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Page 8 of 10 Lu et al. Microstructures 2023;3:2023033 https://dx.doi.org/10.20517/microstructures.2023.28

Figure 5. Interaction energies of kink-kink configurations with a clamped bottom surface. (A) Strain fields of the thin film with a
thickness of Δ = 100 l.u. and wall-wall distances of d = 42 l.u. The strain map was colour-coded according to the atomic-level strain ɛ .
xx
(B) Lattice displacements of fixed bottom surface, lower wall, upper wall, and top surface due to the kink-kink interactions. (C) The
variation of interaction energies as a function of wall-wall distance. The data points in (C) are fitted by using the equation E = E -
kink-kink 0
B
A × d with E ~ 0 (noninteracting kinks), A = 0.832 eV l.u. and B = -1. The scaling exponent of -1 is shown in (D).
0
wedges in transmission electron microscopy investigations where the imaged part of the sample is very thin
but constrained by the thicker part of the specimen. Still, even in clamped samples, such as thin films on
rigid substrates, rather large surface structures, such as ridges and valleys, are observed in our simulation
and could be detected experimentally.

Our simulated results provide a comprehensive understanding of the elastic interactions between kinks in
ferroelastic and ferroelectric domain walls. The existence of a wide crossover regime in free-standing
samples indicate that their domain structures should organise differently and exhibit abnormal behaviours
in response to external fields, leading to unusual functionalities (e.g., superelasticity). Future research on
membranes with in-situ TEM or dynamic piezoresponse force microscopy should provide valuable insights
and investigations on novel nanomaterials.

DECLARATIONS
Authors’ contributions
Conceptualization and performance of molecular dynamic simulations; data curation; writing original draft
preparation: Lu G
Writing, review, and editing: Lu G, Nataf GF, Salje EKH
Review and discussion: Lu G, Hideo K, Ding X, Xu Z, Chu R, Nataf GF, Salje EKH