lang2, the embedded atom method (EAM) was developed by Daw and Baskes3. This approach represents the total energy of the system as two additive terms, a pairwise sum of interactions between atoms, and a term representing the electron density of each atomic site, as shown in Equation 1 below.
Umetallic is the total energy of the system, i and j indicate the unique pairs of atoms within the N atoms of the system, rij is their interatomic separation, V(rij) is a pairwise potential, and F(ρi) is the embedding function for atom i which depends on the electron density, ρi, experienced by that atom.
To evaluate a given atom’s embedding function, one needs to compute the electron density at the position of atom i. This is obtained by a superposition of “atomic densities”, which are described by a density function, φj(r), as shown in Equation 2.
Figure 1. A typical EAM embedding function, F(ρ) (illustrated using the Zr EAM forcefield of Mendelev and Ackland4).
The embedding function, F(ρ), provides an essential degree of freedom in the description of metallic bonding. If this term were linear with respect to varying density, the overall energetic description would be equivalent to a standard two body representation. However, the curvature of the embedding term with varying electron density provides an account of the effects of many body interactions. A common form of the embedding function for an EAM forcefield is shown in Figure 1. Here increasing electron density yields progressively more negative embedding energies, until a minimum value is attained beyond which increasing electron density yields less favorable system energies. Figures 2 and 3 provide views of the density function, employed to compute the electron density at a given site (Figure 2), and the interaction function (Figure 3) which is reminiscent of a typical two-body interaction function.
Computing energies and forces based on Equations 1 and 2 can be achieved rapidly as each of the terms are functions of interatomic separation and such separations and their derivatives with respect to atomic coordinates can be rapidly evaluated. In practice, to avoid restricting the form of the functions employed, and to promote calculation efficiency, numerically splined tabulated look-up tables are employed in most EAM calculations for the necessary functions. The resulting forcefield files are therefore large numerical tables. For individual elements three such tables are required, representing the pairwise function, the embedding function, and the density function.
Figure 2. An EAM density function (illustrated using the Zr EAM forcefield of Mendelev and Ackland4).
Handling alloy systems requires provision for interaction functions describing the pairwise interaction of each element, in addition to embedding, and density functions. For an n-component alloy there will be n(n+1)/2 pairwise interaction functions, n-embedding functions, and, associated density functions. The determination of these functions is challenging, and is complicated by the fact that the creation of a description suitable for a single element provides little information for the behavior of that element in an alloy or compound. Consequently, EAM forcefields are typically developed for specific systems and the description of a given element cannot trivially be combined directly with the description for another element, as assumptions about the two density functions, for example, may not be compatible.
The highly specific nature of EAM forcefields is emphasized by examples such as Mendelev and Ackland’s forcefield derivation work for metallic zirconium4. Here one parameterization is recommended for the exploration of phase stability and another distinct parameterization for the investigation of defects in the hexagonally close packed (hcp) form of α-Zr.
Figure 3. An EAM interaction function (Illustrated using the Zr EAM forcefield of Mendelev and Ackland4).
Despite such specificity, the merit of EAM forcefields is their ability to rapidly and accurately describe the bonding of metallic systems. EAM forcefields allow the simulation of:
- Structures – for example atomic configurations in the vicinity of grain boundaries
- Energies – for example relative polymorph energies and defect energies
- Diffusivity – for example through the use of mean squared displacements of sets of atoms in molecular dynamics trajectories
- Thermal expansivity – for example employing constant pressure simulations as a function of temperature to predict the response of a lattice to a temperature ramp
- Melting of metals and thermodynamic properties of the liquid state
As EAM forcefield calculations are computationally efficient large scale simulations can be readily undertaken, as illustrated in the following sections.
EAM Simulations in the MedeA Environment
The MedeA®environment supports standard ‘Finnis-Sinclair’ format EAM forcefield files, with extensions to permit detailed referencing of the source of the particular EAM description and atom type assignment. Such a file contains named sections expressing atom types, any atom equivalences, the standard Finnis-Sinclair format EAM function tables, information for partial charge assignment, and template information to assign forcefield atom types based on rules concerning topology and element type. This overall format is the standard employed by all MedeA®environment forcefields.
