# Slip-dominated structural transitions

Kanka Ghosh, Oğuz Umut Salman, Sylvain Queyreau, Lev Truskinovsky

► To cite this version:

Kanka Ghosh, Oğuz Umut Salman, Sylvain Queyreau, Lev Truskinovsky. Slip-dominated structural transitions. Physical Review Materials, 2024, 9 (7), pp.073604. 10.48550/arXiv.2409.04066. hal-04696850v2

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Submitted on 2 Feb 2025

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Slip-dominated structural transitions

Kanka Ghosh,¹,* Oguz Umut Salman,¹ Sylvain Queyreau,¹ and Lev Truskinovsky²,†

¹CNRS, LSPM UPR 3407, Université Sorbonne Paris Nord, 93430 Villetaneuse, France

²PMMH, CNRS UMR 7636 ESPCI ParisTech, 10 Rue Vauquelin, 75005 Paris, France

We use molecular dynamics to show that plastic slip is a crucial component of the transformation mechanism of a square-to-triangular structural transition. The latter is a stylized analog of many other reconstructive phase transitions. To justify our conclusions we use a novel atomistically-informed mesoscopic representation of the field of lattice distortions in molecular dynamics simulations. Our approach reveals a hidden alternating slip distribution behind the seemingly homogeneous product phase which points to the fact that lattice invariant shears play a central role in this class of phase transformations. While the underlying pattern of anti-parallel displacements may be also interpreted as microscopic shuffling, its precise crystallographic nature strongly suggests the plasticity-centered interpretation.

Reconstructive phase transitions are the most widespread type of structural transformations in solids. These transitions lack the simplifying group-subgroup relationship and therefore cannot be described by the conventional Landau theory. The development of the equally encompassing theory of reconstructive transitions is still a challenge given that they involve breaking of chemical bonds and are characterized by micro-inhomogeneous configurations with slip, twinning, and stacking faults apparently intertwined [1-7].

The BCC-HCP reconstructive transition is one of the most representative [8-15]. Its mechanism, proposed by Burgers based on crystallographic analysis [16-23], implies the presence of a homogeneous shear and a superimposed alternating shuffling represented by anti-parallel shifting of atomic layers. The origin of the shuffling cannot be addressed based on crystallography only and various attempts to interpret it while referring to structural mechanics, energetics and kinetics can be found in the literature [14, 17, 24-32]. Similar problem exists for the FCC-HCP reconstructive phase transition which can be accomplished crystallographically by the alternating coordinated gliding of Shockley partials on every second close-packed crystallographic plane [33, 34]. The origin of the implied antagonistic displacements still remain obscure despite many insightful attempts to link it to first principles auxiliary computations [33, 35-49].

More generally, the pattern of anti-parallel, crystallographically specific, nanoscale, highly coordinated displacements appear to be a distinguishing feature of reconstructive transitions, which is the main factor placing them outside the classical Landau picture [1, 50, 51]. A fundamental understanding of this phenomenon is then of great theoretical interest and in this Letter we propose its new interpretation. Our conclusions are based on systematic molecular dynamics (MD) studies of a prototypical model which suggest that the crucial non-Landau factor in reconstructive phase transitions is the disguised plastic slip.

Specifically, we study the simplest reconstructive transition between 2D square (S) and triangular (T) latt-

tices [2, 52, 53]. While such square-to-hexagonal phase change is of interest by itself [54-68], it can be considered as a stylized, low dimensional, Bravais lattice analog of both emblematic BCC-HCP and FCC-HCP reconstructive transitions [69-71].

To interpret the results of our MD simulations we shift attention from the conventional focus on the configurations of individual atoms to the original representation of the transformation history in terms of the evolution of atomic neighborhoods. This new approach allows us to map the transformation path into the configurational space of the mesoscopic metric tensors. The purely geometrical periodic tessellation of the latter creates the possibility to distinguish unambiguously between elastic and plastic deformations [2, 53, 72]. The application of such atomistically-informed representation of lattice distortions in the case of S-T transition reveals that its fundamentally non-affine mechanism involves alternating lattice invariant shears which points towards a plasticity-centered interpretation of this reconstructive transition. To corroborate the results of our MD experiments, we also performed a parallel study of a coarse grained mesoscopic model which directly deals with the evolution of atomic neighborhoods [72-76]. The obtained qualitative agreement suggests that the observed slip-dominated mechanism of S-T transition is a robust feature of this class of reconstructive transformations, insensitive to microscopic details.

The molecular dynamics simulations were carried out using LAMMPS [77, 78]. The particle interaction potential was chosen in the form $V(r) = a/r^{12} - c_1 \exp[-b_1(r-r_1)^2] - c_2 \exp[-b_2(r-r_2)^2]$, allowing one to stabilize in 2D both square and triangular lattices [79]; here $r_1$ is the lattice constant and $r_2$ is the second nearest neighbor distance. Periodic boundary conditions were employed and pressure controlled protocol was implemented within isothermal-isobaric ensemble [80]. The computed (kinetic) T-P phase diagram for the direct S-T transition is shown in Fig. 1(a). The predicted negative slope of the coexistence curve agrees with similar numerical experiments [81-83] and is also
<figure><img src="image_1.png" /><figcaption>FIG. 1. MD simulated square-to-triangular transition: (a) kinetic T-P phase diagram showing only the direct transition; (b) the same transition in P-V (area A) performed at T = 10 K (as indicated in (a) by an arrow).</figcaption></figure>

consistent with the data for BCC-HCP transformation
in iron [84, 85]. In Fig. 2 we illustrate the fact that the
originally pure-crystalline square lattice transforms into a
poly-crystalline triangular configuration with specifically
mis-oriented grains separated by dislocation-rich curved
grain boundaries [80].

<figure><img src="image_2.png" /><figcaption>FIG. 2. Multigrain configuration of transformed triangular lattice (T) colored according to the potential energy of each atom. The atomic structure of two triangular grains with 30° mis-orientation is illustrated in the insets.</figcaption></figure>

As a novel way of interpreting the results of MD sim-
ulations, we extracted from individual atomic position
data the local values of the deformation gradients. This
amounts to post-processing the instantaneous MD data
which are then interpreted as representing piecewise lin-
ear strain fields [86–90]. Specifically, by denoting the
reference discrete atomic positions **x**<sup>*i*</sup> and the deformed
atomic positions **y**<sup>*i*</sup>, we can define the effective defor-
mation gradient **F**<sup>*i*</sup> by minimizing the error function
∑ || **Δ**<b>y** − **F**<sup>*i*</sup>**Δ**<b>x** ||<sup>2</sup> with summation over the pairs of
elements inside the chosen neighborhood of a given site
<i>*i*</i> [91–97]. In our case of weakly distorted lattices the
sampling neighborhood could be limited to two comple-
mentary triangular domains [80]. Most importantly, this
approach allows us to compute the MD informed local
metric tensor **C**<sup>*i*</sup> = **F**<sup>*i*</sup><sup>*T*</sup>**F**<sup>*i*</sup>.

