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Lu et al. Microstructures 2023;3:2023033 https://dx.doi.org/10.20517/microstructures.2023.28 Page 7 of 10

dependence is found for interactions between kinks and antikinks [Supplementary Figure 3].

The change of the interaction energy exponent from -2 for small samples to -1 for thick samples extends
over an interval of sample thicknesses between 700 l.u. and 1,400 l.u. In real materials, the interval extends
from 350 nm to 700 nm for a lattice unit of 0.5 nm. The weaker wall-wall interactions could lead to a higher
concentration of domain walls.

Sample on a substrate (thin film)
A thin film on a substrate cannot bend on the fixed surface but can relax the free surface. To explore the
effect of a fixed boundary on the kink interactions, we simulate this scenario by clamping the lower surface
in the limit of hard interfaces with no lattice misfit. The model has a sample size of 601 l.u. in the x direction
and 100 l.u. in the y direction [Figure 5A]. Similar to the cases of samples with free boundaries, the two
parallel domain walls are symmetric with respect to the centre of the sample, and the kink is stabilised at the
centre of each domain wall. We then relaxed the entire sample. The resultant strain fields are shown in
Figure 5A. The strain fields do not extend to the bottom surface, while strain fields deform the top surface.
Figure 5B shows the local displacements of the fixed bottom surface, lower wall, upper wall and top surface
of samples with wall-wall distances of d = 10 l.u., 22 l.u., and 42 l.u. The bottom surface remains flat, while
the top surface shows local ridge-and-valley deformations. These local deformations are in the order of
0.02 l.u. or 2% strain for d = 42 l.u. The local deformations of the domain walls are of the same order of
magnitude, with a sharp singularity at the kinks. The upper wall exhibits a larger deformation than the
lower wall.

The fundamental difference between the clamped and the free samples is that no macroscopic bending can
occur under clamping conditions, while the local deformations are visible in both cases. The decay of the
valley structure away from the centre is exponential and extends over some 50 l.u. The decay resembles the
deformation caused by intersections between domain walls and surfaces [43,44] .

The interaction between the two kinks, and hence between the walls, is shown in Figure 5C. The scaling
function is a power law with an exponent of -1 [Figure 5D]. In straight samples without any macroscopic
-1
bending, the scaling E ~ d holds even though only one surface is clamped and the opposite surface is free
to deform. Similar results were found for the kink-antikink interactions, as shown in
Supplementary Figure 4.

CONCLUSIONS
-1
Kink-kink interactions in bulk samples interact as “monopoles” with a d dependence when they are
separated by the distance d. As the sample size decreases, the interaction for thin samples decays following a
characteristic d trend similar to that of dipoles. This behaviour of any singularity (dielectric, dislocations,
-2
interstitials, etc.) is commonly described analytically by the concept of “image force”. The construction is
based on the calculation of the surface relaxation as having the same energy as if a fictitious image force was
placed outside the sample. Such image forces have also been used to describe the dynamics of dislocation
movements . Our results clarify the role of the crossover regime near d = 1,000 l.u., which is rather wide.
[45]

Detailed investigations of wall profiles are often attempted by transmission electron microscopy where the
typical sample thickness is 50 l.u., well within the “dipolar” range. We demonstrate the role of image forces
that are responsible for the dipolar relaxation by comparing the relaxation patterns of free-standing samples
and thin films. They are closely related to the bending of the sample and much less to the bulging of the
surface to form ridge-and-valley structures. This corresponds not only to substrate effects but also to thin