ECO Driving force

1 	fix eco/force command

fix ID group-ID eco/force u0 eta rcut file

• ID, group-ID are documented in fix command

• u0 = energy added to each atom (energy units)

• eta = cutoff value (usually 0.25)

• rcut = cutoff radius for orientation parameter calculation • file = file that specifies orientation of each grain

Examples: fix gb all eco/force 0.08 0.25 3.524 sigma5.ori 

1.1	Description:

The fix applies a synthetic driving force to a grain boundary which can 

be used for the investigation of grain boundary motion. The affiliation 

of atoms to either of the two grains forming the grain boundary is 

determined from an orientation-dependent order parameter as described 

in (Ulomek). The potential energy of atoms is either increased by an amount 

of 0.5*u0 or -0.5*u0 according to the orientation of the surrounding 

crystal. This creates a potential energy gradient which pushes atoms near 

the grain boundary to orient according to the energetically favorable 

grain orientation. This fix is designed for applications in bicrystal system 

with one grain boundary and open ends, or two opposite grain boundaries in 

a periodic system. In either case, the entire system can experience a 

displacement during the simulation which needs to be accounted for in the 

evaluation of the grain boundary velocity. While the basic method is 

described in (Ulomek), the implementation follows the efficient 

implementation from (Schratt & Mohles). The synthetic potential energy added to an 

atom j is given by the following formulas

\begin{eqnarray}

w(|\vec{r}_{jk}|)=w_{jk}=\left\{\begin{array}{lc}

\frac{|\vec{r}_{jk}|^{4}}{r_{\mathrm{cut}}^{4}}-2\frac{|\vec{r}_{jk}|^{2}}{r_{\mathrm{cut}}^{2}}+1, & |\vec{r}_{jk}|<r_{\mathrm{cut}} \\

0, & |\vec{r}_{jk}|\ge r_{\mathrm{cut}}

\end{array}\right.

\label{eq:envelope}

\end{eqnarray}

\begin{eqnarray}

\chi_{j} & = & \frac{1}{N}\sum_{l=1}^{3}\left\lbrack\left\vert\psi_{l}^{\mathrm{I}}(\vec{r}_{j})\right\vert^{2}-\left\vert\psi_{l}^{\mathrm{II}}(\vec{r}_{j})\right\vert^{2}\right\rbrack

\label{eq:force-long}

\end{eqnarray}

\begin{eqnarray}

\psi_{l}^{\mathrm{X}}(\vec{r}_{j})=\sum_{k\in\mathit{\Gamma}_{j}}w_{jk}\exp\left(\mathrm{i}\vec{r}_{jk}\cdot\vec{q}_{l}^{\mathrm{X}}\right)

\end{eqnarray}

\begin{eqnarray}

u(\chi_{j}) & = & \frac{u_{0}}{2}\left\{\begin{array}{lc}

1, & \chi_{j}\ge\eta\\

\sin\left(\frac{\pi\chi_{j}}{2\eta}\right), &  -\eta<\chi_{j}<\eta\\

-1, & \chi_{j}\le-\eta

\end{array}\right.

\label{eq:energy-mid}

\end{eqnarray}

which are fully explained in (Ulomek) and (Schratt & Mohles).

The force on each atom is the negative gradient of the synthetic potential energy. It 

depends on the surrounding of this atom. An atom far from the grain boundary does not 

experience a synthetic force as its surrounding is that of an oriented single crystal 

and thermal fluctuations are masked by the parameter eta. Near the grain boundary 

however, the gradient is nonzero and synthetic force terms are computed. 

The orientation file specifies the perfect oriented crystal basis vectors for the 

two adjoining crystals. The first three lines for the energetically penalized and the 

last three lines for the energetically favored grain assuming u0 is positive. For 

negative u0 this is reversed. With the rcut parameter, the size of the region around 

each atom which is used in the order parameter computation is defined. It should at 

least include the nearest neighbor shell. For high temperatures or low angle 

grain boundaries, it might be beneficial to increase rcut in order to get a more 

precise identification of the atoms surrounding. However, computation time will 

increase as more atoms considered in the order parameter and force computation. 

It is also worth noting that the cutoff radius must not exceed the communication 

distance for ghost atoms in LAMMPS. Currently however, the method stores results 

for order parameter and force computations in statically allocated arrays to 

increase efficiency such that the user is limited to 24 nearest neighbors. 

This is more than enough in most applications. With file, the input file for 

the 6 oriented crystal basis vectors is specified. Each line of the input file 

contains the three components of a primitive lattice vector oriented according to 

the grain orientation in the simulation box. The first (last) three lines correspond 

to the primitive lattice vectors of the first (second) grain. An example for 

a Σ5⟨001⟩ misorientation is given at the end.

If no synthetic energy difference between the grains is created, u0=0, the 

force computation is omitted. In this case, the order parameter of the 

driving force can be used to track the grain boundary motion throughout the 

simulation.

1.2	Restart, fix_modify, output, run start/stop, minimize info:

No information about this fix is written to binary restart files. The 

fix_modify energy option is supported by this fix to add the potential energy of 

atominteractions with the grain boundary driving force to the system's poten-

tial energy as part of thermodynamic output. 

The total sum of added synthetic potential energy is computed and can be accessed 

by various output options. The order parameter as well as the thermally masked 

output parameter are stored in per-atom arrays and can also be accessed by various 

output commands. 

No parameter of this fix

can be used with the start/stop keywords of the run command. This fix is

not invoked during energy minimization.

1.3	Restrictions:

This fix is part of the MISC package. It is only enabled if LAMMPS was

built with that package. See the Making LAMMPS section for more info.

1.4	Related commands:

fix_modify

Default: none

(Ulomek) Felix Ulomek et al. Modelling Simul. Mater. Sci. Eng. 23 (2015) 025007

(Schratt & Mohles) Adrian A. Schratt and Volker Mohles. Comp. Mat. Sci. 182 (2020) 109774

For illustration purposes, here is an example file that specifies a $\Sigma5\langle001\rangle$ tilt grain boundary. 

This is for a lattice constant of 3.52 Angs.

1.5	File:

    1.671685 0.557228 1.76212

    0.557228 -1.671685 1.76212

    2.228913 -1.114456 0.000000

    0.557228 1.671685 1.76212

    1.671685 -0.557228 1.76212

    2.228913 1.114456 0.000000

log eam.log

dimension 3

boundary p p p

units metal

atom_style atomic

lattice fcc 3.5

read_data examples/eco_force/sigma5.lammps

pair_style eam/fs

pair_coeff * * examples/eco_force/Ni.eam.fs Ni

timestep 0.001

thermo 100

fix integrator all npt temp 750 750 0.1 iso 0 0 0.1

fix eco all eco/force 0.08 0.25 3.6 examples/eco_force/sigma5.ori

velocity all create 750 1

dump save all custom 100 nidata_eam.dump xs ys zs f_eco[1] f_eco[2]

dump_modify save sort id

run 1000

 No newline at end of file

1.671685 0.557228 1.76212

0.557228 -1.671685 1.76212

2.228913 -1.114456 0.000000

0.557228 1.671685 1.76212

1.671685 -0.557228 1.76212

2.228913 1.114456 0.000000

 No newline at end of file