The possibility of mapping the results of MD sim-
ulations into the **C**-space (with coordinates C<sub>11</sub>, C<sub>22</sub>,
and C<sub>12</sub>) is of great interest because its crucial sub-
space, det(**C**) = 1, is naturally tessellated by the action
of the global symmetry group of Bravais lattices. The
latter is a finite strain extension of the crystallographic

<figure><img src="image_3.png" /><figcaption>FIG. 3. Stereographic projection on a Poincare disk of the configurational space of metric tensors **C** with det **C** = 1 given by the formulas x = t(C<sub>11</sub> - C<sub>22</sub>)/2, y = tC<sub>12</sub>, where t = 2(2 + C<sub>11</sub> + C<sub>22</sub>)<sup>-1</sup>, see [80] for details. In (a,b) the reference states are the square phase S and the triangular phase T<sub>1</sub>, respectively. Dark gray indicates the minimal periodicity domain, light gray – the minimal elastic domain; blue lines represent the tessellation induced by the GL(2, **Z**) global symmetry.</figcaption></figure>

point group and can be represented in our case explic-
itly: GL(2, Z) = { **m** , m<sub>IJ</sub> ∈ Z , det(**m**) = ±1}, see for
instance [2, 53, 98–112].

The symmetry-induced periodicity structure in the
space of metric tensors **C** is illustrated in Fig. 3(a,b),
where we identify both, the minimal periodicity (fundamental)
domain and the minimal elastic domain (fundamental
domain extended under the action of the point
group), see [80] and also [75, 76] for more details. The
two panels in Fig. 3 provide equivalent information with
the only difference that in (a) the **C**-space is centered
around the reference square lattice (point S), while in
(b) the reference lattice is triangular (point T<sub>1</sub>) [80]. The
importance of Fig. 3(b) for what follows is that it clearly
indicates that the two triangular lattices T<sub>1</sub> and T<sub>2</sub> be-
long to different elastic domains because they differ by a
lattice invariant shear representing an elementary plastic
slip.

In Fig. 4(a) we show in more detail a fragment of the
**C**-space centered around the point S (taken in this case
as the reference, essentially a zoom in on Fig. 3(a)). One
can see that an unbiased (pressure or temperature in-
duced) S-T transition would be represented in Fig. 4(a)
simultaneously by two paths: S → T<sub>1</sub> and S → T<sub>2</sub>. Both
paths describe pure shear deformations traversing con-
figurations with rhombic symmetry; while those are iso-
choric projections, the actual MD transition is also ac-
companied by a volumetric contraction, see [80]. If we
advance from **C**-space to the larger **F**-space, even more
complex picture emerges, see Fig. 4(b). Thus, the “de-
formation variants” in **C**-space multiply as “orientation
variants” in **F**-space where the same deformation can
correspond to several different orientations of the basis
vectors.

More specifically, observe first that the stretch tensor
**U**<sub>◦</sub>(λ) = √**C**<sub>◦</sub> along the path S → T<sub>1</sub> can be written in
<figure><img src="image_4.png" /><figcaption>FIG. 4. (a) A fragment of the configurational space of metric tensors C showing the original square lattice (point S) and two equivalent versions of the triangular lattice (points T$_{1}$ and T$_{2}$). Solid arrows are directed along the two symmetric pure shear (rhombic) paths S → T$_{1}$ and S → T$_{2}$; the elastic domain is shown in pink; (b) A schematic structure of the four equivalent transformation paths in the extended configurational space of metric tensors F emphasizing rotations needed to recover the geometric compatibility with the square phase.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIG. 5. In (a), (d) we depict the MD simulated fields F$_{12}$ and F$_{21}$ in the transformed triangular phase. Zoomed-in views of fragments from (a,d) are presented in (b,e) where we see the corresponding atomic configurations visualized using OVITO [114]. In (c,f) we show triangulation representations corresponding to zoomed-in fragments from (b,e). In (g,h) we identify the corresponding variants T$_{1,2}^{\pm}$.</figcaption></figure>

the form [113]

$$ \mathbf{U}_{\diamond}(\lambda) = \frac{1}{2} \begin{bmatrix} \lambda + \frac{1}{\lambda} & \lambda - \frac{1}{\lambda} \\ \lambda - \frac{1}{\lambda} & \lambda + \frac{1}{\lambda} \end{bmatrix}, \quad (1) $$

where $\lambda = 1$ at the square phase S and $\lambda = \lambda_{*} = 3^{1/4}$ at the triangular phase T$_{1}$; along the synchronous path S → T$_{2}$ the stretch tensor is $\mathbf{U}_{\diamond}(1/\lambda)$. Note next, that the mappings $\mathbf{U}_{1} = \mathbf{U}_{\diamond}(\lambda_{*})$ and $\mathbf{U}_{2} = \mathbf{U}_{\diamond}(\lambda_{*}^{-1})$, describe area preserving stretching along two opposite diagonals of a square lattice cell with one of the diagonals becoming longer than the other. These mappings, however, do not fully characterize the complete S → T transition because the underlying rigid rotation remains unspecified. For instance, to ensure geometrically compatible lattice orientations, a clockwise rotation $\mathbf{R}^{+}(\vartheta)$ and anti clockwise rotation $\mathbf{R}^{-}(\vartheta)$ with $\vartheta = \pm 15^{\circ}$ have to be added to $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$. Such specification of rotations produces four equivalent triangular lattices $T_{1}^{+}$, $T_{1}^{-}$, $T_{2}^{+}$ and $T_{2}^{-}$ shown schematically in Fig. 4(b). The corresponding deformation gradients are $\mathbf{F}_{1,2}^{\pm} = \mathbf{R}_{1,2}^{\pm}\mathbf{U}_{1,2}$, where, for instance,

$$ \mathbf{R}_{1}^{+} = \frac{1}{\sqrt{\cosh \alpha}} \begin{bmatrix} \cosh(\alpha/2) & \sinh(\alpha/2) \\ -\sinh(\alpha/2) & \cosh(\alpha/2) \end{bmatrix}, \quad (2) $$

with $\alpha = 2 \ln \lambda_{*}$: the rotation aligns in this case the basis vector $\mathbf{e}_{1}$ (already rotated by the mapping $\mathbf{U}_{\diamond}(\lambda_{*})$) with the horizontal direction.

