Sutton Ag EAM Example

This provides an example of using potable to tabulate the Ag model given by Sutton and Chen in [1].

Potential Model

(12)\[E_i = \epsilon \left[ \frac{1}{2} \underset{i \neq j}{\sum \sum} \phi_{\alpha\beta}(r_{ij}) - c \sum_i \sqrt{\rho_i} \right]\]

Where:

  • \(E_T\) is the energy of ths system.
  • \(r_{ij}\) is the separation between atoms \(i\) and \(j\).
  • \(c\) and \(\epsilon\) are adjustable parameters specific to interacting species.
  • Inside the square brackets the first term \(V(r_{ij})\) are the pair potentials.
  • The second is the many body term: \(c \sum_i \sqrt{\rho_i}\). Where \(\rho_i\) is the electron density.

Pair potential form:

\[\phi_{\alpha \beta}(r_{ij}) = (a/r_{ij})^n\]

Where:

  • \(a\) and \(n\) are potential parameters.

This must be multiplied by the \(\epsilon\) term from equation (12) above:

\[\phi_{\alpha \beta}(r_{ij}) = \epsilon (a/r_{ij})^n\]

To make things easier later on, this will be re-expressed as:

\[\phi_{\alpha \beta}(r_{ij}) = \epsilon a^n r_{ij}^n\]

This will allow this functional form to be written using the provided as.exponential potential-form.

Many body terms

Density function:

The density function is:

\[\rho_i = \left( \frac{a}{r_{ij}} \right)^m\]

Again to allow the use of the as.exponential potential-form this will be re-written as:

\[\rho_i = a^m r_{ij}^{-m}\]

Embedding function:

Examining the many-body term from (12) it can be seen that the embedding function is:

\[c \sqrt{\rho_i}\]

Taking the the \(\epsilon\) term from outside the square brackets and pre-multiplying the expression this becomes:

\[\epsilon c \sqrt{\rho_i}\]

Potential parameters

The potential parameters for Ag are:

Table 3 Potential parameters for Ag
Parameter Value
\(m\) 6
\(n\) 12
\(\epsilon\) 2.5415×10-3 eV
\(a\) 4.09
\(c\) 144.41

Potable input

The potable input for this model can be downloaded as Ag_sutton.aspot and will now be described:

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[Tabulation]
target : setfl
#
cutoff_rho : 600
drho : 0.005
#
cutoff : 12.0
dr : 0.001

[EAM-Embed]
Ag : product(as.constant 2.5415e-3, as.sqrt -144.41)

[EAM-Density]
Ag : as.exponential 4681.013008649 -6

[Pair]
Ag-Ag : product(as.constant 2.5415e-3, as.exponential 21911882.787 -12)

Notes:

  • lines 1-8 [Tabulation]:

    • lines 4,5: gives the resolution and extent of the function in [EAM-Embed].
    • lines 7,8: defines resolution and extent of the tables generated for the [Pair] and [EAM-Density] functions.

  • lines 10 and 11 [EAM-Embed]:

    • Defines the embedding function.
    • Note the use of the product() potential modifier to multiply the square root embedding function by the value of \(\epsilon\).

  • lines 13 and 14 [EAM-Density]:

    • Describes the density function.
    • The value of 4681.013008649 is obtained as \(a^m = 4.09^6\).

  • lines 16 and 17 [Pair]:

    • Defines the pair potential component of the model.
    • As above, the product() potential modifier has been used to multiply the function by \(\epsilon\).
    • Here the first parameter to the as.exponential form is \(a^n = 4.09^{12}\) = 21911882.78.

Making and testing the tabulation

To tabulate the potential download the aspot file and run it through potable

potable Ag_sutton.aspot Ag_sutton.eam.alloy

A LAMMPS input file is provided to allow you to test the Ag_sutton.eam.alloy file produced by potable. This input file can be downloaded here: Ag_sutton_fcc.lmpin and will energy minimize the structure and then perform an NPT MD equilibration at T=300K. Frames will be dumped every 1000 timesteps (1ps) and dumped to a LAMMPS dump file named dump.atom(this is suitable for visualisation in Ovito).

In terms of the table file the important part of the LAMMPS input is:

pair_style eam/alloy
pair_coeff * * Ag_sutton.eam.alloy Ag

This tells LAMMPS to accept a setfl formatted file (pair_style eam/alloy). The Ag at the end of the pair_coeff line says that LAMMPS should associate atom type 1 with the Al species label in the table file Ag_sutton_eam.alloy.

Placing both the LAMMPS and table file in the same directory run LAMMPS as follows:

mpirun lammps -in Ag_sutton_fcc.lmpin -log Ag_sutton_fcc.lmpout

Footnotes:

[1]A.P. Sutton, and J. Chen, “Long-range Finnis-Sinclair potentials”, Philos. Mag. Lett. 61 (1990) 139 doi:10.1080/09500839008206493.