Grain boundary-induced martensitic transformations: A phase-field study of
nucleation, size-effect, triple junction-effect, microstructures, and
compatibility at the nanoscale

Anup Basak<EMAIL_ADDRESS>(corresponding author)
Department of Mechanical Engineering,
Indian Institute of Technology Tirupati,
Tirupati, Yerpedu, A.P. 517619, India.

An original thermodynamically consistent large strains-based multiphase phase-
field (PF) approach of Ginzburg-Landau type is developed for studying the
grain boundary (GB)-induced martensitic transformations (MTs) in
polycrystalline materials at the nanoscale considering the structural stresses
within the interfaces. In this general PF approach, $N$ independent order
parameters are used for describing the austenite ($\sf
A$)$\leftrightarrow$martensite ($\sf M$) transformations and $N(>1)$
martensitic variants, and another $M$ independent order parameters are
considered for describing $M(>1)$ grains in the polycrystalline samples. The
change in the GB energy due to its structural rearrangement during MTs is
considered using variable energy for the GB(s) as a function of the order
parameter related to the $\sf A\leftrightarrow\sf M$ transformation. A rich
plot for the temperatures of transformations between the $\sf A$,
premartensite, and $\sf M$ in a bicrystal with a symmetric planar tilt GB are
plotted for the varying austenitic GB width. The strong effects of the
parameters, including the austenitic GB width, change in GB energy due to MTs,
GB misorientation, applied strains, and sample size on heterogeneous
nucleation of the phases and the subsequent complex martensitic
microstructures evolution are explored in various bicrystals with symmetric or
asymmetric planar or circular tilt GBs during the forward and reverse
transformations. The triple junction (TJ) energy and the energy and width of
the adjacent GBs are also shown to strongly influence the nucleation and
microstructures using the tricrystals having three symmetric planar tilt GBs
meeting at $120^{\circ}$ dihedral angles. The compatibility of the
microstructures across the GBs is studied. The elastic and structural stresses
across the GBs and TJ regions are plotted, which is essential for
understanding the role of GBs and TJs in materials failure. The plausible
reasons for the nonintuitive behaviour of the GBs on martensite nucleation
observed in experiments and atomistic simulations are explained.

Keywords: Grain boundary-induced martensitic transformations; Triple junction;
Phase-field approach; Heterogeneous nucleation; Size-effect.

##  1 Introduction

Grain boundary-induced martensitic transformations: Austenite ($\sf
A$)$\leftrightarrow$martensite ($\sf M$) phase transformations play a central
role in various industrially important materials, including shape memory
alloys (SMAs) which can recover large strains and exhibit superelasticity [1,
3, 2, 4]; dual phase steels where a proper mixture of ferrite and martensite
yields suitable mechanical properties [5]; ferroic materials which are used
for actuation and control [6]; various ceramics and minerals [6], etc. In
materially uniform bodies, the heterogeneities, including the dislocations
([7, 8], Chapter 6 of [2]), accumulated plastic strains and shear bands [7,
10, 11, 12, 13, 13, 15], crack tips [16], free surface [8, 9], anti-phase
boundaries [8], or any other regions within the materials with concentrated
stresses [17, 18], etc., are usually the preferred sites for nucleation of
$\sf M$ in a single grain or polycrystalline samples of materials capable of
undergoing martensitic transformations (MTs). The reduction in energies during
MTs at other heterogeneous sites such as the matrix-precipitate interfaces
[19], external surfaces of the samples [9, 20, 21, 22, 18], and heterophase
interfaces [23, 24] also promotes the nucleation of $\sf M$. On the other
hand, point defects such as vacancies, interstitials, and ani-site defects may
suppress the nucleation of $\sf M$ by significantly reducing the
transformation temperatures [25, 26].

The experimental studies have shown that the grain boundaries (GBs) and triple
junctions (TJs) are additional heterogeneities favourable for the $\sf M$
nucleation in the polycrystalline samples of such phase-changing materials
[27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], which was also confirmed through
atomistic simulations [38, 39, 40, 41, 42]. In particular, the parameters such
as the grain orientation distribution (see, e.g. [43, 1, 42, 36], and the
references therein), average grain size [44, 31, 45], type of the GBs (tilt or
twist) [32, 28, 39], GB energy change during MTs [27], stress concentration
within the GBs during transformations [36, 33, 46], degree of reduction in
symmetry from $\sf A$ to $\sf M$ [43, 47], and direction of loadings [22] have
been observed to play an important role on nucleation and evolution of very
rich martensitic microstructures. The MTs in nanocrystalline materials, which
contain a large volume fraction of the GBs and TJs, are greatly influenced by
these heterogeneous sites [48, 49]. The temperatures of $\sf M$ nucleation
from the GBs are usually significantly higher than the homogeneous
transformation temperature of $\sf M$ [28, 29]. The start and finish
temperatures for both the forward ($\sf A\to\sf M$) and the reverse ($\sf
M\to\sf A$) transformations from the tilt GBs are much higher than that for
the twist GBs [32, 28, 38, 39]. The reason for this was attributed to the
local stress fields associated with the intrinsic dislocations in the low
angle tilt GBs [32, 28]. A reduction in the energy of the GB region after $\sf
M$ nucleation was clearly observed in atomistic simulations [50]. While the
atomistic simulations in [38] showed that the GBs are preferred over the
external surface for $\sf M$ nucleation, an opposite result is, however,
reported in [51]. Furthermore, the experiments [31] and recent atomistic
studies [50] showed that not all the tilt GBs could promote MTs. The
heterogeneous nucleation of $\sf M$ from the GBs in polycrystals is clearly a
complex phenomenon and is still far from being well-understood. The TJs have
also been observed to be potential $\sf M$ nucleation sites [33]. The stresses
were found to be concentrated mainly within the GBs and TJ regions causing
nucleation of cracks in the brittle SMAs [33, 46, 36, 52]. Because of this,
great efforts are being made to minimize the GBs and TJs in designing the
phase-changing materials [33, 46, 36, 52]. Because of the additional system
parameters influencing the MTs in polycrystals, a significant difference in
the shape memory effects in polycrystalline SMAs is observed from its single
crystal counterpart (see, e.g. [43] and the references therein).

Figure 1: (a) Schematic of a stress-free austenitic polycrystal. (b) An
exaggerated view of the part (we call it a bicrystal) is highlighted in (a)
and contains a part of the planar GB between grains G1 and $G_{2}$. The
rightmost figure shows the bicrystal after a $\sf PM$/$\sf M$ layer appeared
on the GB and the neighbouring bulk. $\delta_{12}^{gb}$\- austenitic GB width
between grains $G_{1}$ and $G_{2}$; $\delta_{0M}$\- width of $\sf A$-$\sf M$
interface.

Although multiple reasons have been observed behind the GBs to promote $\sf M$
nucleation, as discussed above, a complete understanding of the GB-induced MTs
is still missing. The general understanding of any GB-induced phase
transformation is that a thin layer of a new phase (e.g. a precipitate, a
disordered phase like premelt/melt, an ordered phase like $\sf M$, etc.)
nucleates within a GB region (maybe locally) at an energetically favourable
thermomechanical condition, which is driven by a reduction in the total energy
of the transformed region (see, e.g. Sec. 6.3.2.2 of [53], [54, 56, 55], and
the references therein for explanation about GB-induced premelting/melting).
We explain the GB-induced MTs here through a schematic shown in Fig. 1(b),
where a stress-free austenitic bicrystal is shown with a planar GB of width
$\delta_{12}^{gb}$. Here we see the bicrystal as a small region between the
grains $G_{1}$ and $G_{2}$ from a large polycrystal shown in Fig. 1(a). The
suffix $12$ in $\delta_{12}^{gb}$ is now obvious. At a thermo-mechanical
condition within the stability regime of $\sf A$ phase, a layer of
premartensite (denoted by $\sf PM$), or $\sf M$ phase (say, a single variant)
of thickness $w$ nucleates within the austenitic GB and grains as shown in
Fig. 1(b) if

$\gamma_{12}^{M}+2\gamma_{0M}+(\psi^{\theta}_{M}+\Delta\psi^{e}-\psi^{\theta}_{0})w<\gamma_{12}^{A},$
(1.1)

where $\sf PM$ is an intermediate state between $\sf A$ and $\sf M$ (see, e.g.
Chapter 6 of [2] and [20]), $\gamma_{12}^{A}$ and $\gamma_{12}^{M}$ are the
energies of the GB when they are in $\sf A$ and $\sf M$ phases, respectively,
$\gamma_{0M}$ is the $\sf A$-$\sf M$ interface energy,
$\psi^{\theta}_{0}(\theta)$ and $\psi^{\theta}_{M}(\theta)$ are the thermal
energies of $\sf A$ and $\sf M$ phases, respectively, $\theta>0$ is the
absolute temperature, and $\Delta\psi^{e}$ is the change in average elastic
energy of the transformed region. We have taken twice $\gamma_{0M}$ in
inequality (1.1) as there are two $\sf A$-$\sf M$ interfaces in the
transformed region. The criterion (1.1) should be seen from the micron scale
and when the GB and $\sf A$-$\sf M$ widths, which vary between a few angstroms
to a few nanometres, are negligible compared to $w$. Note that we have assumed
a single variant appears in the transformed region in the relation (1.1); the
interfacial energies between the variants should be taken into account therein
if multiple variants are considered. The above criterion in the nanoscale,
when the interfacial widths and the transformed region thickness $w$ are
comparable, can be expressed as (see [55, 57] for the nanoscale criterion of
GB-induced premelting/melting)

$\displaystyle\gamma_{12}^{M}+2\gamma_{0M}+(\gamma_{12}^{A}-\gamma_{12}^{M}-2\gamma_{0M})\exp\left(-\frac{w}{\delta}\right)+(\psi^{\theta}_{M}+\Delta\psi^{e}-\psi^{\theta}_{0})w$
$\displaystyle<$ $\displaystyle\gamma_{12}^{A}$
$\displaystyle\Rightarrow\,\,\Delta\gamma_{12}^{gb}+2\gamma_{0M}-(\Delta\gamma_{12}^{gb}+2\gamma_{0M})\exp\left(-\frac{w}{\delta}\right)+(\psi^{\theta}_{M}+\Delta\psi^{e}-\psi^{\theta}_{0})w$
$\displaystyle<$ $\displaystyle 0,$ (1.2)

where $\delta$ is the width of the interfaces (both GB width and $\sf A$-$\sf
M$ interface width are assumed to be identical in inequality (1.2)), and
$\Delta\gamma_{12}^{gb}=\gamma_{12}^{M}-\gamma_{12}^{A}$ is the change in GB
energy due to MT. When $w\gg\delta$, the inequality (1.2) coincides with the
criterion given by inequality (1.1) for the micron scale. Based on the
nanoscale criterion (1.2), we identify that the following two GB parameters
which determine the thermo-mechanical condition for nucleation of $\sf M$ from
the GBs in polycrystals: (i) the change in GB energy $\Delta\gamma_{12}^{gb}$
due to MTs, and (ii) austenitic GB width.

Note that in the inequalities (1.1) and (1.2), we have considered different
energies for the GB in $\sf A$ and $\sf M$ phases are based on the atomistic
results of [50] on NiTi alloy bicrystals, which is arising because of the
structural changes of the GBs due to the transformations [50]. Although no
experimental studies demonstrating the change in GB energies due to MTs are
there in the literature to the best of the author’s knowledge, the atomistic
simulations in [50] clearly showed a decrease in the GB energy during the $\sf
A\to\sf M$ PT. Another exciting and counter-intuitive result reported in [50]
is that a $\Sigma 9$ austenitic GB promoting $\sf M$ nucleation has much lower
energy ($0.22$ times) than another $\Sigma 9$ GB which does not promote
nucleation. Furthermore, the drop in the average energy in the former sample
due to $\sf A\to\sf M$ transformation is higher than that of the later one at
the respective transformation temperatures (see Table-1, Figs. 6a and 8a of
[50]). We also note that the $\Sigma 9$ GB promoting nucleation has a smaller
width than the other (see Table-1 and Fig. 2 of [50]). It is important to
mention here that the size (width) of the surface/interface and its change in
energy due to the PT have been observed to strongly influence the nucleation
of the phases during the solid$\leftrightarrow$melt transformations induced by
the external surface [58, 59] or the GBs [56, 57, 55], and also the MTs
induced by the external surfaces [60, 61] or the matrix-precipitate interfaces
[62]. It is thus of utmost importance to investigate the role of the similar
parameters, i.e. the austenitic GB width and the change in GB energy due to
MTs for the understanding of the MTs in polycrystalline samples, which is yet
to be done.