In Fig. 5(a,d) we show that all four variants $T_{1,2}^{\pm}$ have been indeed observed in our MD experiments. One can see that the two representations of the same atomic configuration, reached at the end of the S-T transformation, feature alternating rows/columns of positive and negative components of the deformation gradients, $F_{12}^{\pm}$ and $F_{21}^{\pm}$, see Fig. 5(b,e). In other words, we observe mixtures of alternating states, either $T_{1}^{+}$ and $T_{2}^{-}$, in Fig.

$5(\text{g})$, or $T_{1}^{-}$ and $T_{2}^{+}$, in Fig. 5(h). In Fig. 5(c,f) we illustrate the associated Delaunay triangulations visualizing the non-affine deformation behind the apparently homogeneous grains. The parallel numerical experiments using athermal molecular statics fully corroborate these rather remarkable observations, see [80] for details.

The main conclusion is that the standard representation of MD data, showing in the case of the S-T transition a polycrystal with mis-oriented homogeneous grains, is deceptive. Instead, our new way of representing MD data reveals crystallographically specific nano-twinning disguised as rigid rotations. Given that the underlying antiparallel atomic displacements correspond exactly to lattice invariant shears, it is natural to interpret the resulting pattern as representing alternating plastic slips. The emerging depiction of the transformation path complements and broadens previous studies of the mechanism of the S-T transition [1, 2, 18, 51, 53, 62, 81, 83, 115–120]. The apparently overlooked peculiarity of the S-T transition is the possibility to compose elementary variants of the product phase at the atomic level to obtain the final configuration with no overall shear deformation and no ‘surface’ energy penalty inside any of the grains.

Additional insights can be obtained if we present the obtained strain distribution against the energy landscape $\phi(\mathbf{C})$ in the C-space. To construct such a landscape it is sufficient to apply homogeneous deformation C to a mesoscopic set of atoms, while accounting for all pairwise interactions, and then use the Cauchy-Born rule [121, 122] and write $\phi(\mathbf{C}) = \frac{1}{2\Omega} \sum_{\mathbf{x}} \sum_{\mathbf{x}^{c} \in \mathcal{N}(\mathbf{x})} V(\sqrt{R_{i}C_{ij}R_{j}})$, where $R_{i}$ are the vectors representing reference points and the internal summations extend over all points $\mathbf{x}^{c}$ belonging to the cut-off neighborhood $\mathcal{N}(\mathbf{x})$. The globally symmetric potential $\phi(\mathbf{C})$ emerges if we extend by GL(2, $\mathbb{Z}$) periodicity the results of computations
<figure><img src="image_6.png" /></figure>

FIG. 6. Gradual progression of the S→T transformation in C-space at three different stages corresponding to points (1-3) indicated in Fig. 1(b). Fragments of the initial square and the transformed rhombic and triangular configurations are shown in the insets. The energy landscape is visible at the background.

performed in a single elastic domain.

In Fig. 6, we show a fragment of the computed energy landscape around the reference configuration S which includes the two target configurations $T_1$ and $T_2$. We mapped into the same C-space the atomic strains from our MD simulations while showing separately the three stages (1-3) of the S-T transition indicated in Fig. 1(b). Specifically, Fig. 6(a) shows the beginning of the transformation when all atomic strains populate the marginally stable square configuration S located at the origin. At the intermediate stage of the transformation, shown in Fig. 6(b), we observe spreading of atomic strains along the two symmetric rhombic (pure shear) paths towards two energy wells representing the variants $T_1$ and $T_2$. Note that both paths are pursued simultaneously and, as a result, the transformed triangular phase emerges as comprised of strains populating both energy wells $T_1$ and $T_2$, see Fig. 6(c). A broader picture, representing also a small number of elements which end up in distant energy wells, can be found in [80].

Further analysis of the stage-by-stage transformation process in the physical space, see Fig. 7(a), shows that the alternating micro-slips (represented by oscillatory fields $F_{12}$ or $F_{21}$) develop layer-by-layer through back and forth transverse propagation of Shockley partials; the possibility of such nano-scale ‘zipping’ by moving surface steps is well known, see for instance [123]. At larger scale the transformation will appear as proceeding through front propagation which leaves behind a pattern of anti-parallel displacements disguised as a rigid rotation of a perfect triangular lattice, see Fig. 7(b–e).

Using the constructed multi-well periodic potential $\phi(C)$ we can now develop a coarse grained description of the system. The simplest way to obtain a regularized theory is to assume that such potential describes mechanical response of elastic finite elements whose size introduces a mesoscopic cut-off spatial scale [72, 73, 75, 121]. The piece-wise affine deformation of the elements can be then presented in the form $y(x) = y_{ij}N_{ij}(x)$, where $y_{ij}$

<figure><img src="image_7.png" /></figure>

FIG. 7. (a) Layer-by-layer propagation of the deformation gradient (here only $F_{21}$ component) during S-T transformation; (b-e) transient spatial configurations of the deformation gradient showing the formation of an ideally periodic pattern.

is the deformation of the 2D network of discrete nodes and $N_{ij}(x)$ are the linear shape functions. The problem reduces to finite dimensional parametric minimization of the energy functional $W = \int_{\Omega_0} \phi(C)d\Omega_0$, where $\Omega_0$ is the computational domain, see [80] for details. The outcome of such mesoscopic modeling of the S-T transformation process is illustrated in Fig. 8. As in our molecular simulations we started with a perfect square lattice brought to a marginally stable state. In the emerging polycrystalline configuration, see Fig. 2, we again observe a texture of triangular grains with mis-orientation of 30°. The boundaries of the grains are similarly dislocation-rich even if now the dislocation cores are blurred at the cut-off scale of elastic finite elements. The transformation is again advancing along two concurrent rhombic (pure shear) paths with eventual lock-in on the higher symmetry configurations $T_1$ and $T_2$ [80]. In the final configuration, the apparent rotations are again achieved through alternating crystallographically specific slips inside the adjacent planes with the formation of the same two types of variant mixtures ($T_1^+, T_2^-$) and ($T_1^-, T_2^+$), see Fig. 8.

<figure><img src="image_8.png" /></figure>

FIG. 8. Outcome of the coarse grained modeling of S-T transition: (a) The post-transformation polycrystalline texture; (b-c) two fragments of mis-oriented triangular grains, with (b) presenting ($T_1^+, T_2^-$) variant mixture and (c) representing ($T_1^-, T_2^+$) variant mixture.

