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Lu et al. Microstructures 2023;3:2023033 https://dx.doi.org/10.20517/microstructures.2023.28 Page 3 of 10

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interactions U(r) = 8(r - 2) (green springs), where r is the distance between atoms. The first- and third-
nearest interactions are related to the elastic interactions and constitute the elastic background in
ferroelastic materials. The model parameters are inspired by SrTiO with the energy scale determined by T
c
3
= 105 K and typical ferroelastic shear angle of 2°. The atomic mass is M = 50 amu. The relevant strain
components ε , ε , and ε are calculated from the appropriate interatomic distances relative to those of the
yy
xx
xy
monodomain. The sample was relaxed for ca. 10 computational steps. All simulations were performed
5
using the LAMMPS program , which minimises the potential energy of the total system. Although our
[35]
formulated interatomic potential is simple, the main elastic properties of ferroelastic materials have been
considered such that the simulated results can successfully reveal the elastic interactions between kinks,
which would also be predicted by advanced methods, such as analytical potentials and DFT calculations,
[36]
where more parameters and physical processes are considered. Two different boundary conditions are used
in our simulations. The first one is the open or Dirichlet boundary condition applied in both x and y
directions, where sample relaxations, including shape changes and rotations, are allowed. The initial lattice
parameters in x and y directions are set to a = 1 l.u. and relax to 1.0001297 l.u. and 0.9995027 l.u. in x and y
directions, respectively. A single domain wall with a kink residing at its centre was constructed to
investigate the size dependence of tilt angles and self-energies of the kink [Figure 1A and B]. Two parallel
domain walls containing one kink in each wall, i.e., a kink-kink pair and a kink-antikink pair [Figure 1A]
with various separations [Figure 1C] were constructed to investigate the effect of the sample size on the kink
interactions.

The second boundary condition was constructed with the bottom layer fixed while all other surfaces were
free to relax. This configuration represents an extreme case of hard interfacial bonding without any lattice
defect between the sample and the substrate. This situation is encountered in thin films on “hard”
substrates. All other parameters were identical to the first case of open boundary conditions. The kinks were
initially created inside domain walls, and the system was then relaxed using a conjugate gradient method
followed by 5 × 10 (5 × 10 ps) molecular dynamics (MD) simulation steps to obtain the full ferroelastic
3
6
domain structure. Ferroelastic domain structures were obtained by averaging structural snapshots every
1,000 MD steps (1 ps). To avoid the movement of kinks in the domain wall, the temperature was kept very
low at T = 0.001 K using a Nosé-Hoover thermostat . All simulations were performed using the LAMMPS
[37]
[38]
code. The atomic configurations were displayed using the OVTIO software .

RESULTS AND DISCUSSION
Free standing sample (membrane, lamella)
We first construct a single domain wall with a stable kink located at the centre within a cell with open
boundary conditions with a constant lateral size (L ) of 1,601 l.u. in the x direction and variable vertical sizes
x
(L ) between 101 l.u. and 1,601 l.u. in the y direction (sample thickness indicated by Δ in Figure 2A). The
y
kink-induced distortion is observed in the strain ɛ map, as shown in Figure 2A. The strain fields generated
xx
by the kinks manifest obvious compressive and tensile regimes on the top and bottom of the kink, similar to
those of dislocations . A local bending near the kink, together with a macroscopic tilt of the domain wall
[39]
on the left and right sides of this kink, is observed in Figure 2A and B. Such local bending can also be
[40]
discussed under the framework of the Helfrich model . The tilt angle is defined as the macroscopic angle
between the domain wall and the horizontal direction, with the angles θ and θ shown in Figure 2B. All
1
2
atomic layers from the bottom (green lines in Figure 2A) to the top surface (blue lines in Figure 2A) show
similar tilt angles. Figure 2C shows the tilt angles as a function of the sample thickness. Both θ and θ
1
2
decrease as the system thickness increases, following θ ~ Δ . All these results are in agreement with our
-1
previous results on kink interactions .
[30]