Phase-field studies of heterogeneous nucleation: The phase-field (PF)
approaches of Ginzburg-Landau (similar to Allen-Cahn approach) provide an
appropriate framework for studying various structural transformations in
materials, including the MTs by homogeneous or heterogeneous nucleation and
the subsequent growth of the phases ([73, 60, 61, 87, 63, 64, 65, 66, 68, 69,
70, 71, 72, 74, 75, 67, 88, 89, 90, 91, 76, 77, 78, 79, 80, 81, 82, 83, 84,
85] and the references therein). The surface-effect on the PTs within the
Ginzburg-Landau theory was first introduced by Lipowsky [92] to the best of
the author’s knowledge, and an analytical treatment for a semi-infinite domain
was given. The analytical treatment for a finite-sized sample was presented in
[93], where variable energy for the heterogeneity as a quadratic function of
the order parameter was considered, which is, however, oversimplified. A more
realistic and non-contradictory variable surface energy, which is quartic in
the order parameter, was introduced in [60]. The model was used to study the
surface-induced multivariant MTs in [61]. Several PF approaches have been
further developed and successfully used for studying surface-induced melting
from the flat surfaces [94] and curved surfaces of nanoparticles [58] and
nanovoids [59]; GB and TJ-induced solid$\leftrightarrow$melt transformations
(e.g. [56, 55] and the references therein); surface-induced paraelectric-
ferroelectric PT [21]; matrix-precipitate interface induced MTs [62] in single
grains; see [96] for a review. The studies in [58, 59, 55, 61, 62] have shown
that the width and the change in energy of the corresponding surface/interface
during the PTs strongly influence the respective transformation temperatures,
which may vary with the characteristic size of the heterogeneities in a
complex and nonintuitive way. The TJ energy also plays a vital role in
premelt/melt nucleation, and kinetics during the forward and reverse PTs [55].

The small strains-based multiphase phase-field approaches have been developed
to study MTs in polycrystalline materials in [97, 98, 102, 106, 107, 105, 101,
108, 109, 110, 111, 103, 104, 100, 112, 113, 114, 99, 115]. The only PF model
used for polycrystalline samples by Malik et al. [117, 116] considers a
geometric nonlinear total strain, which was additively decomposed into
inelastic and elastic parts. Such a strain decomposition is well-known to have
limitations and is applicable only in some special cases (see Sec. 8.2 of
[118] and the references therein). Both concentration based [97, 98, 102, 106,
105, 101, 108, 109, 110, 111, 103, 104, 100, 112, 113, 114, 99, 115] and
transformation strains [98, 102, 111, 103] based order parameters have been
used for describing the $\sf A$ and the $N$ variants ${\sf M}_{1}\,{\sf
M}_{2},\ldots,{\sf M}_{N}$. In [111], the order parameters were indexed in a
way to take care of both the phase changes and the variation of the grains.
However, the remaining papers for polycrystals cited here did not introduce
any order parameter for describing the grains. A single constraint, which is
the summation of all the order parameters (volume fraction) is equal to the
unity, was used in [97, 104, 105]. Such a single constraint cannot assure that
each of the MTs can be systematically described by a single order parameter
only, which is necessary for identifying the relations between the system
parameters and the measurable material parameters; see [87]. The other papers
[102, 106, 107, 101, 108, 109, 110, 111, 103, 100, 112, 113, 114, 99, 115,
117, 116] also did not consider all necessary constraints which allow
controlling all the transformation paths and calibration of all the system
parameters; see [87] for a detailed discussion. Various strategies were
adopted to initiate the MTs in those papers as follows. While the random noise
in the order parameters [102, 103, 104, 105, 119, 109, 110, 111] or in the
displacements [120, 121] was considered to trigger nucleation of $\sf M$ from
the GBs, artificial nuclei were placed on the GBs in [100, 107] for that
purpose. In the PF model of [111], a relaxation parameter for the misfit
strain between $\sf A$ and $\sf M$ variants within the GB is considered to
promote nucleation. Nucleation was also observed due to the stress
concentration within dislocation pileups near the GBs and shear bands (see,
e.g. [15] and the references therein) and due to the geometric constraint
within the samples in [106]. In [97, 98, 116, 117, 101, 99], where the primary
objective was to study the effect of resistance provided by the GBs on the
growth of the variant plates, artificial nuclei were placed inside the grains
to initiate MTs. These studies have also shown that with a reduction in the
average grain size in the polycrystals, the barrier energy for the nucleation
of $\sf M$ increases, drastically affecting the subsequent transformations.
The GB texture also was shown to greatly influence the nucleation and growth
of the variants [111, 100, 102, 112, 113].

It is to be mentioned that none of the existing papers on MTs in
polycrystalline solids [97, 98, 15, 117, 102, 106, 105, 101, 108, 109, 110,
81, 111, 103, 104, 100, 112, 113, 116, 114, 99, 115] studied the role of the
GB width on the temperature of nucleation of the phases and the subsequent
microstructure evolution. The change in GB energy due to MTs, an important
material parameter discussed above, is also yet to be incorporated to develop
a robust model. Furthermore, the role of the TJ energy on GB-induced MTs is
not yet studied. The PF models in [97, 98, 15, 102, 106, 105, 101, 108, 109,
110, 111, 103, 104, 100, 112, 113, 114, 99, 115] considered small strains,
whereas the MTs usually involve large strains, and hence a large strains-based
general PF model for MTs in polycrystals is yet to be developed. Recall that
the only large strains-based model in [117, 116] has issues with the kinematic
model, as pointed out above. The structural stresses [79, 87], which play an
important role in MTs [60, 61] and are still missing in all the existing
models for polycrystals cited here, must be considered for completeness.

Contributions of the paper: The contributions of the paper are as follows:

(I) A thermodynamically consistent original and non-contradictory multiphase
PF approach for studying GB-induced $\sf A\leftrightarrow\sf M$
transformations at the nanoscale is developed for polycrystalline solids
capable of undergoing MTs with $M(>1)$ grains and considering $N(>1)$
martensitic variants. The present model considers the following aspects which
are missing in the existing ones for MTs in polycrystals [97, 98, 15, 117,
102, 106, 105, 101, 108, 109, 110, 81, 111, 103, 104, 100, 112, 113, 116, 114,
99, 115]:

(a) Variable energy of the GB region as a function for the order parameter
related to ${\sf A}\leftrightarrow{\sf M}$ transformation is considered, which
incorporates the change in GB energy due to the structural transformation
between $\sf A$ and $\sf M$ phases within the free energy of the system.

(b) The order parameters related to the variants and those related to the GBs
are constrained to particular hypersurfaces in the respective order parameter
spaces, which is essential to control the transformation paths between the
variants and between the grains and also calibration of all the system
parameters [87].

(c) Our model considers large strains, and hence a multiplicative
decomposition for the total deformation gradient into elastic and
transformational parts is used.

(d) The structural part of the total interfacial stress tensor is considered
for all the interfaces.

(e) An excess energy for the TJ regions connecting the GBs is considered which
is necessary to control the energy and size of the TJ regions.

(f) A non-contradictory and new kinetic model for the coupled Ginzburg-Landau
equations related to the MTs and GBs are derived.

(II) The numerical results are presented for (a) bicrystals with a planar
symmetric or asymmetric tilt GB, and a bicrystal with a circular symmetric
tilt GB; and (b) tricrystals consisting of three planar symmetric tilt GBs
meeting the interior of the sample at $120^{\circ}$ dihedral angles. The
results for both forward and reverse transformations are presented. The strong
effect of the width of the GBs in stress-free $\sf A$ polycrystals on $\sf M$
nucleation, temperature hysteresis, and microstructure evolution is studied. A
plot for the temperatures of transformations between $\sf A$, $\sf PM$, and
$\sf M$ within the GBs for varying austenitic GB width is presented for a
bicrystal with a planar symmetric GB. The role of change in the GB energy due
to MTs, energy of the TJ, GB misorientation angle, applied strain, and the
sample size is studied. The compatibility of the microstructures across the
GBs is studied. The plausible reasons for various nonintuitive effects of the
GBs on MTs observed in the experiments, and atomistic studies are discussed.

(III) The distribution of the elastic and structural stresses within the
bicrystals and tricrystals are studied, which is essential for failure
analysis and design of materials.

The paper is organized as follows. The multiphase PF model is developed in
Sec. 2; the material parameters are listed in Sec. 3; the numerical results
are presented in Sec. 4; we conclude the paper in Sec. 5.

Notation: We denote the inner product and multiplication between two arbitrary
second-order tensors as ${\boldsymbol{A}}:{\boldsymbol{D}}=A_{ab}D_{ba}$ and
$({\boldsymbol{A}}\cdot{\boldsymbol{D}})_{ab}=A_{ac}D_{cb}$, respectively,
where the repeated indices denote Einstein’s summation, and $A_{ab}$ and
$D_{ab}$ are the components of the tensors in a right-handed Cartesian basis
$\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3}\\}$. The
Euclidean norm of $\boldsymbol{A}$ is defined as
$|\boldsymbol{A}|=\sqrt{{\boldsymbol{A}}:{\boldsymbol{A}}^{T}}$;
${\boldsymbol{I}}$ is the second order identity tensor and its, say, $ab^{th}$
component is denoted by the Kronecker delta $\delta_{ab}$;
$\boldsymbol{A}^{-1}$, $\boldsymbol{A}^{T}$, $det\,\boldsymbol{A}$, and
$tr\,\boldsymbol{A}$ denote inversion, transpose, determinant, and trace of
$\boldsymbol{A}$, respectively. The dyadic product between two arbitrary
vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ is denoted as
$\boldsymbol{a}\otimes\boldsymbol{b}$. We denote the reference, stress-free
intermediate, and current configurations of a body by $\Omega_{0}$,
$\Omega_{t}$, and $\Omega$, respectively. The volumes from the reference and
current configurations are denoted by $V_{0}$ and $V$, and their boundaries by
$S_{0}$ and $S$, respectively. The symbols $\nabla_{0}$ and $\nabla$ represent
the gradient operators in $\Omega_{0}$ and $\Omega$, respectively;
$\nabla_{0}^{2}:=\nabla_{0}\cdot\nabla_{0}$ and
$\nabla^{2}:=\nabla\cdot\nabla$ are the Laplacian operators in $\Omega_{0}$
and $\Omega$, respectively. The symbol $:=$ stands for equality by definition.

##  2 System of coupled phase field and mechanics equations

In this section, we present our novel multiphase PF approach to GB-induced MTs
in polycrystals. The coupled mechanics and PF equations are derived, followed
by an introduction to the order parameters.