We can now try to draw some parallels between the observed mechanism of the S-T transition in 2D and, for instance, the mechanism of the reconstructive BCC-HCP transition in 3D [85, 124]. While the latter involves volume preserving pure shear deformation in addition to shuffling, those two phenomena appear to be well separated in time and therefore can be decoupled [125]. We
can then, following [18], neglect the Landau-type compo-
nent of the transformation by associating the primary or-
der parameter with the shuffle. Note that a shuffle would
have naturally emerged in S-T transition if in our recon-
struction of atomistic deformation gradients we had used
a double unit cell [126]. Usually the BCC-HCP shuffle is
perceived as proceeding via softening of an optical mode
with the formation of an intermediate orthorhombic con-
figuration with an idea that such lowering of symmetry is
maintained until the system locks-in in the higher sym-
metry configuration [127]. Our analysis suggests that
it can be viewed, in the first approximation, as a pat-
tern of anti-parallel shifts of consecutive planes inside
a single unit-cell (probably with some homogeneous ad-
justment layer-wise). The large transformation strain in
the lock-in conditions drives the scale of such apparent
micro-twinning to atomic dimensions.

The proposed analogy should be viewed with caution
as the BCC-HCP transition in 3D is still different from
the S-T transition in 2D. Thus, it is not clear whether
the experimentally confirmed path for BCC-HCP tran-
sition [17] can be decomposed into full plastic slips or
instead represented by alternating stacking faults result-
ing from the passage of only partial dislocations. Thus,
the reconstructive FCC-HCP transition appear to be an
example of the latter possibility as in this case the HCP
phase emerges from an anti-parallel coordinated gliding
of Shockley partials [38–40, 128, 129]. In the setting of
S-T transformation the implied nano-scale stacking fault
laminate [33, 48] would correspond to the layering of the
type T₁-S-T₂. Since in our case the *S* phase is unstable,
such ‘partially’ plastic laminates are not observed with
partials appearing only transiently as it is clear from our
Fig. 7(a).

To conclude, we showed that tracking the history of
atomic-scale metric tensors in MD simulations offers a
unique perspective on the intricate micro-pattern forma-
tion during reconstructive phase transitions. The pro-
posed interpretation of MD numerical experiments re-
veals previously hidden details of the deformation paths
allowing one to analyze systematically the underlying re-
lation between elastic and inelastic modes. The micro-
mechanism, shown to be operative during the prototypi-
cal S-T transition, contains some generic elements which
are expected to be common for most reconstructive tran-
sitions including the iconic BCC-HCP and FCC-HCP
transitions.

Acknowledgments. The authors are grateful to D. Gra-
tias, C. Denoual and Y.P. Pellegrini for helpful sugges-
tions in the course of this work. K.G and O.U.S. were
supported by Grant ANR-19-CE08-0010. L.T. acknowl-
edges the support of the Grants ANR-17-CE08-0047-02,
ANR-21-CE08-MESOCRYSP and ERC-H2020-MSCA-
RISE-2020-101008140.

* kanka.ghosh@lsqm.cnrs.fr

† lev.truskinovsky@espci.fr

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Slip-dominated structural transitions:
Supplemental Material

Kanka Ghosh,<fnref n="1" />,* Oguz Umut Salman,<fnref n="1" /> Sylvain Queyreau,<fnref n="1" /> and Lev Truskinovsky<fnref n="2" />,†

SIMULATION DETAILS

We model phase transition under hydrostatic pressure from a square lattice (S, plane space group p4mm) to a close-packed hexagonal lattice, interpreted here triangular lattice (T, plane space group p6mm).

*MD simulations.* We used Boyer potential presented in the main paper with $c_1 = c_2 = 2$ and $b_1 = b_2 = 8$ as in [1]. Then the choice $r_2/r_1=1.425$ (where $r_2$ and $r_1$ are second nearest neighbor distance and lattice parameter respectively) ensures that a square lattice with a lattice constant of 1.0659 Å is the ground state. We simulated $10^4$ atoms with periodic boundary conditions (PBC) and used a cutoff distance $r_c = 2.5$ Å. First the square crystal is equilibrated within NVT ensemble which takes $10^5$ time steps with the step size $\Delta t = 0.0001$ ps. Then, the pressure control protocol is implemented within isothermal-isobaric (NPT) ensemble till the square phase is *marginalized*. A broad range of temperatures (10 K - 900 K) and pressures ($0.1 \times 10^7$ bar - $2.4 \times 10^7$ bar) was considered to construct the T-P phase diagram presented in the main text. We used velocity Verlet algorithm and performed $10^7$ MD steps (= 1 ns) to study the S-T transition at T = 10 K to avoid melting.

*MS simulations.* We used an athermal molecular statics protocol which implies that the effects of finite temperature were negligible. PBC were maintained throughout the simulation. Initially, a stable planar square crystal was prepared with $10^4$ atoms using the same potential as in our MD simulations [1]. However, due to the fact that in athermal molecular statics thermodynamic pressure is an ambiguous concept and only virial pressure can be computed confidently, we had change the potential parameter $r_2/r_1$ from 1.425 to 1.3 to *marginalize* the square lattice while stabilizing triangular lattice. To induce the S-T transition we could then introduce a small disturbance by displacing all the atoms at random distances (about 0.9 % of the lattice parameter) along both *x* and *y* directions. The conjugate gradient algorithm was employed to conduct energy minimization.

*Mesoscopic simulations.* In our mesoscopic simulations we again used the same potential with square phase *marginalized* at the value of parameter $r_2/r_1 = 1.3$. Within the framework of our mesoscopic tensorial model (MTM), the globally periodic Landau potential $\phi(\mathbf{C})$ was

constructed using the Cauchy-Born rule from the same interatomic potential as in our MD and MS numerical experiments [1].

The energy was extended by symmetry beyond the minimal periodicity domain. It implies the representation of an arbitrary metric tensor **C** in the form $\tilde{\mathbf{C}} = \mathbf{m}^T \mathbf{C} \mathbf{m}$ where **m** is a unimodular integer-valued matrix and $\tilde{\mathbf{C}}$ is an image of **C** inside the minimal periodicity domain in the sense that $\phi(\tilde{\mathbf{C}}) = \phi(\mathbf{C})$.

In Figure S1 we illustrate the **C**-space (Poincaré disk, see the detailed explanations below) with the superimposed computed strain energy landscape for three values of the parameter $r_2/r_1$ as it gradually changes from 1.425 to 1.3. One can see that the energy minima (indicated via dark blue) eventually shift from square configuration (see ‘S’ in Fig. S1(a)) to triangular configuration (see equivalent triangular minima ‘T$_1$’ and ‘T$_2$’ in Fig. S1(c)).