In addition to standard Finnis-Sinclair EAM forcefields, the MedeA®environment also supports the EAM parameterization described by Zhou and co-workers5. Here, mixing rules have been implicitly included in the design of the forcefield, and any combination of the elements: Cu, Ag, Au, Ni, Pd, Pt, Al, Pb, Fe, Mo, Ta, W, Mg, Co, Ti, or Zr may be handled. It is likely that this generality results in diminished accuracy in some circumstances (see for example6). However, when the effects of alloy formation are of interest, in the creation of layered metallic structures, for example, this description is highly effective5.
EAM Simulation in the MedeA®environment
An illustrative EAM calculation performed in the MedeA®environment. Here a supercell containing 4,000 atoms was initially constructed, and molecular dynamics employed to melt the upper half of this model. A sequence of 2 nano-second simulations were then employed to simulate the interface between the solid and liquid as a function of temperature. This task can be conveniently accomplished using a MedeA-Flowchart interface, which can be configured to carry out the necessary building and simulation tasks automatically for any desired metal or model type.
Figure 4 shows a typical configuration from the simulation. In this case the copper system was described with the Zhou EAM forcefield5. Then for a range of temperatures, the molecular dynamics trajectory was computed from a half crystalline and half molten starting point and the relative extents of the two regions examined at the culmination of the calculation.
The simulations reveal that below 1375K, the molten region retreats in the dynamics calculations, indicating that the crystalline system is more stable and crystallizes from such a melt. Above 1375K, however, the molten region grows relative to the crystalline region. We can conclude that in this simple EAM description and simulation regime, the melting temperature of copper is around 1375K, which compares quite well with the experimental value of 1358K, and emphasizes the fact that the simple EAM description captures much of the physics of metallic systems such as copper.
Figure 4. A copper solid-liquid interface for the molecular dynamics based simulation of melting.
These calculations may be refined, of course, for example, through the inclusion of thermal expansivity effects. However, the melting point calculation illustrated by Figure 4 required around 2 hours of computation time on a modest Linux cluster, demonstrating that such calculations can be rapidly conducted and providing a measure of the types of properties which may be computed. Of course, simulations permit the exploration of a wide range of properties, and are able to address conditions which are difficult to observe directly experimentally, providing a valuable complement to practical investigation.
EAM in the MedeA Environment
The range of properties accessible via EAM calculations is large. Some example properties, obtained using the Zhou5 copper description are listed in Table 1. In addition to these static properties, as the simulation of melting points illustrates, it is also possible to obtain dynamic properties, such as diffusion coefficients.
|Vacancy formation energy||1.28 eV||1.29 eV |
|Bulk modulus||133 GPa||140 GPa|
Table 1. Selected properties for copper obtained using the Zhou EAM description and their comparison with experiment.
One of the most important attributes of the EAM description is that it permits large scale simulations spanning significant time scales. Thus it is possible to simulate a wide range of phenomena using the method which are inaccessible to first-principles procedures. The MedeA®environment support for EAM forcefields makes the use of this important simulation method broadly accessible for the first time.
MedeA Modules for this Application Note
This application note refers to the capabilities of the MedeA®platform using the following integrated modules of the MedeA®software environment:
- The standard MedeA®framework including crystal structure builders and geometric analysis tools, as well as:
- MedeA LAMMPS interface
- MedeA LAMMPS-EAM
- MedeA LAMMPS-Diffusion
- MedeA JobServer
- MedeA TaskServers
Calculations based on empirical energy functions and parameters derive from early analysis of vibrational properties of molecular systems, leading to the use of the term forcefield to describe functional form and its parameterization. ↩
J. K. Nørskov, Phys. Rev. B 20, 446 (1979; J. K. Nørskov and N. D. Lang, Phys. Rev. B 21, 2131 (1980) ↩
M.S. Daw, M. Baskes, Phys. Rev. B 29, 6443 (1984) ↩
M.I. Mendelev, G.J. Ackland, Phil. Mag. Lett. 87, 349 (2007) ↩
X. W. Zhou, R. A. Johnson, H.N.G. Wadley, Phys. Rev. B. 69, 144113 (2004) ↩
M.F. Francis, M.N. Neurock, X.W. Zhou, J.J. Quan, H.N.G. Wadley, E.B. Webb III, J. Appl. Phys. 104, 034310 (2008) ↩
W. Trifthauser, J.D. McGervey, Appl. Phys. 6, 177 (1975) ↩