###  2.1  Order parameters

We introduce $N+1$ order parameters
$\eta_{0},\,\eta_{1},\eta_{2},\ldots,\eta_{N}$ related to the $\sf A$ and $\sf
M$ phases, where $\eta_{0}$ describes ${\sf A}\leftrightarrow{\sf M}$ PTs, and
$\eta_{1},\eta_{2},\ldots,\eta_{N}$ describe $N$ martensitic variants. We
consider $0\leq\eta_{0}\leq 1$ and assign $\eta_{0}=0$ in $\sf A$ and
$\eta_{0}=1$ in $\sf M$. For the order parameters relatd to the variants, we
consider $0\leq\eta_{i}\leq 1$, and assign $\eta_{i}=1$ in ${\sf M}_{i}$ and
$\eta_{i}=0$ in ${\sf M}_{j}$ for all $i,j=1,2,\ldots,N$ and $j\neq i$. We
accept a particle as $\sf A$, $\sf PM$, and $\sf M$ if $\eta_{0}$ therein lies
within $0\leq\eta_{0}<0.05$, $0.05\leq\eta_{0}\leq 0.95$, and
$0.95<\eta_{0}\leq 1$, respectively. The order parameters
$\eta_{1},\eta_{2},\ldots,\eta_{N}$ are taken to be constrained to lie on the
plane [87]

$\sum_{i=1}^{N}\eta_{i}-1=0,$ (2.1)

which is necessary for controlling the transformation paths between the
variants; see [87] for details. We introduce another $M$ order parameters
$\xi_{1},\,\xi_{1},\,\ldots,\xi_{M}$, which continuously vary between $0$ and
$1$, and describe $M$ grains in a polycrystal such that $\xi_{I}=1$ in $G_{I}$
and $\xi_{I}=0$ in $G_{J}$ for all $I,J=1,2,\ldots,M$ and $J\neq I$. The
denote the GB between grains $G_{I}$ and $G_{J}$ by $\Gamma_{IJ}$. We assume
these order parameters also satisfy the following constraint for controlling
the transformation path between the grains (see, e.g. [87, 55]):

$\sum_{I=1}^{M}\xi_{I}-1=0.$ (2.2)

The set of the order parameters related to the MTs is denoted as
$\tilde{\eta}=\\{\eta_{0},\eta_{1},\ldots,\eta_{i},\ldots,\eta_{N}\\}$, and a
subset for the $\sf M$ variants
$\tilde{\eta}_{M}=\\{\eta_{1},\ldots,\eta_{i},\ldots,\eta_{N}\\}$ is
introduced following [87]. Similarly, for the order parameters related to the
grains, we introduce the set
$\tilde{\xi}=\\{\xi_{1},\xi_{2},\ldots,\xi_{I},\ldots,\xi_{M}\\}$. The set of
the gradient of the order parameters in $\Omega$ are introduced as
$\tilde{\eta}^{\nabla}=\\{\nabla\eta_{0},\nabla\eta_{1},\ldots,\nabla\eta_{i},\ldots,\nabla\eta_{N}\\}$
and
$\tilde{\xi}^{\nabla}=\\{\nabla\xi_{1},\nabla\xi_{2},\ldots,\nabla\xi_{I},\ldots,\nabla\xi_{M}\\}$.

###  2.2 Kinematics

The position vector of a particle in the deformed configuration $\Omega$ is
given by
$\boldsymbol{r}(\boldsymbol{r}_{0},t)=\boldsymbol{r}_{0}+\boldsymbol{u}(\boldsymbol{r}_{0},t)$,
where $\boldsymbol{r}_{0}$, $\boldsymbol{u}$, and $t$ denote the particle’s
position vector in $\Omega_{0}$, the displacement vector, and time instance,
respectively. We consider the following multiplicative decomposition for the
total deformation gradient $\boldsymbol{F}:=\nabla_{0}\boldsymbol{r}$ [79]:

${\boldsymbol{F}}=\boldsymbol{F}_{e}\cdot\boldsymbol{F}_{t}=\boldsymbol{V}_{e}\cdot\boldsymbol{R}\cdot\boldsymbol{U}_{t},$
(2.3)

where we have used the decompositions
$\boldsymbol{F}_{e}=\boldsymbol{V}_{e}\cdot\boldsymbol{R}_{e}$ and
$\boldsymbol{F}_{t}=\boldsymbol{R}_{t}\cdot\boldsymbol{U}_{t}$, the subscripts
$e$ and $t$ stand for the elastic and transformational parts, respectively,
$\boldsymbol{U}_{t}$ is the symmetric right transformation stretch tensor,
$\boldsymbol{V}_{e}$ is the symmetric left elastic stretch tensor,
$\boldsymbol{R}_{e}$ and $\boldsymbol{R}_{t}$ are the elastic and
transformational rotations, respectively, and
$\boldsymbol{R}=\boldsymbol{R}_{e}\cdot\boldsymbol{R}_{t}$ is the resultant
lattice rotation. We designate $J=det\,\boldsymbol{F}$,
$J_{t}=det\,\boldsymbol{F}_{t}=det\,\boldsymbol{U}_{t}$, and
$J_{e}=det\,\boldsymbol{F}_{e}=det\,\boldsymbol{V}_{e}$. Hence $J=J_{t}J_{e}$
by Eq. (2.3). We will use the Lagrangian total and elastic strains defined as
(see, e.g. Chapter 2 of [122])

${\boldsymbol{E}}=\frac{1}{2}(\boldsymbol{F}^{T}\cdot\boldsymbol{F}-{\boldsymbol{I}}),\quad\text{and}\quad{\boldsymbol{E}}_{e}=\frac{1}{2}(\boldsymbol{F}_{e}^{T}\cdot\boldsymbol{F}_{e}-{\boldsymbol{I}}),$
(2.4)

and the spatial total and elastic strain tensors defined as (see e.g. Chapter
2 of [122])

${\boldsymbol{b}}=\frac{1}{2}(\boldsymbol{F}\cdot\boldsymbol{F}^{T}-{\boldsymbol{I}}),\quad\text{and}\quad{\boldsymbol{b}}_{e}=\frac{1}{2}(\boldsymbol{F}_{e}\cdot\boldsymbol{F}_{e}^{T}-{\boldsymbol{I}}).$
(2.5)

####  2.2.1 Transformation stretch tensor

Within each grain, we assume the transformation stretch tensor as a linear
combination of the Bain tensor multiplied with the nonlinear interpolation
functions related to $\eta_{0}$ and $\eta_{i}$ ($i=1,\ldots,N$) [87]

${\boldsymbol{U}}_{t}^{(I)}={\boldsymbol{F}}_{t}^{(I)}=\boldsymbol{I}+\varphi(a_{\varepsilon},\eta_{0})\sum_{i=1}^{N}(\boldsymbol{U}_{ti}^{(I)}-\boldsymbol{I})\,\phi_{i}\quad\text{for
all }I=1,\ldots,M,$ (2.6)

where $\boldsymbol{U}_{ti}^{(I)}$ and
$\boldsymbol{\varepsilon}_{ti}^{(I)}:=\boldsymbol{U}_{ti}^{(I)}-\boldsymbol{I}$
are the Bain stretch tensor and Bain strain tensor, respectively, for
$\mathsf{M}_{i}$ in the grain $G_{I}$ obtained using the relation

${\boldsymbol{U}}_{ti}^{(I)}=\boldsymbol{Q}^{(I)}\cdot{\boldsymbol{U}}_{ti}\cdot\boldsymbol{Q}^{(I)T}\quad\text{for
all }i=1,\ldots,N,\text{ and for all }I=1,\ldots,M.$ (2.7)

In Eq. (2.6), ${\boldsymbol{U}}_{ti}$ is the Bain stretch tensor for ${\sf
M}_{i}$ with respect to a reference orthonormal basis attached to a $\sf A$
unit cell, and $\boldsymbol{Q}^{(I)}$ is the orientation of the austenitic
grain $G_{I}$ with respect to that reference basis. The interpolation
functions $\varphi(a,\eta_{0})$ and $\phi_{i}(\eta_{i})$ are taken as [87]

$\varphi(a,\eta_{0})=a\eta_{0}^{2}+(4-2a)\eta_{0}^{3}+(a-3)\eta_{0}^{4},\quad\text{and}\quad\phi_{i}=\eta_{i}^{2}(3-2\eta_{i})=\varphi(3,\eta_{i}),\quad\text{for
all }i=1,\ldots,N,$ (2.8)

which satisfy the necessary conditions derived from the thermodynamic
equilibrium of each of the homogeneous phases in each grain [87] as follows

$\displaystyle\varphi(a,0)=0,\quad\varphi(a,1)=1,\quad\text{and}\quad\frac{\partial\varphi(a,0)}{\partial\eta_{0}}=\frac{\partial\varphi(a,1)}{\partial\eta_{0}}=0;$
$\displaystyle\phi_{i}(0)=0,\quad\phi_{i}(1)=1,\quad\text{and}\quad\frac{\partial\phi_{i}(0)}{\partial\eta_{i}}=\frac{\partial\phi_{i}(1)}{\partial\eta_{i}}=0,\quad\text{for
all }i=1,\ldots,N.$ (2.9)

The parameter $a$ in Eqs. (2.8)1 and (2.9)1 is a constant lying in the range
$0\leq a\leq 6$ [79]. We consider the following rule for the the
transformation stretch tensor $\boldsymbol{U}_{t}$ introduced in Eq. (2.3) for
the polycrystal:

${\boldsymbol{U}}_{t}={\boldsymbol{F}}_{t}=\boldsymbol{I}+\sum_{I=1}^{M}({\boldsymbol{U}}_{t}^{(I)}-\boldsymbol{I})\phi_{I}^{\xi}(\xi_{I})=\boldsymbol{I}+\varphi(a_{\varepsilon},\eta_{0})\sum_{I=1}^{M}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{ti}^{(I)}\,\phi_{i}(\eta_{i})\phi_{I}^{\xi}(\xi_{I}),\quad\text{where}$
(2.10) $\phi_{I}^{\xi}(\xi_{I})=\xi_{I}^{2}(3-2\xi_{I}),$ (2.11)

and it satisfies the following conditions in the corresponding grain

$\phi_{I}^{\xi}(0)=0,\quad\phi_{I}^{\xi}(1)=1,\quad\text{and}\quad\frac{\partial\phi_{I}^{\xi}(0)}{\partial\xi_{I}}=\frac{\partial\phi_{I}^{\xi}(1)}{\partial\xi_{I}}=0,\quad\text{for
all }I=1,\ldots,M.$ (2.12)

###  2.3 Free energy

The Helmholtz free energy per unit mass of the polycrystalline sample is
considered as

$\displaystyle\psi(\boldsymbol{F},\boldsymbol{E}_{e},\tilde{\eta},\tilde{\xi},\theta,\tilde{\eta}^{\nabla},\tilde{\xi}^{\nabla})=\frac{J_{t}}{\rho_{0}}\psi_{e}(\boldsymbol{E}_{e},\tilde{\eta},\tilde{\xi},\theta)+J\breve{\psi}^{\theta}(\tilde{\eta},\theta)+\tilde{\psi}^{\theta}(\tilde{\eta},\theta)+{\psi}^{p}(\tilde{\eta})+J\psi^{\nabla}(\tilde{\eta}^{\nabla})+J\psi_{\xi}(\tilde{\xi},\tilde{\xi}^{\nabla},\eta_{0}),$
(2.13)

where $\psi_{e}$ is the strain energy per unit volume of $\Omega_{t}$ and it
is considered here to have the standard quadratic form in
$\boldsymbol{E}_{e}$, $\rho_{0}$ is the density of material in $\Omega_{0}$;
$\breve{\psi}^{\theta}$ is the specific energy barrier related to ${\sf
A}\leftrightarrow{\sf M}$ and ${\sf M}_{i}\leftrightarrow{\sf M}_{j}$
transformations, $\tilde{\psi}^{\theta}$ is the specific thermal energy,
$\psi^{\nabla}$ is the specific gradient energy accounting for the interfacial
energies between $\sf A$-$\sf M$ and between the variants, $\psi^{p}$ is the
specific energy for penalizing variant-variant transformation path deviations
from the specified paths, and the $\psi^{\xi}$ is the total of the specific
barrier energy between the grains and the energies of the GBs and TJs. The
first five terms on the right-hand side of Eq. (2.13) were earlier used in
[87] for modeling MTs in single grains. The additional last term in Eq. (2.13)
accounts for the excess free energy of the heterogeneities, including the GB
and TJ regions and the interactions between these heterogeneities and the MTs
in the polycrystal. The expressions for all the energies are as follows [79,
87]:

$\displaystyle\psi_{e}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\mathbb{C}:\boldsymbol{E}_{e}):\boldsymbol{E}_{e},\,\text{where
}{\boldsymbol{\mathbb{C}}}(\tilde{\eta},\tilde{\xi})=\sum_{I=1}^{M}\left[(1-\varphi(a,\eta_{0})){\boldsymbol{\mathbb{C}}}_{0}^{(I)}+\varphi(a,\eta_{0})\sum_{i=1}^{N}\phi_{i}(\eta_{i}){\boldsymbol{\mathbb{C}}}_{i}^{(I)}\right]\phi_{I}^{\xi}(\xi_{I});$
$\displaystyle\breve{\psi}^{\theta}$ $\displaystyle=$
$\displaystyle[A_{0M}+(a_{\theta}-3)\Delta\psi^{\theta}(\theta)]\eta_{0}^{2}(1-\eta_{0})^{2}+\varphi(a_{b},\eta_{0})\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}{A}_{ij}^{\eta}\eta_{i}^{2}\eta_{j}^{2};$
$\displaystyle\tilde{\psi}^{\theta}$ $\displaystyle=$
$\displaystyle{\psi^{\theta}}_{0}(\theta)+\eta_{0}^{2}(3-2\eta_{0})\Delta\psi^{\theta}(\theta);$
$\displaystyle\psi^{p}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}K_{ij}^{\eta}(\eta_{i}+\eta_{j}-1)^{2}\eta_{i}^{2}\eta_{j}^{2};$
$\displaystyle\psi^{\nabla}$ $\displaystyle=$
$\displaystyle\frac{1}{2\rho_{0}}\left[\beta_{0M}|\nabla\eta_{0}|^{2}-\tilde{\varphi}(a_{\beta},a_{0},\eta_{0})\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}\beta_{ij}^{\eta}\nabla\eta_{i}\cdot\nabla\eta_{j}\right],\quad\text{where
}\beta_{ij}^{\eta}=\beta_{ji}^{\eta},\,\,\text{and}$
$\displaystyle\tilde{\varphi}(a_{\beta},a_{c},\eta_{0})$ $\displaystyle=$
$\displaystyle\varphi(a_{\beta},\eta_{0})+a_{c}=a_{\beta}\eta_{0}^{2}+(4-2a_{\beta})\eta_{0}^{3}+(a_{\beta}-3)\eta_{0}^{4}+a_{c};$
$\displaystyle\psi^{\xi}$ $\displaystyle=$
$\displaystyle\psi^{\xi}_{loc}+\psi^{\xi}_{\nabla},\qquad\text{where}\quad\psi^{\xi}_{\nabla}=-\sum_{I=1}^{M-1}\sum_{J=I+1}^{M}\frac{\beta_{IJ}^{gb}(\eta_{0})}{2\rho_{0}}\nabla\xi_{I}\cdot\nabla\xi_{J},\quad\beta_{IJ}^{gb}=\beta_{JI}^{gb}$
$\displaystyle\psi^{\xi}_{loc}$ $\displaystyle=$
$\displaystyle\sum_{I=1}^{M-1}\sum_{J=I+1}^{M}\left[A_{IJ}^{gb}(\eta_{0})\xi_{I}^{2}\xi_{J}^{2}+K_{IJ}^{gb}(\xi_{I}+\xi_{J}-1)^{2}\xi_{I}^{2}\xi_{J}^{2}\right]+\sum_{H=1}^{M-2}\sum_{I=H+1}^{M-1}\sum_{J=I+1}^{M}K_{HIJ}\xi_{H}^{2}\xi_{I}^{2}\xi_{J}^{2}.$

The material parameters introduced in the energies (Eq. (LABEL:MF2)) will now
be defined. The fourth order tensor $\mathbb{C}$ is the elasticity tensor of
the polycrystalline sample; $\mathbb{C}_{0}^{(I)}$ and $\mathbb{C}_{i}^{(I)}$
are the elasticity tensors of $\sf A$ phase and the variant ${\sf M}_{i}$,
respectively, in grain $G_{I}$, and they should be obtained using e.g.
(Chapter 1 of [123])

$(\mathbb{C}_{l}^{(I)})_{abcd}=Q^{(I)}_{ae}Q^{(I)}_{bf}Q^{(I)}_{cg}Q^{(I)}_{dh}(\mathbb{C}_{l})_{efgh}\qquad\text{for
}l=0,1,\ldots,N,$ (2.15)

where $\mathbb{C}_{l}$ is the elasticity tensor of the $\sf A$ ($l=0$) and
$\sf M_{i}$ ($l=i=1,\ldots,N$) with respect to the reference basis; $A_{0M}$
and ${A}_{ij}^{\eta}$ are the coefficients for barrier energies between ${\sf
A}$-${\sf M}$ and ${\sf M}_{i}$-${\sf M}_{j}$, respectively; $A_{IJ}^{gb}$ is
the barrier energy coefficient between the grains $G_{I}$ and $G_{J}$.
Assuming the specific heat at constant pressure to be identical for both $\sf
A$ and $\sf M$, we consider the thermal energy difference between these two
phases as $\Delta\psi^{\theta}=\psi^{\theta}_{M}-\psi^{\theta}_{0}=-\Delta
s\,(\theta-\theta_{e})$, where $\Delta s$ is the change in entropy due to
${\sf A}$ to ${\sf M}$ transformation; $\theta_{e}$ is the thermodynamic
equilibrium temperature between $\sf A$ and $\sf M$. The coefficient
$K_{ij}^{\eta}$ is a positive constant which determines the deviation of the
path for ${\sf M}_{i}\leftrightarrow{\sf M}_{i}$ transformations from the
lines $\eta_{i}+\eta_{j}=1$, $\eta_{i}=0$, and $\eta_{j}=0$ for all
$i,j=1,\ldots,N$ and $i\neq j$ (see [87] for a detailed explanation);
$K_{IJ}^{gb}$ has a similar meaning as $K_{ij}^{\eta}$ has but is related to
the transformations between the grains $G_{I}$ and $G_{J}$. The constant
coefficient $K_{HIJ}$ is used for penalizing the TJs between the grains
$G_{H}$, $G_{I}$, and $G_{J}$, i.e. it allows control of the energy and size
of the TJs region (also see [124, 87, 55]). The gradient energies considered
in Eq. (LABEL:MF2) are similar to that of [125]; see [126] for a comparative
study for various gradient energies. The parameters $\beta_{0M}$,
$\beta_{ij}^{\eta}$, and $\beta_{IJ}^{gb}(\eta_{0})$ are the gradient energy
coefficients for the ${\sf A}$-${\sf M}$, ${\sf M}_{i}$-${\sf M}_{j}$, and
$\Gamma_{IJ}$ interfaces, respectively. The constant parameters
$a_{\theta},a_{b},a_{\beta}$ lie within the range $0\leq
a_{\theta},a_{b},a_{\beta}\leq 6$ [79, 87], and the constant $a_{c}$ in Eq.
(LABEL:MF2)5,6 is taken as $0<a_{c}\ll 1$; see [87] for introducing $a_{c}$.
The energies $\breve{\psi}^{\theta}$, $\psi^{\nabla}$, and $\psi^{\xi}$ are
multiplied with $J=det\,\boldsymbol{F}$, and the gradients of all the order
parameters are taken in the current configuration $\Omega$ in Eq. (2.13) to
obtain the desired expression for the structural stresses given by Eqs.
(2.21), and (2.24) (see [79, 87] for further discussion on the structural
stresses). The material parameters, which are dependent on the grain
orientations, at any point in the sample are determined using the following
interpolation

$B(\tilde{\eta},\tilde{\xi},\theta,{\boldsymbol{F}})=\sum_{I=1}^{M}\left[(1-\varphi(a,\eta_{0}))B_{0}^{(I)}+\varphi(a,\eta_{0})\sum_{i=1}^{N}\phi_{i}B_{i}^{(I)}\right]\phi_{I}^{\xi}(\xi_{I}),$
(2.16)

where $B_{0}^{(I)}$ and $B_{i}^{(I)}$ are the material properties of
$\mathsf{A}$ and $\mathsf{M}_{i}$, respectively, in $G_{I}$, and
$\varphi(a,\eta_{0})$, $\phi_{i}(\eta_{i})$, and $\phi_{I}^{\xi}(\xi_{I})$ are
given by Eqs. (2.8) and (2.11).

####  2.3.1 GB energy

The energy of the GBs $\Gamma_{IJ}$ appeared in Eqs. (LABEL:MF2)7,8,9 is
assumed to be isotropic, i.e. it is a function of GB misorientations and
independent of the GB inclination (see, e.g. Chapter 1 of [53]), and it is
taken as

$\gamma_{IJ}^{gb}(\Theta_{IJ},\eta_{0})=\gamma_{IJ}^{A}(\Theta_{IJ})+\Delta\gamma_{IJ}^{gb}(\Theta_{IJ})\,\varphi(a,\eta_{0}),$
(2.17)

where $\Theta_{IJ}$ is the list of three misorientation angles associated with
$\Gamma_{IJ}$, $\Delta\gamma_{IJ}^{gb}=\gamma_{IJ}^{M}-\gamma_{IJ}^{A}$ is the
change in GB energy due to MT, $\varphi(a,\eta_{0})$ is given by Eq. (2.8)1,
and the parameters $\gamma_{IJ}^{A}$ and $\gamma_{IJ}^{B}$ are energies of GB
$\Gamma_{IJ}$ in $\sf A$ and $\sf M$ phases, respectively. From Eq. (2.17),
$\gamma_{IJ}^{gb}=\gamma_{IJ}^{A}$ when $\eta_{0}=0$ (in $\sf A$), and
$\gamma_{IJ}^{gb}=\gamma_{IJ}^{M}$ when $\eta_{0}=1$ (in $\sf M$). The energy
of the heterogeneities similar to Eq. (2.17) was earlier used to study the
surface-induced solid$\leftrightarrow$melt PTs in nanoparticles [58] and
nanovoids [59], surface-induced MTs [61]), matrix-precipitate interface-
induced MTs [62], and GB-induced solid$\leftrightarrow$melt PTs in
polycrystals [55].

###  2.4 Dissipation rate, coupled mechanics and Ginzburg-Landau equations

We start with the following local form of the dissipation inequality, whose
detailed derivation is given in Appendix A:

$\rho_{0}{\mathcal{D}}=\boldsymbol{P}_{d}:\dot{\boldsymbol{F}}^{T}+X_{0}\dot{\eta}_{0}+\sum_{i=1}^{N}X_{i}^{\eta}\dot{\eta}_{i}+\sum_{I=1}^{M}X_{I}^{\xi}\dot{\xi}_{I}\geq
0\quad\text{in }\Omega_{0},$ (2.18)

where ${\mathcal{D}}$ is the total power dissipation per unit mass,
$\boldsymbol{P}_{d}$ is the dissipative or viscous first Piola-Kirchhoff
stress tensor (see Eq. (2.20)), the overdot denotes material time derivative,
and $X_{l}^{\eta}$ and $X_{I}^{\xi}$ are the conjugate thermodynamic forces
(see Eq. (2.31)) for the rates of the order parameters $\dot{\eta}_{l}$ (for
$l=0,\ldots,N$) and $\dot{\xi}_{l}$ (for $I=1,\ldots,M$), respectively. The
exact expressions for $\boldsymbol{P}_{d}$, $X_{l}^{\eta}$, and $X_{I}^{\xi}$
are given in Secs. 2.4.1 and 2.4.2. We simplify the present theory by assuming
that the power dissipation due to the dissipative stress, the evolution of the
phases, and evolution of the grains are independent, and hence we decouple the
inequality (2.18) into the following three, which respect the original
inequality (2.18):

$\rho_{0}{\mathcal{D}}_{d}=\boldsymbol{P}_{d}:\dot{\boldsymbol{F}}^{T}\geq
0;\qquad\rho_{0}{\mathcal{D}}_{\eta}=X^{\eta}_{0}\dot{\eta}_{0}+\sum_{i=1}^{N}X_{i}^{\eta}\dot{\eta}_{i}\geq
0;\qquad\text{and}\qquad\rho_{0}{\mathcal{D}}_{\xi}=\sum_{I=1}^{M}X_{I}^{\xi}\dot{\xi}_{I}\geq
0;\quad\text{in }\Omega_{0}.$ (2.19)

####  2.4.1  Stresses and mechanical equilibrium

The dissipative first Piola-Kirchhoff stress tensor ${\boldsymbol{P}}_{d}$ in
the inequality (2.19)1 is defined as

${\boldsymbol{P}}_{d}:=\boldsymbol{P}-{\boldsymbol{P}}_{e}-\boldsymbol{P}_{st},$
(2.20)

where $\boldsymbol{P}$ is the total Piola-Kirchhoff stress tensor, and
$\boldsymbol{P}_{e}$ and $\boldsymbol{P}_{st}$ are the elastic and structural
parts of $\boldsymbol{P}$, respectively, and they are given by