<figure><img src="image_9.png" /><figcaption>FIG. S1: Strain energy landscape (color coded) in the con-figurational **C**-space (Poincaré disk) at three different values of the Boyer parameter $r_2/r_1$. Letters indicate the square (S) and the two relevant triangular minima ($T_1$, $T_2$). The colors represent the strain energy level.</figcaption></figure>

We associated the elastic energy $\phi(\mathbf{C})$ with the node **x** assuming that $\mathbf{C} = \nabla\mathbf{y}^T\nabla\mathbf{y}$. The piecewise linear deformation field $\mathbf{y}(\mathbf{x})$ was discretized using shape functions as described in the main text which turned the problem into finite dimensional, parameterized by displacements of the 2D network of discrete nodes identified by their integer-valued coordinates *ij*. Parametric minimization of the discretized energy functional $W = \int_{\Omega_0} \phi d\Omega_0$ was accomplished using a variant of conjugate gradient optimization known as the L-BFGS algorithm [2]. This algorithm seeks solutions to the equilibrium equations $\partial W/\partial \mathbf{u}_{ij} = \int_{\Omega_0} \mathbf{P} \nabla N_{ij} d\Omega_0 = 0$, where $\mathbf{P} = \partial \phi/\partial \nabla \mathbf{y}$, $\mathbf{u}_{ij}$ denote the values of displacement at node *ij*, $N_{ij}$ is the shape function at node *ij* and $\Omega_0$ is defined as the computational domain. In our numerical experiments we used (discretized) free boundary conditions $\mathbf{P} \cdot \mathbf{N} = 0$,

<footnote n="1">¹CNRS, LSPM UPR 3407, Université Sorbonne Paris Nord, 93430 Villetaneuse, France</footnote>

<footnote n="2">²PMMH, CNRS UMR 7636 ESPCI ParisTech, 10 Rue Vauquelin,75005 Paris, France</footnote>
where **N** is the normal to the surface at the reference state. The S-T transformation was again initiated by incrementally changing the potential parameter r₂/r₁ from the value 1.425 to the value 1.3. This ensured the eventual shift in the nature of the ground state configuration from square to triangular.

## DEFORMATION GRADIENT

In Figure S2, we show the mapping which schematically describes the deformation of an atomic neighborhood.

<figure><img src="image_10.png" /><figcaption>FIG. S2: The schematic description of the deformation of the chosen 'atomic neighborhood'; (a), (b) show the reference and the deformed states, respectively.</figcaption></figure>

Suppose that in a two-dimensional lattice a reference point is represented by a vector **X** = {X₁, X₂} while its deformed position is represented by the vector **x** = {x₁, x₂}. Then, the deformation gradient is Fᵢⱼ = ∂xᵢ/∂Xⱼ, where the indexes i, j refer to deformed and reference coordinate systems, respectively. If **R**<sup>αβ</sup> and **r**<sup>αβ</sup> are the vectors connecting atom α with its neighbors (β or β₁, β₂, ..., βₙ for n neighbors) in the reference and in the actual configurations respectively, the approximate deformation gradient obtained by minimizing mean-square difference between the actual displacements of the neighboring atoms relative to the chosen central atom and the relative displacements, that they would have had if they were in a region of uniform strain, can be written as F<sup>α</sup><sub>ij</sub> = ω<sup>α</sup><sub>ij</sub> (η<sup>α</sup><sub>ij</sub>)<sup>-1</sup>, where ω<sup>α</sup><sub>ij</sub> = Σ<sup>n</sup><sub>β=1</sub> r<sup>αβ</sup><sub>i</sub> R<sup>αβ</sup><sub>j</sub> and η<sup>α</sup><sub>ij</sub> = Σ<sup>n</sup><sub>β=1</sub> R<sup>αβ</sup><sub>i</sub> R<sup>αβ</sup><sub>j</sub> [3]. As a representative atomic neighborhood we found sufficient to choose two non collinear nearest neighbors and one of the second nearest neighbors as the averaging domain, see Fig. S2. In other words, the chosen atomic neighborhood of a particle is made of two complementary triangular domains. Atomistic strain tensor **C** is recovered from the deformation gradient via the formula **C** = **F**<sup>T</sup>**F**.

## LATTICE INVARIANT SHEARS

Lattice invariant shear is a crystallographically specific affine volume preserving deformation which transforms an infinite lattice into itself. General lattice invariant shears are described by the deformation gradients

represented by the integer valued unimodular matrices {**m**, m<sub>IJ</sub> ∈ Z, det(**m**) = 1}, see [4, 5] for the details.

For instance, consider a square lattice (S) which is homogeneously deformed with the deformation gradient given by the formula **F**(α) = 1 + α **e**₁ ⊗ **e**₂, where α denotes the amplitude of shear, **e**₁, **e**₂ represent unit base vectors along x and y directions, respectively. Such deformation describes one parametric family of simple shears.

<figure><img src="image_11.png" /><figcaption>FIG. S3: (a) Schematic representation of the strain energy φ(**C**) as a function of a simple shear amplitude α (see the text). The insets schematically depict the equivalent square lattice configurations S₂, S and S₁. (b) Schematic presentation of an edge dislocation.</figcaption></figure>

Our Fig. S3(a) presents schematic structure of the strain energy landscape φ(**C**) as a function of shear amplitude α. As one increases the parameter α from zero, which corresponds to a minimum of the energy, the function φ(α) first reaches its maximum but then decreases reaching again a symmetry related minimum at α = 1. At this point an equivalent lattice is obtained. Similarly, negative increment of α transforms at α = -1 the original lattice configuration into yet another symmetric configuration with exactly the same energy. The insets of Fig. S3(a) schematically describe three equivalent lattice configuration S, S₁ and S₂ corresponding to three equivalent energy minima. Note that if we remove the bonds in those insets and leave only atoms, the corresponding three atomic configurations will be indistinguishable. Furthermore, our Fig. S3(b) presents a schematic structure of an edge dislocation which can be viewed as a 'domain wall' between a sheared (S₁ in this case) and undeformed (S) 'phases'.

An image of the same edge dislocation in the Poincaré disk (see the definition below) with the superimposed GL(2, Z) invariant energy landscape is shown in Fig. S4. Here our simple shear parametrized by α is represented by a curve which contains equivalent square lattices S, S₁ and S₂ related via 'lattice invariant simple shears'. Similarly, the equivalent triangular lattices T₁ and T₂ are related via two symmetric 'lattice invariant simple shears' only one of which is shown in Fig. S4(b).
<figure><img src="image_12.png" /><figcaption>FIG. S4: (a) Schematic presentation of an edge dislocation linking equivalent square lattices on a Poincaré disk. (b) Similar edge dislocation in the case of triangular lattice. Dashed lines represent examples of lattice invariant simple shear paths. The colors represent the strain energy level.</figcaption></figure>

## CONFIGURATIONAL SPACE

The 3D space of symmetric metric tensors with co-ordinates $(C_{11}, C_{22}, C_{12})$ contains a subspace describing isochoric deformations and selected by the condition $C_{11}C_{22} - C_{12}^2 = 1$. This is our 2D configurational space. The action of the infinite discrete symmetry group $GL(2, \mathbb{Z})$ naturally tessellates the configurational space into periodicity domains. A convenient approach to visualize globally the subspace of metric tensors $\mathbf{C}$ with $\det(\mathbf{C}) = 1$ is to use a stereographic projection of the corresponding hyperbolic surface in the $(C_{11}, C_{22}, C_{12})$ space onto a unit disk, which is known as the Poincaré disk, see Fig. S5.

