A
Ensemble Creation
B
Crossover
C
Mutation
D
Exchange
NEW GENERATION
PRE-SELECTION
TOURNAMENT STAGE
E
Selection
FIG. 1: Schematic representation of the genetic algorithm. (A) An ensemble of supercells
containing the grain boundaries is created. The randomized regions in each supercell are shaded
pink and are ‘enclosed’ on both sides by undisturbed, bulk crystal. (B) During crossover, groups of
atoms are exchanged between randomly chosen pairs of configurations. The shaded boxes represent
the cuboids described in the text. (C) Atoms are displaced by an amount inversely proportional
to their order parameter during the mutation step. Although the diagram implies that only atoms
in the disordered region are displaced, atoms in the bulk crystal region also move. Their displacements are much smaller however, owing to their much higher order parameter. (D) Two atoms
to be exchanged in a configuration are chosen with probability proportional to their disorder. (E)
The next generation of individuals is created using tournament selection. In the diagram, the tournament size is two and darker levels of shading indicate higher energy configurations. The ‘winner’
(lower energy configuration) of each tournament becomes a member of the next generation.
1
2.2
2
1.8
1.6
-2
σ (J m )
1.4
1.2
Γ TiO = 1
2
1
Γ TiO = -1
2
Γ TiO = 0, SrO3 terminated
0.8
2
Γ TiO = 0, Ti terminated
2
0.6
0.4
0.2
0
-1.5
-1.25
-1
-0.75
-0.5
0
μTiO -g
2
-0.25
0
(eV)
TiO2
FIG. 2: Grain boundary energies as a function of µTiO2 for the Σ3(111)[1̄10] interface. The area
As of the supercell parallel the boundary plane has been set equal to 1 in specifying the boundary
excesses Γi .
2.5
-2
σ (J m )
2
1.5
1
Γ TiO = 0, SrTiO termination
2
Γ TiO = 0, O2 termination
0.5
2
Γ TiO = 2
2
Γ TiO = 1
2
0
-1.5
-1
-0.5
0
µ TiO - g
2
TiO2
0
(eV)
FIG. 3: Grain boundary free energies as a function of µTiO2 for the Σ3(112)[1̄10] system.
2
[11-1]
[112]
[-110]
FIG. 4: The ΓTiO2 = 2 Σ3(112)[1̄10] grain boundary. The arrows indicate the position of the two
equivalent grain boundaries in the supercell. Blue spheres indicate Sr atoms, red spheres indicate
O atoms and yellow spheres indicate Ti atoms. The Ti and O atoms facing each other at the
interface are in the same plane along the [1̄10] tilt axis.
3