$\displaystyle{\boldsymbol{P}}_{e}$ $\displaystyle=$ $\displaystyle
J_{t}\frac{\partial\psi_{e}(\boldsymbol{F}_{e})}{\partial\boldsymbol{F}_{e}}\cdot\boldsymbol{F}_{t}^{-T},\,\,\text{and}\,\,{\boldsymbol{P}}_{st}={\boldsymbol{P}}_{st}^{\eta}+{\boldsymbol{P}}_{st}^{\xi},\quad\text{where}$
$\displaystyle{\boldsymbol{P}}_{st}^{\eta}$ $\displaystyle=$ $\displaystyle
J\rho_{0}(\breve{\psi}^{\theta}+\psi^{\nabla}){\boldsymbol{F}}^{-T}-J\rho_{0}\sum_{l=0}^{N}\left(\nabla\eta_{l}\otimes\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{l}}\right)\cdot\boldsymbol{F}^{-T};$
$\displaystyle{\boldsymbol{P}}_{st}^{\xi}$ $\displaystyle=$ $\displaystyle
J\rho_{0}\psi^{\xi}{\boldsymbol{F}}^{-T}-J\rho_{0}\sum_{I=1}^{M}\left(\nabla\xi_{I}\otimes\frac{\partial\psi^{\xi}_{\nabla}}{\partial\nabla\xi_{I}}\right)\cdot\boldsymbol{F}^{-T}.$
(2.21)

Considering the dissipative stress to be negligible in the present theory
(also see [87]), we express the total Piola-Kirchhoff stress tensor using Eq.
(2.20) as

$\boldsymbol{P}={\boldsymbol{P}}_{e}+{\boldsymbol{P}}_{st}.$ (2.22)

Recalling that the density of material in $\Omega$ is given by
$\rho=J^{-1}\rho_{0}$, and the relation between the first Piola-Kirchhoff and
Cauchy stress tensors is (see Chapter 3 of [123])

${\boldsymbol{\sigma}}=J^{-1}{\boldsymbol{P}}\cdot{\boldsymbol{F}}^{T},$
(2.23)

we get the total, elastic, and structural Cauchy stresses as (recall that
$\boldsymbol{\sigma}_{d}$ is neglected)

$\displaystyle\boldsymbol{\sigma}=\boldsymbol{\sigma}_{e}+\boldsymbol{\sigma}_{st};\qquad\boldsymbol{\sigma}_{e}=J^{-1}_{e}\frac{\partial\psi_{e}(\boldsymbol{F}_{e})}{\partial{\boldsymbol{F}}_{e}}\cdot{\boldsymbol{F}}_{e}^{T}=J_{t}\,{\boldsymbol{F}}_{e}\cdot\frac{\partial\psi_{e}(\boldsymbol{E}_{e})}{\partial{\boldsymbol{E}}_{e}}\cdot{\boldsymbol{F}}_{e}^{T};\qquad\boldsymbol{\sigma}_{st}=\boldsymbol{\sigma}_{st}^{\eta}+\boldsymbol{\sigma}_{st}^{\xi},\quad\text{where}$
$\displaystyle\boldsymbol{\sigma}_{st}^{\eta}=J\rho(\breve{\psi}^{\theta}+\psi^{\nabla})\boldsymbol{I}-J\rho\
\sum_{l=0}^{N}\nabla\eta_{i}\otimes\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{i}},\qquad\boldsymbol{\sigma}_{st}^{\xi}=J\rho\psi^{\xi}{\boldsymbol{I}}-J\rho\sum_{I=1}^{M}\nabla\xi_{I}\otimes\frac{\partial\psi^{\xi}_{\nabla}}{\partial\nabla\xi_{I}}.$
(2.24)

Using Eqs. (2.13) and (LABEL:MF2) in Eqs. (2.24)1,2,3,4,5 the structural
stresses can be finally expressed as

$\displaystyle{\boldsymbol{P}}_{st}^{\eta}$ $\displaystyle=$ $\displaystyle
J\rho_{0}(\breve{\psi}^{\theta}+\psi^{\nabla}){\boldsymbol{F}}^{-T}-J\beta_{0M}\left(\nabla\eta_{0}\otimes\nabla\eta_{0}\right)\cdot\boldsymbol{F}^{-T}+\frac{J\tilde{\varphi}(a_{\beta},a_{c},\eta_{0})}{2}\sum_{i=1}^{N}\sum_{j=1,\neq
i}^{N}\beta^{\eta}_{ij}\left(\nabla\eta_{i}\otimes\nabla\eta_{j}\right)\cdot\boldsymbol{F}^{-T};$
$\displaystyle{\boldsymbol{P}}_{st}^{\xi}$ $\displaystyle=$ $\displaystyle
J\rho_{0}\psi^{\xi}{\boldsymbol{F}}^{-T}+\frac{J}{2}\sum_{H=1}^{M}\sum_{I=1,\neq
H}^{M}\beta^{gb}_{HI}(\eta_{0})\left(\nabla\eta_{H}\otimes\nabla\eta_{I}\right)\cdot\boldsymbol{F}^{-T};$
(2.26) $\displaystyle\boldsymbol{\sigma}_{st}^{\eta}$ $\displaystyle=$
$\displaystyle
J\rho(\breve{\psi}^{\theta}+\psi^{\nabla})\boldsymbol{I}-\beta_{0M}\nabla\eta_{0}\otimes\nabla\eta_{0}+\frac{\tilde{\varphi}(a_{\beta},a_{c},\eta_{0})}{2}\sum_{i=1}^{N}\sum_{j=1,\neq
i}^{N}\beta^{\eta}_{ij}\nabla\eta_{i}\otimes\nabla\eta_{j};\,\,\text{and}$
$\displaystyle\boldsymbol{\sigma}_{st}^{\xi}$ $\displaystyle=$ $\displaystyle
J\rho\psi^{\xi}\boldsymbol{I}+\sum_{H=1}^{M}\sum_{I=1,\neq
H}^{M}\frac{\beta^{gb}_{HI}(\eta_{0})}{2}\nabla\eta_{H}\otimes\nabla\eta_{I}.$
(2.27)

For isotropic elastic response, the Cauchy elastic stress tensor can
alternatively be expressed as [79]

${\boldsymbol{\sigma}}_{e}=J_{e}^{-1}\boldsymbol{V}_{e}^{2}\cdot\frac{\partial\psi_{e}(\boldsymbol{b}_{e})}{\partial\boldsymbol{b}_{e}}.$
(2.28)

The total stresses satisfy the mechanical equilibrium equations, which are
obtained using the balance of linear momentum, when the inertia and body
forces are neglected, as [79]

$\nabla_{0}\cdot\boldsymbol{P}=\boldsymbol{0}\quad\text{in
}\Omega_{0},\quad\text{or
equivalently,}\quad\nabla\cdot\boldsymbol{\sigma}=\boldsymbol{0}\quad\text{in
}\Omega.$ (2.29)

The boundary conditions (BCs) on the displacement boundary $S_{0u}\subset
S_{0}$ and on the traction boundary $S_{0t}\subset S_{0}$ are given by

$\boldsymbol{u}=\boldsymbol{u}^{sp}\quad\text{on
}S_{0u},\quad\text{and}\quad\boldsymbol{p}^{sp}=\boldsymbol{P}\cdot\boldsymbol{n}_{0}\quad\text{on
}S_{0t},$ (2.30)

where $S_{0}=S_{0u}\cup S_{0t}$, $\boldsymbol{n}_{0}$ is the outward unit
normal to $S_{0}$, and $\boldsymbol{p}^{sp}$ is the traction specified. The
exact displacement and traction BCs for the numerical calculations will be
specified while presenting the results in Sec. 4.

####  2.4.2 Thermodynamic forces and Ginzburg-Landau equations

Thermodynamic forces: The thermodynamic forces $X_{0}^{\eta}$, $X_{i}^{\eta}$,
and $X_{I}^{\xi}$ appearing in inequalities (2.19)2,3 are given by

$\displaystyle X_{0}^{\eta}$ $\displaystyle=$
$\displaystyle\left(\boldsymbol{P}_{e}^{T}\cdot\boldsymbol{F}_{e}-J_{t}\psi_{e}\boldsymbol{F}_{t}^{-1}\right):\frac{\partial\boldsymbol{F}_{t}}{\partial\eta_{0}}-J_{t}\left.\frac{\partial\psi_{e}}{\partial\eta_{0}}\right|_{\boldsymbol{F}_{e}}-\rho_{0}J\frac{\partial(\breve{\psi}^{\theta}+\psi^{\nabla}+\psi^{\xi})}{\partial\eta_{0}}-\rho_{0}\frac{\partial(\tilde{\psi}^{\theta}+\psi^{p})}{\partial\eta_{0}}+$
$\displaystyle\nabla_{0}\cdot\left(\rho_{0}J\boldsymbol{F}^{-1}\cdot\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{0}}\right);$
$\displaystyle X_{i}^{\eta}$ $\displaystyle=$
$\displaystyle\left(\boldsymbol{P}_{e}^{T}\cdot\boldsymbol{F}_{e}-J_{t}\psi_{e}\boldsymbol{F}_{t}^{-1}\right):\frac{\partial\boldsymbol{F}_{t}}{\partial\eta_{i}}-J_{t}\left.\frac{\partial\psi_{e}}{\partial\eta_{i}}\right|_{\boldsymbol{F}_{e}}-\rho_{0}J\frac{\partial(\breve{\psi}^{\theta}+\psi^{\nabla})}{\partial\eta_{i}}-\rho_{0}\frac{\partial(\tilde{\psi}^{\theta}+\psi^{p})}{\partial\eta_{i}}+$
$\displaystyle\nabla_{0}\cdot\left(\rho_{0}J\boldsymbol{F}^{-1}\cdot\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{i}}\right)\quad\text{for
}i=1,2,\ldots,N;$ $\displaystyle X_{I}^{\xi}$ $\displaystyle=$
$\displaystyle\left(\boldsymbol{P}_{e}^{T}\cdot\boldsymbol{F}_{e}-J_{t}\psi_{e}\boldsymbol{F}_{t}^{-1}\right):\frac{\partial\boldsymbol{F}_{t}}{\partial\xi_{I}}-J_{t}\left.\frac{\partial\psi_{e}}{\partial\xi_{I}}\right|_{\boldsymbol{F}_{e}}-\rho_{0}J\frac{\partial\psi_{loc}^{\xi}}{\partial\xi_{I}}+\nabla_{0}\cdot\left(\rho_{0}J\boldsymbol{F}^{-1}\cdot\frac{\partial\psi^{\xi}_{\nabla}}{\partial\nabla\xi_{I}}\right),\,\text{for
}I=1,\ldots,M.$ (2.31)

The thermodynamic forces given in Eq. (2.31)1,2,3 can also be expressed as
functions of the Cauchy stresses and the gradients in the current
configuration as