The geometrically minimal periodicity domain in the configurational space, known as the fundamental domain is $\mathcal{D} = \{2C_{12} \le \min(C_{11}, C_{22})\}$, see for instance a dark gray triangular area in Fig. 3(a) in the main text; it corresponds to the ‘minimal’ choice for the lattice vectors $\tilde{e}_1, \tilde{e}_2$, selected by the Lagrange reduction algorithm [6–13]. The three boundaries of the fundamental domain $\mathcal{D}$ can be presented explicitly in the parametric form:

$$ \mathbf{C} = \begin{pmatrix} \alpha^2 & 0 \\ 0 & \frac{1}{\alpha^2} \end{pmatrix}, 0 < \alpha \le 1, \quad (1) $$

$$ \mathbf{C} = \begin{pmatrix} \alpha^2/4 + 1/\alpha^2 & -\alpha^2/4 + 1/\alpha^2 \\ -\alpha^2/4 + 1/\alpha^2 & \alpha^2/4 + 1/\alpha^2 \end{pmatrix}, 0 < \alpha \le \sqrt{2} \quad (2) $$

and

$$ \mathbf{C} = \begin{pmatrix} \alpha^2 & \alpha^2/2 \\ \alpha^2/2 & \alpha^2/4 + 1/\alpha^2 \end{pmatrix}, 0 < \alpha \le \gamma. \quad (3) $$

The $GL(2, \mathbb{Z})$ copies (replicas) of these boundaries, constitute the tessellation of the configurational space, represented in Fig. S5 by the thin black lines which are divided (artificially) into solid and dashed ones for easier identification. To construct such copies one needs

to apply to (1-3) the group action $\tilde{\mathbf{C}} = \mathbf{m}^T \mathbf{C} \mathbf{m}$, where $\{\mathbf{m}, m_{IJ} \in \mathbb{Z}, \det(\mathbf{m}) = \pm 1\}$.

The elastic domain, which would describe fully the elastic response of a single-well material as well as the behavior of a solid undergoing Landau type structural phase transition, is also known in the literature as the maximal Ericksen-Pitteri neighborhood [6, 10, 11]. It can be obtained from $\mathcal{D}$ by applying discrete transformations forming the crystallographic point group $P(\mathbf{e}_1)$, which contains only rigid rotations and is used to characterize material symmetries within classical continuum elasticity [6, 11].

<figure><img src="image_13.png" /><figcaption>FIG. S5: (a) Three dimensional hyperbolic surface $C_{11}C_{22} - C_{12}^2 = 1$ in the configurational space of metric tensors $C_{11}, C_{22}, C_{12}$ projected onto a Poincaré disk: the points A, B on such a surface are mapped to the points $A_1, B_1$ on the disk. (b) Poincaré disk: thin lines indicate the boundaries of minimal periodicity domains; points on the disk describing equivalent square and triangular lattices are marked by black squares and red triangles, respectively.</figcaption></figure>

In the main paper we presented a structure of the $GL(2, \mathbb{Z})$ tessellation of the Poincaré disk in the two cases: when the reference state corresponds to a square lattice S or to a triangular lattice $T_1$.

To obtain such a representation in the case when the reference state is a square lattice, we first define the corresponding reference basis $\mathbf{e}_1 = \{1, 0\}, \mathbf{e}_2 = \{0, 1\}$. The deformed basis is then $\mathbf{f}_i = \mathbf{Fe}_i$, where $i = 1, 2$ and $\mathbf{F}$ is the deformation gradient. Under the assumption that $\det \mathbf{F} = 1$ we can introduce metric tensors $\mathbf{C} = \mathbf{F}^T \mathbf{F}$ describing different states in our configurational space. These metric tensors are then projected onto the Poincaré disk using the rectangular coordinates on the disk: $x = t(C_{11} - C_{22})/2, y = tC_{12}$, where $t = 2(2 + C_{11} + C_{22})^{-1}$, see Fig. S5.

To obtain a similar representation in the case when the reference state is a triangular lattice, we first write the basis vectors of the triangular lattice $T_1$ in the coordinates of the basis of the square lattice S: $\mathbf{h}_1 = \{\gamma, 0\}$ and $\mathbf{h}_2 = \{\gamma/2, \gamma\sqrt{3}/2\}$, where $\gamma = (4/3)^{1/4}$. We then introduce a matrix $\mathbf{H}$ whose columns are the vectors $\mathbf{h}_{1,2}$. Since $\mathbf{h}_j = \mathbf{He}_j$ the metric tensor in this new (triangular lattice) basis takes the form $\mathbf{C}' = \mathbf{H}^{-T} \mathbf{C} \mathbf{H}^{-1}$ where $\mathbf{C}$ is the metric tensor in the old (square lattice) basis. Now the components of the tensor $\mathbf{C}'$ can be stereographically
projected into the Poincaré disk using the same mapping
as before which gives rise to a tessellation presented in
Fig. 3(b) in the main text. While in both cases the fun-
damental domains $\mathcal{D}$ have the same triangular shape, the
elastic domains are different as the point group involves
four rotations when the reference lattice is square and six
rotations when it is triangular, see the light gray areas in
Fig. 3(a,b) in the main text.

DIRECT TRANSITION S→T

*Molecular dynamics.* We begin with highlighting that our Fig. 2 and Fig. 5 in the main text represent the outcome of the same MD numerical experiment, see their direct juxtaposition in Fig. S6. Both figures show the same transformed triangular crystal with the only difference is that in Fig. 2 atoms are colored according to their potential energies per atom, see the reproduction in Fig. S6(a), whereas in Fig. 5, which we present now as two separate figures Fig. S6 (b,c), atoms inside two different triangular phase grains are colored according to the value of the particular components of the deformation gradient: F₁₂ in Fig. S6(b) and F₂₁ in Fig. S6(c). More specifically, the fragment A in Fig. S6(a) is shown in Fig. S6(b) emphasizing the component F₁₂ because it presents an example of a horizontal micro-layering (crystallographically specific finite shuffling). Instead, the fragment B in Fig. S6(a) is represented in Fig. S6(c) emphasizing the component F₂₁ as the micro-layering in this case is vertical. Note that the selective coloring of the grains in Fig. S6(b,c) is chosen for convenience, in particular the gray areas there do not indicate the presence of the square phase which has completely disappeared in the shown product (triangular) phase: the gray areas in Fig. S6(b,c) indicate instead that either F₁₂=0 or F₂₁ = 0 in the corresponding regions.

