ECO Driving force
adds an energy to each atom depending on the surrounding crystal orientation, in order to drive grain boundaries
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vmohles
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doc/src/fix_eco_force.rst
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doc/src/fix_eco_force.rst
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1 fix eco/force command
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fix ID group-ID eco/force u0 eta rcut file
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• ID, group-ID are documented in fix command
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• u0 = energy added to each atom (energy units)
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• eta = cutoff value (usually 0.25)
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• rcut = cutoff radius for orientation parameter calculation • file = file that specifies orientation of each grain
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Examples: fix gb all eco/force 0.08 0.25 3.524 sigma5.ori
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1.1 Description:
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The fix applies a synthetic driving force to a grain boundary which can
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be used for the investigation of grain boundary motion. The affiliation
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of atoms to either of the two grains forming the grain boundary is
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determined from an orientation-dependent order parameter as described
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in (Ulomek). The potential energy of atoms is either increased by an amount
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of 0.5*u0 or -0.5*u0 according to the orientation of the surrounding
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crystal. This creates a potential energy gradient which pushes atoms near
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the grain boundary to orient according to the energetically favorable
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grain orientation. This fix is designed for applications in bicrystal system
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with one grain boundary and open ends, or two opposite grain boundaries in
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a periodic system. In either case, the entire system can experience a
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displacement during the simulation which needs to be accounted for in the
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evaluation of the grain boundary velocity. While the basic method is
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described in (Ulomek), the implementation follows the efficient
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implementation from (Schratt & Mohles). The synthetic potential energy added to an
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atom j is given by the following formulas
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\begin{eqnarray}
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w(|\vec{r}_{jk}|)=w_{jk}=\left\{\begin{array}{lc}
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\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}} \\
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0, & |\vec{r}_{jk}|\ge r_{\mathrm{cut}}
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\end{array}\right.
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\label{eq:envelope}
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\end{eqnarray}
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\begin{eqnarray}
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\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
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\label{eq:force-long}
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\end{eqnarray}
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\begin{eqnarray}
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\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)
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\end{eqnarray}
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\begin{eqnarray}
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u(\chi_{j}) & = & \frac{u_{0}}{2}\left\{\begin{array}{lc}
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1, & \chi_{j}\ge\eta\\
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\sin\left(\frac{\pi\chi_{j}}{2\eta}\right), & -\eta<\chi_{j}<\eta\\
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-1, & \chi_{j}\le-\eta
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\end{array}\right.
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\label{eq:energy-mid}
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\end{eqnarray}
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which are fully explained in (Ulomek) and (Schratt & Mohles).
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The force on each atom is the negative gradient of the synthetic potential energy. It
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depends on the surrounding of this atom. An atom far from the grain boundary does not
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experience a synthetic force as its surrounding is that of an oriented single crystal
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and thermal fluctuations are masked by the parameter eta. Near the grain boundary
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however, the gradient is nonzero and synthetic force terms are computed.
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The orientation file specifies the perfect oriented crystal basis vectors for the
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two adjoining crystals. The first three lines for the energetically penalized and the
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last three lines for the energetically favored grain assuming u0 is positive. For
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negative u0 this is reversed. With the rcut parameter, the size of the region around
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each atom which is used in the order parameter computation is defined. It should at
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least include the nearest neighbor shell. For high temperatures or low angle
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grain boundaries, it might be beneficial to increase rcut in order to get a more
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precise identification of the atoms surrounding. However, computation time will
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increase as more atoms considered in the order parameter and force computation.
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It is also worth noting that the cutoff radius must not exceed the communication
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distance for ghost atoms in LAMMPS. Currently however, the method stores results
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for order parameter and force computations in statically allocated arrays to
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increase efficiency such that the user is limited to 24 nearest neighbors.
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This is more than enough in most applications. With file, the input file for
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the 6 oriented crystal basis vectors is specified. Each line of the input file
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contains the three components of a primitive lattice vector oriented according to
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the grain orientation in the simulation box. The first (last) three lines correspond
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to the primitive lattice vectors of the first (second) grain. An example for
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a Σ5⟨001⟩ misorientation is given at the end.
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If no synthetic energy difference between the grains is created, u0=0, the
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force computation is omitted. In this case, the order parameter of the
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driving force can be used to track the grain boundary motion throughout the
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simulation.
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1.2 Restart, fix_modify, output, run start/stop, minimize info:
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No information about this fix is written to binary restart files. The
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fix_modify energy option is supported by this fix to add the potential energy of
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atominteractions with the grain boundary driving force to the system's poten-
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tial energy as part of thermodynamic output.
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The total sum of added synthetic potential energy is computed and can be accessed
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by various output options. The order parameter as well as the thermally masked
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output parameter are stored in per-atom arrays and can also be accessed by various
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output commands.
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No parameter of this fix
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can be used with the start/stop keywords of the run command. This fix is
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not invoked during energy minimization.
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1.3 Restrictions:
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This fix is part of the MISC package. It is only enabled if LAMMPS was
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built with that package. See the Making LAMMPS section for more info.
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1.4 Related commands:
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fix_modify
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Default: none
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(Ulomek) Felix Ulomek et al. Modelling Simul. Mater. Sci. Eng. 23 (2015) 025007
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(Schratt & Mohles) Adrian A. Schratt and Volker Mohles. Comp. Mat. Sci. 182 (2020) 109774
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For illustration purposes, here is an example file that specifies a $\Sigma5\langle001\rangle$ tilt grain boundary.
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This is for a lattice constant of 3.52 Angs.
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1.5 File:
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1.671685 0.557228 1.76212
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0.557228 -1.671685 1.76212
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2.228913 -1.114456 0.000000
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0.557228 1.671685 1.76212
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1.671685 -0.557228 1.76212
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2.228913 1.114456 0.000000
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