$\displaystyle X_{0}^{\eta}$ $\displaystyle=$
$\displaystyle\left(J\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma}_{e}\cdot\boldsymbol{F}-J_{t}\psi_{e}\boldsymbol{I}\right):\boldsymbol{F}_{t}^{-1}\cdot\frac{\partial\boldsymbol{F}_{t}}{\partial\eta_{0}}-J_{t}\left.\frac{\partial\psi_{e}}{\partial\eta_{0}}\right|_{\boldsymbol{F}_{e}}-\rho_{0}J\frac{\partial(\breve{\psi}^{\theta}+\psi^{\nabla}+\psi^{\xi})}{\partial\eta_{0}}-\rho_{0}\frac{\partial(\tilde{\psi}^{\theta}+\psi^{p})}{\partial\eta_{0}}+$
$\displaystyle
J\nabla\cdot\left(\rho_{0}\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{0}}\right);$
$\displaystyle X_{i}^{\eta}$ $\displaystyle=$
$\displaystyle\left(J\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma}_{e}\cdot\boldsymbol{F}-J_{t}\psi_{e}\boldsymbol{I}\right):\boldsymbol{F}_{t}^{-1}\cdot\frac{\partial\boldsymbol{F}_{t}}{\partial\eta_{i}}-J_{t}\left.\frac{\partial\psi_{e}}{\partial\eta_{i}}\right|_{\boldsymbol{F}_{e}}-\rho_{0}J\frac{\partial(\breve{\psi}^{\theta}+\psi^{\nabla})}{\partial\eta_{i}}-\rho_{0}\frac{\partial(\tilde{\psi}^{\theta}+\psi^{p})}{\partial\eta_{i}}+$
$\displaystyle
J\nabla\cdot\left(\rho_{0}\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{i}}\right),\quad\text{for
}i=1,\ldots,N;$ $\displaystyle X_{I}^{\xi}$ $\displaystyle=$
$\displaystyle\left(J\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma}_{e}\cdot\boldsymbol{F}-J_{t}\psi_{e}\boldsymbol{I}\right):\boldsymbol{F}_{t}^{-1}\cdot\frac{\partial\boldsymbol{F}_{t}}{\partial\xi_{I}}-J_{t}\left.\frac{\partial\psi_{e}}{\partial\xi_{I}}\right|_{\boldsymbol{F}_{e}}-\rho_{0}J\frac{\partial\psi_{loc}^{\xi}}{\partial\xi_{I}}+J\nabla\cdot\left(\rho_{0}\frac{\partial\psi^{\xi}_{\nabla}}{\partial\nabla\xi_{I}}\right),\quad\text{for
}I=1,\ldots,M,$ (2.32)

where Eq. (2.23), and the relations
$\frac{\partial\psi^{\nabla}}{\partial\nabla_{0}\eta_{l}}=\boldsymbol{F}^{-1}\cdot\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{l}}$
[79], and $\displaystyle\nabla_{0}\cdot(J\boldsymbol{F}^{-T})=\boldsymbol{0}$
(see Sec. 2.2.2 of [123] for the proof) have been used. The thermodynamic
forces given in Eqs. (2.31) and (2.32) can also be compactly expressed as

$\displaystyle X_{l}^{\eta}$ $\displaystyle=$
$\displaystyle-\rho_{0}\frac{\partial\psi}{\partial\eta_{l}}+\nabla_{0}\cdot\left(\rho_{0}J\frac{\partial\psi^{\nabla}}{\partial\nabla_{0}\eta_{l}}\right)=-J\rho\frac{\partial\psi}{\partial\eta_{l}}+J\nabla\cdot\left(\rho
J\frac{\partial\psi^{\nabla}}{\partial\nabla\eta_{l}}\right)\quad\text{for
}l=0,1,2,\ldots,N;$ $\displaystyle X_{I}^{\xi}$ $\displaystyle=$
$\displaystyle-\rho_{0}\frac{\partial\psi}{\partial\xi_{I}}+\nabla_{0}\cdot\left(\rho_{0}J\frac{\partial\psi^{\xi}_{\nabla}}{\partial\nabla_{0}\xi_{I}}\right)=-J\rho\frac{\partial\psi}{\partial\xi_{I}}+J\nabla\cdot\left(\rho
J\frac{\partial\psi^{\xi}_{\nabla}}{\partial\nabla\xi_{I}}\right)\quad\text{for
}I=1,2,\ldots,M.$ (2.33)

Ginzburg-Landau equations: In deriving the Ginzburg-Landau equations, we
further simplify our theory by decoupling the inequality (2.19)2 into the
following sufficient conditions:

$\rho_{0}D_{\eta 0}=X^{\eta}_{0}\dot{\eta}_{0}\geq 0,\qquad\text{
and}\qquad\rho_{0}{\mathcal{D}}_{\eta
i}=\sum_{i=1}^{N}X_{i}^{\eta}\dot{\eta}_{i}\geq 0.$ (2.34)

Using the inequality (2.34)1, we derive

$\dot{\eta}_{0}=L_{0M}X_{0}^{\eta},$ (2.35)

where $L_{0M}\geq 0$ is the kinetic coefficient for $\sf A\leftrightarrow\sf
M$ transformations. Following [87, 78] we introduce the rates of the order
parameters as

$\displaystyle\dot{\eta}_{i}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}\dot{\eta}_{ij},\qquad\text{where
}\dot{\eta}_{ij}=-\dot{\eta}_{ji}\text{ and }\dot{\eta}_{ii}=0\quad\text{(no
sum) for all }i,j=1,\ldots,N;$ $\displaystyle\dot{\xi}_{I}$ $\displaystyle=$
$\displaystyle\sum_{J=1}^{M}\dot{\xi}_{IJ},\qquad\text{where
}\dot{\xi}_{IJ}=-\dot{\xi}_{JI}\text{ and }\dot{\xi}_{II}=0\quad\text{(no sum)
for all }I,J=1,\ldots,M$ (2.36)

for driving the kinetic equations for the order parameters $\eta_{i}$
($i=1,\ldots,N$) and $\xi_{I}$ ($I=1,\ldots,M$). Using Eq. (2.36) into the
dissipation inequalities (2.34)2 and (2.19)3 we can show that the
corresponding dissipations can be rewritten as (see [87] for the proof)

$\displaystyle\rho_{0}{\mathcal{D}}_{\eta
i}=\sum_{j=1}^{N-1}\sum_{i=j+1}^{N}X_{ij}^{\eta}\dot{\eta}_{ij}\geq
0,\quad\text{and}\quad\rho_{0}{\mathcal{D}}_{\xi}=\sum_{J=1}^{M-1}\sum_{I=J+1}^{M}X_{IJ}^{\xi}\dot{\xi}_{IJ}\geq
0,$ (2.37)

where $X_{ij}^{\eta}=X_{i}^{\eta}-X_{j}^{\eta}$ and
$X_{IJ}^{\xi}=X_{I}^{\xi}-X_{J}^{\xi}$. Using the inequality (2.37)1 we derive
the kinetic equations for the evolution of the variants as

$\displaystyle\dot{\eta}_{ij}=L_{ij}^{\eta}(X_{i}^{\eta}-X_{j}^{\eta}),$
(2.38)

where $L_{ij}^{\eta}\geq 0$ is the kinetic coefficient for ${\sf
M}_{i}\leftrightarrow{\sf M}_{j}$ transformations, and
$L_{ij}^{\eta}=L_{ji}^{\eta}$ by the Onsager’s reciprocity theorem [127].
Following [128] we consider the coefficient as

$\displaystyle L_{ij}^{\eta}\left\\{\begin{array}[]{@{}lr@{}}\neq 0&\text{if
}(X_{i}^{\eta}-X_{j}^{\eta})\geq
0\quad\text{and}\quad\\{0\leq\eta_{i}<1\,\,\&\,\,0<\eta_{j}\leq 1\\}\\\ \neq
0&\text{if }(X_{i}^{\eta}-X_{j}^{\eta})\leq
0\quad\text{and}\quad\\{0<\eta_{i}\leq 1\,\,\&\,\,0\leq\eta_{j}<1\\}\\\
=0&\text{if }(X_{i}^{\eta}-X_{j}^{\eta})\geq
0\quad\text{and}\quad\\{\eta_{i}=1\,\,\text{or}\,\,\eta_{j}=0\\}\\\
=0&\text{if }(X_{i}^{\eta}-X_{j}^{\eta})\leq
0\quad\text{and}\quad\\{\eta_{i}=0\,\,\text{or}\,\,\eta_{j}=1\\};\end{array}\right.$
(2.43)

also see [78]. Substituting Eq. (2.38) into Eq. (2.36)1, the Ginzburg-Landau
equations for all $N$ order parameters $\eta_{1},\ldots,\eta_{N}$ are obtained
as

$\displaystyle\dot{\eta}_{i}=\sum_{j=1,j\neq
i}^{N}L_{ij}^{\eta}(X_{i}^{\eta}-X_{j}^{\eta})\qquad\text{for
}i=1,2,\ldots,N.$ (2.44)

The kinetic coefficient, for example, $L_{ij}^{\eta}$ in Eq. (2.44), is
considered as a piecewise constant function of the driving force
$X_{i}^{\eta}-X_{j}^{\eta}$ given by Eq. (2.43) rather than considering it as
a constant similar to that of [87]. We now show that considering
$L_{ij}^{\eta}$ as constant leads to contradictions. Without the loss of
generality, let us assume a martensitic region ($\eta_{0}=1$) where three
variants (denoted by $\sf M_{1}$, $\sf M_{2}$, and $\sf M_{3}$) are evolving.
The order parameters for the variants are $\eta_{1}$, $\eta_{2}$, and
$\eta_{3}$, which are constrained by $\eta_{1}+\eta_{2}+\eta_{3}=1$ as per Eq.
(2.1). We can treat $\eta_{1}$, and $\eta_{2}$ as two independent order
parameters, and the corresponding Ginzburg-Landau equations using from Eq.
(2.44) are written as

$\displaystyle\dot{\eta}_{1}=L_{12}^{\eta}(X_{1}^{\eta}-X_{2}^{\eta})+L_{13}^{\eta}(X_{1}^{\eta}-X_{3}^{\eta}),\quad\text{and}\quad\dot{\eta}_{2}=L_{12}^{\eta}(X_{2}^{\eta}-X_{1}^{\eta})+L_{23}^{\eta}(X_{2}^{\eta}-X_{3}^{\eta}),$
(2.45)

and $\dot{\eta}_{3}=-\dot{\eta}_{1}-\dot{\eta}_{2}$. We now consider a
subregion within the $\sf M$ region where only the variants $\sf M_{1}$ and
$\sf M_{2}$ are coexisting, and $\sf M_{3}$ is absent. The order parameters
$\eta_{1}$ and $\eta_{2}$ must be determined using the Ginzburg-Landau
equation $\dot{\eta}_{1}=L_{12}^{\eta}(X_{1}^{\eta}-X_{2}^{\eta})$ and the
constraint $\eta_{1}+\eta_{2}=1$, which are, however, possible to achieve from
Eqs. (2.45)1 and (2.45)2 if and only if $L_{13}^{\eta}=L_{23}^{\eta}=0$ in
that subregion. Such conditions clearly cannot be achieved with a constant
$L_{ij}^{\eta}$, but when $L_{ij}^{\eta}$ is a piecewise function, for
example, given by Eq. (2.43). The last two conditions in Eq. (2.43) say that
it cannot be transformed to any other variant when, say, $\sf M_{i}$ is
absent.

Similarly, using the dissipation inequality (2.19)3, we derive the Ginzburg-
Landau equations for $\xi_{1},\ldots,\xi_{M}$ as

$\displaystyle\dot{\xi}_{I}=\sum_{J=1,J\neq
I}^{M}L_{IJ}^{\xi}(X_{I}^{\xi}-X_{J}^{\xi})\qquad\text{for }I=1,2,\ldots,M,$
(2.46)

where the kinetic coefficients $L_{IJ}^{\xi}=L_{JI}^{\xi}\geq 0$ for
transformations between the grains $G_{I}$ and $G_{J}$ is taken similar to
that for $L_{ij}^{\eta}$ given by Eq. (2.43), where obviously we read
$X_{i}^{\eta}$ and $X_{j}^{\eta}$ as $X_{I}^{\xi}$ and $X_{J}^{\xi}$,
respectively.

Boundary conditions: Assuming that the external surface energy of the sample
(say, $V_{0}$) remains unchanged during the MTs, we consider the following
Neumann BC for $\eta_{0},\eta_{1},\ldots,\eta_{N}$ expresses on the both
undeformed and deformed surfaces of the body (see [79] for derivation):

$\displaystyle\rho_{0}\frac{\partial\psi}{\partial\nabla_{0}\eta_{l}}\cdot\boldsymbol{n}_{0}$
$\displaystyle=$ $\displaystyle 0\quad\text{on }S_{0}\quad\text{for
}l=0,1,\ldots,N;$
$\displaystyle\rho\frac{\partial\psi}{\partial\nabla\eta_{l}}\cdot\boldsymbol{n}$
$\displaystyle=$ $\displaystyle 0\quad\text{on }S\quad\text{for
}l=0,1,\ldots,N;$ (2.47)

where $\boldsymbol{n}$ is the unit outward normal to the surface in the
deformed body. Similarly, assuming the surface energy does not change during
the evolution of the GBs, the Neumann BC for $\xi_{1},\ldots,\xi_{M}$ is
expressed as

$\displaystyle\rho_{0}\frac{\partial\psi}{\partial\nabla_{0}\xi_{I}}\cdot\boldsymbol{n}_{0}$
$\displaystyle=$ $\displaystyle 0\quad\text{on }S_{0}\quad\text{for
}I=1,\ldots,M;$
$\displaystyle\rho\frac{\partial\psi}{\partial\nabla\xi_{I}}\cdot\boldsymbol{n}$
$\displaystyle=$ $\displaystyle 0\quad\text{on }S\quad\text{for
}I=1,\ldots,M.$ (2.48)

The initial conditions (ICs) on the order parameters for numerical
calculations will be specified while discussing the results in Sec. 4.