<figure><img src="image_14.png" /><figcaption>FIG. S6: Transformed triangular polycrystal in MD
numerical experiment with atoms colored using: (a)
potential energies per atom, (b) F<sub>12</sub> and (c) F<sub>21</sub>. Two
fragments A and B are also shown as insets.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>FIG. S7: The S-T transition in the C-space. (a) 3D his-
togram representation of the strain distribution in triangular
phase; note that the ‘counts’ axis has a logarithmic scale.
(b) 2D histogram representation of the same strain distribu-
tions shown against the corresponding energy landscape. (c)
A highly deformed atomic fragment around the grain bound-
ary; both deformation gradient distribution and the deformed
triangulation network are shown in the two insets. (d) The
distribution of detC in the initial square (red) and the final
triangular lattices (blue).</figcaption></figure>

In MD simulations the distribution of the values of
**C** over the atoms representing individual neighborhoods
evolves during the S-T transition. The transformation
starts when the values of **C** in all nodes are the same
which means that the distribution is fully localized at
the origin representing unstressed square lattice S. At the
end of the transformation when the T phase is nominally
reached, the configurational points spread in the **C**-space.

Our Fig. S7 illustrates the final strain distribution
projected on detC =1 surface. The three dimensional
histogram representation of the strain distribution, see
Fig. S7(a), indicates that most of the elements are in
either T₁ or T₂ energy wells. This is also seen in 2D his-
togram of the same atomistic strains shown in the Fig.
S7(b). Note however, that in addition to the most popu-
lated T₁ and T₂ triangular configurations, several other
locations outside the T₁ and T₂ energy wells within the
C-space are also occupied. Some of them correspond to
grain boundaries like the one in the fragment shown in
Fig. S7(c). In Fig. S7(d) we illustrate the fact that the
S-T transformation is accompanied by a volumetric con-
traction. Specifically, the original S phase with det C =
1 transforms into the final T phase with det C = 0.55.

*Molecular statics.* Simulations of S-T transition using
athermal molecular statics (MS) exhibited all the main
ingredients of the transformation mechanism observed
in MD experiments. In particular, our Fig. S8(a–h)
shows that the lattice scale alternate plastic slips involv-
ing both, atomic rows and atomic columns, are recovered.
Thus, we observe in neighboring grains the same alter-
nating mixtures of configurations T₁⁺, T₂⁻ (realized via
alternating F₁₂⁺) and of configurations T₁⁻, T₂⁺ (realized
via alternating $F_{21}^{\pm}$), see Fig. S8(g,h). The evolution of the strain populations inside the C-space indicates the same mechanism involving concurrent symmetric pure shears. Specifically, our Fig. S8(i–k) shows that the atomic strains spread via rhombic valleys towards the triangular energy minima $T_1$ and $T_2$.

<figure><img src="image_16.png" /><figcaption>FIG. S8: (a), (d) The distribution of deformation gradient components $F_{12}$ and $F_{21}$ in the transformed triangular phase obtained in MS simulations. (i)–(k) Evolution of the atomistic strain distribution in the C-space.</figcaption></figure>

A minor difference is that in our MS experiments the S-T transformation proceeded in almost isochoric conditions. Indeed, due to lack of thermodynamic control parameters in the athermal MS protocol, we had to induce the S-T transition by changing the potential parameter $r_2/r_1$ from 1.425 to 1.3. To capture the structural transformation, radial distribution function $g(r) = \frac{1}{\pi r^2 N \rho} \sum_{i=1}^{N} \sum_{j \neq i}^{N} \langle \delta(\mathbf{r} - |\mathbf{r}_j - \mathbf{r}_i|) \rangle$ is calculated for our 2D crystal, where $\rho = N/A$ is the density, $N$ is the total number of atoms and $A$ is the area of the system. The distance between the atoms $i$ and $j$ is represented by the difference $\mathbf{r}_j - \mathbf{r}_i$. As a result of the athermal S-T transition, the lattice constant changed from 1.0659 Å ($r_0$ in Fig. S9(a)) in the ‘S’ phase to 1.14 Å ($r_{eq}$ in Fig. S9(b)) in the ‘T’ phase, see Fig. S9(a), (b). The corresponding areas for the square unit cell is $r_0^2 = 1.1361$ Å$^2$ and for the triangular lattice (rhombic cell) is $\frac{\sqrt{3}}{2} r_{eq}^2 = 1.125$ Å$^2$. The ratio of the unit cells is then $\frac{\text{area}\square}{\text{area}\triangle} = 1.009$. Therefore, the unit cell areas of the initial ‘S’ phase and the transformed ‘T’ phase are almost identical in MS numerical experiments.

However, it is known that the S-T transition must necessarily come with the increase of the packing fraction as, for instance, triangular/hexagonal lattice is closed-packed while its square counterpart is not. Specifically, the packing fraction, defined as the ratio of the area occupied by the circles inside the unit cell and that of the area of the unit cell, in ‘S’ phase is $\pi a^2 / a^2 = \pi/4 \approx 0.785$ while in ‘T’ phase it is $\frac{1}{2} \pi \frac{a^2}{4} / \sqrt{3} \frac{a^2}{4} = \frac{\pi}{2\sqrt{3}} \approx 0.907$, see the distribution of the coordination numbers $n_c$ in Figure S9(c).

<figure><img src="image_17.png" /><figcaption>FIG. S9: Radial distribution functions $g(r)$ for the initial square lattice (a) and for the transformed triangular crystal (b). In (c) we show the variation during the S-T transition of the fractions of atoms ($N/N_{tot}$) characterized by different coordination ($n_c$). A fragment of the misoriented triangular grains with atoms colored according to their coordination is shown in (d).</figcaption></figure>

Therefore, in view of the similarities of the parameters of the unit cells, the S-T transition should result in the formation of ‘void space’ that is realized through the formation of less dense grain boundaries, see the fragment shown in Fig. S9(d). In particular, we see that the atoms constituting grain boundaries with $n_c = 5, 7$ are differently coordinated vis a vis atoms inside the triangular grains where $n_c = 6$. Our numerical experiments show that around 5 % of atoms constitute effectively loose grain boundaries against 95 % of atoms inside the close packed grains.

**Mesoscopic tensorial model.** The finite element simulation of the S-T transition started again with a finite element configuration describing a marginally stable phase S. The mechanism of the transformation into a T phase followed closely what we have already seen in our MD and MS simulations. For example, in Fig. S10 we show two snapshots of the distribution of MTM strains in C-space. Similar to what we have seen in our atomistic

<figure><img src="image_18.png" /><figcaption>FIG. S10: S-T transformation in MTM numerical experiments: (a,b) fragments of the C-space illustrating two successive stages of S-T transition; (c) the histogram showing the distribution of the values of C in the final state.</figcaption></figure>
simulations, the transformed triangular lattice evolves
as a mixture of two triangular configurations $T_1$ and $T_2$.
Thus, we again observed alternating layer-wise propaga-
tion (both horizontal and vertical) of triangular phase.
These results show that within the coarse-grained de-
scription the front-propagation-based mechanism of S-T
reconstructive transition producing micro-plastically de-
formed triangular phase is preserved.