The complete system of coupled mechanics and PF equations derived in this
section is collected and enlisted in Appendix B.

##  3 Material parameters identification

We now calibrate the material parameters involved in the kinematic model for
the transformation stretch, free energy, GB energy, Ginzburg-Landau equations
(see Eqs. (2.10), (LABEL:MF2), (2.17), (2.35), (2.43), and (2.46)) using the
atomistic simulation and experimental data from the literature. Following [79,
87], we simplify the Ginzburg-Landau equation (2.35) for $\sf
A\leftrightarrow\sf M$ transformations in a single grain (say, $\xi_{1}=1$ and
$\xi_{I}=0$ for $I=2,\ldots,M$) for a system with $\sf A$ and a single $\sf M$
variant and a planar interface between them as (neglecting mechanics)

$\displaystyle\dot{\eta}_{0}=L_{0M}\left[-\rho_{0}\Delta\psi^{\theta}(6\eta_{0}-6\eta_{0}^{2})-\rho_{0}A_{0M}(2\eta_{0}-6\eta_{0}^{2}+4\eta_{0}^{3})+\beta_{0M}\frac{\partial^{2}\eta_{0}}{\partial
r_{02}^{2}}\right],$ (3.1)

where the parameters $A_{0M}$, $\beta_{0M}$, and $L_{0M}$ are assumed to be
constants. The solution to Eq. (3.1) is [80, 88]

$\displaystyle\eta_{0}(r_{02},t)$ $\displaystyle=$ $\displaystyle
0.5+0.5\operatorname{tanh}[3(r_{02}-r_{0c}-c_{0M}t)/\delta_{0M}],\quad\text{where}$
$\displaystyle\delta_{0M}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{18\beta_{0M}}{\rho_{0}A_{0M}}},\quad\gamma_{0M}=\frac{\beta_{0M}}{\delta_{0M}},\quad\text{and}\quad
c_{0M}=L_{0M}\delta_{0M}\rho_{0}\Delta\psi^{\theta}(\theta),$ (3.2)

are the width, energy, and speed, respectively, of the $\sf A$-$\sf M$
interface, and $r_{0c}$ is the position of the centre of the interface where
$\eta_{0}=0.5$. Note that the model parameters $A_{0M}$, $\beta_{0M}$, and
$L_{0M}$ can be determined using the directly measurable quantities (either
experimentally or by atomistic simulations) $\delta_{0M}$, $\gamma_{0M}$,
$c_{0M}$, and $\Delta\psi^{\theta}(\theta)$ from Eqs. (3.2)2,3,4.

In a similar manner, we can calibrate the parameters related to the ${\sf
M}_{i}\leftrightarrow{\sf M}_{j}$ transformations (for all
$i,j=1,\ldots,N;\,i\neq j$) and $G_{I}\leftrightarrow G_{J}$ transformations
(for all $I,J=1,\ldots,M;\,I\neq J$). We consider the equations for ${\sf
M}_{i}\leftrightarrow{\sf M}_{j}$ transformations in a single grain, and
$G_{I}$-$G_{J}$ transformations in a stress-free austenitic bicrystal
($\eta_{0}=0$), which are given by (obtained using Eqs. (C.26)2 and (C.26)3,
respectively, and neglecting mechanics)

$\displaystyle\dot{\eta}_{i}$ $\displaystyle=$ $\displaystyle
L_{ij}^{\eta}\left[-\rho_{0}A_{ij}^{\eta}(2\eta_{i}-6\eta_{i}^{2}+4\eta_{i}^{3})+\beta_{ij}^{\eta}\frac{\partial^{2}\eta_{i}}{\partial
r_{02}^{2}}\right],\,\,\text{and}$ $\displaystyle\dot{\xi}_{I}$
$\displaystyle=$ $\displaystyle
L_{IJ}^{gb}\left[-\rho_{0}A_{IJ}^{gb}(0)(2\xi_{I}-6\xi_{I}^{2}+4\xi_{I}^{3})+\beta_{IJ}^{gb}(0)\frac{\partial^{2}\xi_{I}}{\partial
r_{02}^{2}}\right],$ (3.3)

respectively, where there is no sum on $i,\,j,\,I,\,J$, and we have used the
constrains $\eta_{i}+\eta_{j}=1$ and $\xi_{I}+\xi_{J}=1$. The ${\sf
M}_{i}$-${\sf M}_{j}$ interface and GB $\Gamma_{IJ}$ in Eqs. (3.3)1 and (3.3)2
are considered as planar. The solution for Eqs. (3.3)1 and (3.3)2 are [80, 88]

$\displaystyle\eta_{i}(r_{02},t)=0.5+0.5\operatorname{tanh}[3(r_{02}-r_{0c})/\delta_{ij}^{\eta}],\quad\text{and}\quad\xi_{I}(r_{02},t)=0.5+0.5\operatorname{tanh}[3(r_{02}-r_{0c})/\delta_{IJ}^{gb}],$
(3.4)

respectively, where $r_{0c}$ is the location of the middle of the respective
interfaces, and the width, energy, and speed of these respective interfaces
are

$\displaystyle\delta_{ij}^{\eta}=\sqrt{\frac{18\beta_{ij}^{\eta}}{\rho_{0}A_{ij}^{\eta}}},\quad\gamma_{ij}^{\eta}=\frac{\beta_{ij}^{\eta}}{\delta_{ij}^{\eta}},\quad
c_{ij}^{\eta}=0;\quad\text{and}\quad\delta_{IJ}^{gb}=\sqrt{\frac{18\beta_{IJ}^{gb}(0)}{\rho_{0}A_{IJ}^{gb}(0)}},\quad\gamma_{IJ}^{gb}(0)=\frac{\beta_{IJ}^{gb}(0)}{\delta_{IJ}^{gb}},\quad
c_{IJ}^{gb}=0.$ (3.5)

For the numerical calculations presented in Sec. 4, we consider the material
constants for Ni-rich NiAl alloy, which exhibit cubic ($\mathsf{A}$) to
tetragonal ($\sf M$) transformations [1, 8]. The corresponding Bain tensors
for variants $\sf M_{1}$, $\sf M_{2}$, and $\sf M_{3}$, in the standard
$\\{\boldsymbol{c}_{1},\boldsymbol{c}_{2},\boldsymbol{c}_{3}\\}$ basis
attached to a cubic $\sf A$ unit cell in a reference grain, are (see Chapter 4
of [1])

$\boldsymbol{U}_{t1}=diag(\chi,\alpha,\alpha),\quad\boldsymbol{U}_{t2}=diag(\alpha,\chi,\alpha),\quad\text{and}\quad\boldsymbol{U}_{t3}=diag(\alpha,\alpha,\chi),$
(3.6)

respectively, where $\alpha=0.92$, and $\chi=1.22$ for NiAl alloy [8], and the
argument of $diag(\cdot)$ shows the diagonal elements of the tensor. For all
the simulations, we will consider $a=a_{b}=a_{\beta}=a_{\theta}=3$,
$a_{c}=10^{-3}$, $\delta_{0M}=1.5$ nm, $\gamma_{0M}=0.2$ N/m,
$\delta_{12}^{\eta}=0.75$ nm, and $\gamma_{12}^{\eta}=0.03$ N/m (see, e.g.
[65, 87] and the references therein for typical values of the widths and
energies). Hence using Eqs. (3.2)2,3,4 and (3.5) we have $\rho_{0}A_{0M}=2400$
MJ/m3, $\rho_{0}A_{12}^{\eta}=720$ MJ/m3, $\beta_{0M}=3\times 10^{-10}$ N, and
$\beta_{12}^{\eta}=2.25\times 10^{-11}$ N. A typical austenitic GB energy
$\gamma^{A}_{IJ}=0.9$ N/m [129] is considered for all the calculations,
whereas different values of $\Delta\gamma_{IJ}^{gb}$ (and hence different
$\gamma^{M}_{IJ}$) are considered, and they would be specified for the
simulations at hand in Sec. 4; $I,J=1,2,3$. In this paper, the results for the
austenitic GB width $\delta_{IJ}^{gb}$ between $0.5$ nm and $1.7$ nm are
presented; see, e.g. [130] and Chapter 7 of [131] for typical values. Based on
the atomistic simulation results of [8], we take the critical temperatures for
${\sf A}\to{\sf M}$ and ${\sf M}\to{\sf A}$ transformations for NiAl alloy as
$\theta^{c}_{{\sf A}\to{\sf M}}=240$ K and $\theta^{c}_{{\sf M}\to{\sf
A}}=490$ K, respectively. The thermodynamic equilibrium temperature between
$\sf A$ and $\sf M$ is considered as $\theta_{e}=0.5(\theta^{c}_{{\sf
A}\to{\sf M}}+\theta^{c}_{{\sf M}\to{\sf A}})$; see, e.g. [132] and the
references therein. Thus, $\theta_{e}=365$ K for NiAl alloy. In [132], the
critical temperatures are shown to be related to the parameters
$\rho_{0}A_{0M}$, $\theta_{e}$, and $\rho_{0}\Delta s$ by $\theta^{c}_{{\sf
A}\to{\sf M}}=\theta_{e}+\rho_{0}A_{0M}/(3\rho_{0}\Delta s)$ and
$\theta^{c}_{{\sf M}\to{\sf A}}=\theta_{e}-\rho_{0}A_{0M}/(3\rho_{0}\Delta
s)$, respectively, using which we get $\rho_{0}\Delta s=-6.4$ MJ/(m3K). We
assume the isotropic elastic response of the phases, and identical Lamé
constants ($\lambda,\,\mu$) are assumed for $\sf A$, $\sf M_{1}$, and $\sf
M_{2}$: $\lambda=74.6$ GPa, and $\mu=72$ GPa [87]. The kinetic coefficients of
the phases are taken as $L_{0M}=L_{12}^{\eta}=2600$ (Pa-s)-1 [87].

Figure 2: Schematic of the unit cells of cubic austenite and tetragonal
martensitic variants ${\sf M}_{1}$ and ${\sf M}_{2}$ in the
$\boldsymbol{c}_{1}$-$\boldsymbol{c}_{2}$ plane, where
$\\{\boldsymbol{c}_{1},\boldsymbol{c}_{2},\boldsymbol{c}_{3}\\}$ is an
orthonormal basis attached to the cubic $\sf A$ unit cell.