REVERSE TRANSITION T→S

Even though in much less detail, we have also per-
formed numerical experiments involving the reverse tran-
sition from triangular to square phase. More specifically,
we performed cyclic loading producing a hysteresis loop
and exhibiting a succession of two S→T→S reconstruc-
tive transitions.

<figure><img src="image_19.png" /></figure>

FIG. S11: S→T→S reconstructive transition in preliminary MD simulations. (a) P-V(area A) phase diagram of the complete compression-expansion cycle. With the letters S, T, and S₀ we denote : pristine square, transformed triangular lattice under compression and transformed square lattice after the subsequent expansion. Insets in (b), (c), and (d) show microscopic configurations with their associated multi-grain compositions.

In Fig. S11(a) we illustrate the hysteretic nature of
the square to triangular transition in MD by showing the
complete compression-expansion cycle on the P-V (area
A) plane. Specifically, as in our main discussion, we start
with a defect-free (pristine) 2D square crystal, see Fig.
S11(b), and employ isotropic compression (by increasing
hydrostatic pressure) to induce the transformation into a
triangular phase. The resulting configuration of misori-
ented triangular grains, see Fig. S11(c), is then subjected
to isotropic expansion via the reduction of pressure. This
brings the system back into the square phase which is
now also represented by a texture of misoriented square
grains, see Fig. S11(d). The insets reveal that as a re-
sult of the S→T→S cycle the original highly symmetric
'cold' square phase S develops into a multi-grain 'hot'
configuration S₀ which is obviously more generic and en-
tropically favored.

To corroborate these results, we also performed slightly

different athermal MS numerical experiments, see Fig.
S12. Here we studied the transition from a pristine tri-
angular phase to a multi-grain square phase by changing
the value of parameter r₂/r₁ from 1.3 to 1.425. As we
have already mentioned, this modifies the ground state
of the lattice from triangular to the square symmetry.
The resulting grain texture exhibits the same misorien-
tation angles as in our MD simulation and also reveals a
hidden alternating slip distribution behind the apparent
homogeneous rotations of the product phase (not shown
here). A systematic study of the S-T transition including
the comparison with the results of the MTM simulations
is underway and the discussion of the detailed structure
of the resulting hysteretic cycle is left for a separate study.

<figure><img src="image_20.png" /></figure>

FIG. S12: Preliminary modeling of T→S transition using athermal MS simulations: (a) pristine triangular lattice (T), (b) transformed square (S) lattice represented by a texture of misoriented grains. Fragments of ‘T’ and ‘S’ grains with their associated orientations are shown in the insets.

GRAIN MISORIENTATION

Here we clarify how the misorientation between the
grains was quantified in the main text. We focus then on
the direct S-T transition and deal with the grain struc-
ture inside the product triangular (hexagonal) phase.
Note first that the reference orientation of the triangular
basis is chosen in such a way that **e**₁ is horizontal and
**e**₂ forms with it a 60° angle, see Fig. S13. The mis-
orientation angle characterizes the rotation which relates
the basis vectors in two neighboring grains. Our Fig.
S13(a) shows a fragment of the product triangular lat-
tice featuring two grains separated by a grain boundary
and it is clear from this figure that the basis vectors in
the left grain need a 30° rotation to transform into the
basis vectors on the right.

However, as we explain in the main text, the same mis-
orientation angle can also be computed analytically. For
instance, we show below how one can evaluate the relative
rotation of the T<sub>1</sub><sup>+</sup> and T<sub>1</sub><sup>-</sup> components of the microtwin
patterns representing two neighboring triangular grains
shown in Fig. S13.
<figure><img src="image_21.png" /><figcaption>FIG. S13: (a) A fragment of the product polycrystalline tri- angular phase showing two misoriented grains separated by a grain boundary; (b) basis vector e₁ and e₂ in both grains. The identification of elementary cells in (b) shows that they are related via a 30° rotation.</figcaption></figure>

Recall that the coexisting variants T₁⁺ and T₁⁻ are ob- tained from the S phase via deformation gradients F₁⁺ = R₁⁺U₁, where

$$
\mathbf{U}_1 = \frac{1}{2} \left[ \begin{array}{cc} \lambda_* + \frac{1}{\lambda_*} & \lambda_* - \frac{1}{\lambda_*} \\ \lambda_* - \frac{1}{\lambda_*} & \lambda_* + \frac{1}{\lambda_*} \end{array} \right], \quad (4)
$$

with $\lambda_* = 3^{1/4}$ and

$$
\mathbf{R}_1^{\pm} = \frac{1}{\sqrt{\cosh \alpha}} \begin{bmatrix} \cosh(\alpha/2) & \pm \sinh(\alpha/2) \\ \mp \sinh(\alpha/2) & \cosh(\alpha/2) \end{bmatrix}, \quad (5)
$$

with $\alpha = 2 \ln(\lambda_*)$. The corresponding rotation angles are
$\vartheta_{\pm}$ where

$$
\vartheta_{\pm} = \sin^{-1} \left[ \pm \frac{\sinh(\alpha/2)}{\sqrt{\cosh(\alpha)}} \right] \frac{180^{\circ}}{\pi} = \pm 15^{\circ}, \quad (6)
$$

where clockwise and counter clockwise rotations are de-
noted via '+' and '-' respectively, see Fig. S13 (b). Note
that the rotations **R**<sub>1</sub><sup>+</sup>(**v**) and **R**<sub>1</sub><sup>-</sup>(**v**) align the basis vec-
tors **e**<sub>1</sub> and **e**<sub>2</sub> of the T<sub>1</sub> lattice with the horizontal and
vertical directions, respectively. Therefore, the misorien-
tation between the variants T<sub>1</sub><sup>+</sup> and T<sub>1</sub><sup>-</sup> is exactly 30° as
we have already seen in Fig. S13 (a).

Similarly, analysis for the coexisting variants T₂⁺ and
T₂⁻ shows that the misorientation between them is again
30°, see for instance Fig. 4(b) in the main text. We
reiterate that similar to variants T₁⁺ and T₁⁻, the variants
T₂⁺ and T₂⁻ reside in neighboring grains separated by the
same grain boundary.

<footnote>* kanka.ghosh@lspm.cnrs.fr</footnote>

<footnote>† lev.truskinovsky@espci.fr</footnote>

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