##  4 Numerical results and discussions

We now present the numerical results. Assuming the plane stress condition and
isotropic elastic response, the results for the 2D bicrystals and tricrystals
with tilt GBs are presented in Secs. 4.1 and 4.2, respectively, considering
the PTs between $\sf A$ and two variants. We consider $\sf M_{1}$ and $\sf
M_{2}$ as those two variants without a loss of generality. The schematic of
the unit cells is shown in Fig. 2 in the
$\boldsymbol{c}_{1}$-$\boldsymbol{c}_{2}$ plane, where
$\boldsymbol{c}_{3}$-axis is parallel to the outward normal to that plane. The
governing equations for both the bicrystals and tricrystals, obtained using
the general system of equations derived in Sec. 2, are enlisted in Appendix C.
Figures 3(a) and (b) show two typical undeformed austenitic bicrystals
($V_{0}$) with a planar GB and a circular GB, respectively. Figure 3(c) shows
a typical undeformed austenitic tricrystal ($V_{0}$) with three planar GBs
with identical energy and width, which are meeting in the middle of the sample
at $120^{\circ}$ dihedral angles according to the equilibrium condition at the
TJ (see, e.g. Sec. 3.3.3. of [2]). The exact sample size and the GB
parameters, including $\gamma_{IJ}^{M}$, $\delta^{gb}_{IJ}$,
$\rho_{0}K_{IJ}^{gb}$, and $\rho_{0}K_{HIJ}$ are specified while presenting
the simulation results. The initial stress-free austenitic bicrystals are
simply obtained by using the analytical solution for $\xi_{1}$ given by Eq.
(3.4)2. The Eqs. (C.26)3, (C.26)4, and (C.26)5 have been solved, neglecting
mechanics until the stationary distribution of the order parameters $\xi_{1}$,
$\xi_{2}$, and $\xi_{3}$ are obtained to get all the stress-free austenitic
tricrystals (see e.g. Fig. 3(c)), where we have used the Neumann BC given by
Eq. (C.32)3 on all the surfaces, and
$L^{\xi}_{12}=L^{\xi}_{13}=L^{\xi}_{23}=100$ (Pa-s)-1 when the coefficients
are non-zero; see the discussion of Sec. 2.4.2. The nonlinear finite element
procedure based on the algorithm presented in [63] has been developed, and it
is implemented using the open-source package deal.ii [133]. The quadratic
quadrilateral elements are used for all the simulations. The mesh sizes in all
the samples are so chosen that at least four grid points are present across
all the interfaces to ensure the mesh-independent results. The time
derivatives in the Ginzburg-Landau equations (C.22)1,2 for the bicrystals and
(C.26)1,2 for the tricrystals are discretized using the Euler’s backward
difference scheme of order one, and a time step of size $\Delta t=10^{-14}$ s
is used for all the calculations. While solving the equilibrium equations
given by Eq. (C.21) and the PF equations for the bicrystals (C.22)1,2 or
tricrystals (C.26)1,2, we have applied the following BCs during both the
forward and reverse MTs: (i) the external surfaces are free of traction in
their tangential directions; (ii) all the external surfaces are supported by
rollers, i.e. the normal displacement at each boundary is zero, except for the
example of Sec. 4.1.5, and it will be specified there itself; (iii) the
homogeneous Neumann BCs for $\eta_{0}$ and $\eta_{1}$ given by Eqs. (C.32)1
and (C.32)2 are applied on the surfaces. The same time discretization scheme
and time step size mentioned above are used for solving Eqs. (C.26)3,4,5 for
obtaining the stress-free tricrystals. The orientation of the basis
$\\{\boldsymbol{c}_{1},\boldsymbol{c}_{2}\\}$ for each austenitic grain is
shown in Figs. 3(a,b,c). A fixed basis
$\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3}\\}$ is also
considered, where $\boldsymbol{e}_{3}$ being parallel to the
$\boldsymbol{c}_{3}$-axis. The Bain tensors $\boldsymbol{U}_{ti}^{(I)}$
($i=1,2$) for grain $G_{I}$, introduced in Eq. (2.10), is obtained using Eq.
(3.6) in Eq. (2.7), considering

$[Q^{(I)}]=\begin{bmatrix}\cos\vartheta_{I}&\sin\vartheta_{I}&0\\\
-\sin\vartheta_{I}&\cos\vartheta_{I}&0\\\ 0&0&1\\\ \end{bmatrix},$ (4.1)

where $\vartheta_{I}$ is the angle of the $\boldsymbol{c}_{1}$-axis from the
$\boldsymbol{e}_{1}$-axis for $G_{I}$ (see Fig. 3), and it is considered
positive if anticlockwise. In the 2D samples, each of the GBs $\Gamma_{IJ}$
has only one misorientation angle, i.e. the list $\Theta_{IJ}$ introduced in
Eq. (2.17) has one element, which we define as

$\vartheta_{IJ}=\vartheta_{I}-\vartheta_{J}.$ (4.2)

Figure 3: Typical stress-free austenitic samples ($V_{0}$) of $30$
nm$\times$30 nm size: bicrystals with (a) a planar tilt GB and (b) a circular
tilt GB of $15$ nm diameter; (iii) tricrystal with planar tilt GBs meeting at
$120^{\circ}$ dihedral angles at the TJ. We used $\gamma_{12}^{A}=0.9$ N/m and
$\delta_{IJ}^{gb}=1$ nm. For the tricrystal,
$\rho_{0}K_{IJ}^{gb}=\rho_{0}K_{123}=800$ GPa. Orientation of the basis
$\\{\boldsymbol{c}_{1},\boldsymbol{c}_{2}\\}$ for each grain is shown;
$\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2}\\}$ is a fixed reference frame.

###  4.1 Results for bicrystals

The results for the bicrystals are presented here. The typical austenitic
initial samples are shown in Fig. 3(a) and (b). The effect of the GB width
$\delta_{12}^{gb}$, the difference in GB energy $\Delta\gamma_{12}^{gb}$,
misorientation $\vartheta_{12}$, applied strains, sample size, and GB
curvature are studied for samples with a symmetric tilt GB during the forward
and reverse transformations. A bicrystal with an asymmetric planar tilt GB is
also considered. The role of compatibility of the Bain strains across the GB
is studied.

Figure 4: Transformation temperature $\theta$ versus a nondimensional width
ratio $\bar{\Delta}_{12}=\delta_{12}^{gb}/\delta_{0M}$ in a $30$ nm $\times$
30 nm bicrystal with a planar symmetric tilt GB. We used $\gamma_{12}^{A}=0.9$
N/m, $\Delta\gamma_{12}^{gb}=-0.5$ N/m, $\vartheta_{12}=40^{\circ}$, and no
external strain.

####  4.1.1  GB size (width)-effect

Figure 5: Free energy per unit volume (see Eq. (2.13)) variation with
$\eta_{0}$ neglecting mechanics at $\theta^{c}_{A\to
M}\leq\theta\leq\theta^{c}_{M\to A}$ in a (a) homogeneous grain, and (b) in
the middle of a long planar GB of a bicrystal, where $\xi_{1}=0.5$. We used
$\gamma_{12}^{A}=0.9$ N/m, $\Delta\gamma_{12}^{gb}=-0.5$ N/m, and
$\delta_{12}^{gb}=1$ nm.

We consider an austenitic bicrystal of $30$ nm $\times$ $30$ nm size with a
planar symmetric tilt GB, where $\vartheta_{1}=-\vartheta_{2}=20^{\circ}$, and
hence $\vartheta_{12}=40^{\circ}$ by Eq. (4.2) (also see Fig. 3(a)). We vary
$\delta_{12}^{gb}$ between $0.5$ nm and $1.7$ nm, while
$\Delta\gamma_{12}^{gb}$ is fixed to $-0.5$ N/m. The GB energy in the $\sf M$
phase is thus $\gamma_{gb}^{M}=0.4$ N/m.

Transformation temperature variation with $\bar{\Delta}_{12}$:

The variation of the temperatures for transformations between $\sf A$, $\sf
PM$, and $\sf M$ within the GB region for varying non-dimensional GB width
$\bar{\Delta}_{12}:=\delta_{12}^{gb}/\delta_{0M}$ is shown in Fig. 4, where
the value for $\delta_{0M}$ is fixed and is given in Sec. 3. The curve
indicated by $\sf A\to\sf PM$ represents the temperature (denoted by
$\theta_{\sf A\to\sf PM}$) at which a $\sf PM$ layer appears in a jump-like
manner in the GB region with $0.05\leq max(\eta_{0})\leq 0.95$. However, $0.1$
K above the corresponding $\theta_{\sf A\to\sf PM}$, the entire sample was
austenitic. For obtaining $\theta_{\sf A\to\sf PM}$, the ICs on $\eta_{0}$ and
$\eta_{1}$ were taken as $0\leq\eta_{0}(\boldsymbol{r}_{0},t=0)\leq 0.02$ and
$0\leq\eta_{1}(\boldsymbol{r}_{0},t=0)\leq 0.1$, both distributed randomly all
over the sample. The initial sample was hence in the $\sf A$ phase (see Sec.
2.1). As the sample temperature is further decreased from $\theta_{\sf A\to\sf
PM}$, the $\sf PM$ layer thickness increases, and at the temperatures
corresponding to the curve indicated by $\sf PM\to\sf M$ and denoted by
$\theta_{\sf PM\to\sf M}$, the GB region and the adjacent part of the grains
transform to complete $\sf M$ with $0.95<max(\eta_{0})\leq 1$ in a jump-like
manner. However, that region was $\sf PM$ only above $0.1$ K of $\theta_{\sf
PM\to\sf M}$. We see from Fig. 4 that all the $\theta_{\sf A\to\sf PM}$ and
$\theta_{\sf PM\to\sf M}$ lie within the temperature regime of stable $\sf A$,
and they are significantly higher than the critical temperature $\theta_{\sf
A\to\sf M}^{c}=240$ K. To explain the reason, we note that the local part of
the free energy per unit volume of the bicrystal within the GB region is given
by

$\rho_{0}\psi^{loc}(\eta_{0},\theta)=\rho_{0}A_{0M}\eta_{0}^{2}(1-\eta_{0})^{2}-\rho_{0}\Delta
s\,(\theta-\theta_{e})\eta_{0}^{2}(3-2\eta_{0})+\frac{36}{\delta_{12}^{gb}}\left[\gamma^{A}_{12}+\Delta\gamma_{12}^{gb}\eta_{0}^{2}(3-2\eta_{0})\right]\xi_{1}^{2}(1-\xi_{1})^{2},$
(4.3)

which we have obtained from Eqs. (2.13) and (LABEL:MF2) by neglecting
mechanics and considering the transformations between $\sf A$ and a single
$\sf M$ variant only. In Eq. (4.3), we have also used the relation

$A_{12}^{gb}(\eta_{0})\xi_{1}^{2}(1-\xi_{1})^{2}+0.5\beta_{12}^{gb}(\eta_{0})\left(\frac{\partial\xi_{1}}{\partial
r_{02}}\right)^{2}=\frac{36\gamma_{12}^{gb}(\eta_{0})}{\delta_{12}^{gb}}\xi_{1}^{2}(1-\xi_{1})^{2},$

which can be easily proved using the analytical solution for $\xi_{1}(r_{02})$
given by Eq. (3.4)2, and Eqs. (3.5)4,5. Note that we have also used
$\partial\xi_{1}/\partial r_{01}=0$. The last term on the right-hand side of
Eq. (4.3) vanishes within the homogeneous grains where $\xi_{1}=0$ or $1$. We
show the local energy $\rho_{0}\psi^{loc}$ plots within the particles of a
homogeneous grain and in the middle line of the planar GB where $\xi_{1}=0.5$
and the energy is given by Eq. (4.3) is the maximum in Figs. 5(a) and 5(b),
respectively, for the temperatures between $\theta^{c}_{A\to M}=240$ K and
$\theta^{c}_{M\to A}=490$ K. We have used $\gamma_{12}^{A}=0.9$ N/m,
$\Delta\gamma_{12}^{gb}=-0.5$ N/m, and $\delta_{12}^{gb}=1$ nm and all other
material parameters are listed in Sec. 3. From Fig. 5(b), it is obvious that
the barrier energy between $\sf A$ ($\eta_{0}=0$) and $\sf M$ ($\eta_{0}=1$)
on the $\xi_{1}=0.5$ line vanishes at $\theta$ much above the
$\theta^{c}_{A\to M}$, at which the barrier energy vanishes in the homogeneous
grains (see Fig. 5(a)). Hence the instability temperatures at which $\sf PM$
or $\sf M$ nucleates in the GB region lie within the stability regime of $\sf
A$, and it is determined by the values of $\delta_{IJ}^{gb}$ and
$\Delta\gamma_{IJ}^{gb}$. The actual scenario, however, is more complex due to
the interaction of the thermal problem with the mechanics, which is the case,
for example, in Fig. 4 also.

We now determine the temperature for the reverse ($\sf PM\to\sf A$)
transformation, which is denoted as $\theta_{\sf PM\to\sf A}$ and indicated by
$\sf PM\to\sf A$ curve in Fig. 4. For that, we have used the sample obtained
at $\theta_{\sf A\to\sf PM}$ as the initial sample and increased its
temperature. As $\theta$ increases, the $\sf PM$ layer thickness decreases
along with a decrease in the value of $max(\eta_{0})$. However, at
$\theta=\theta_{\sf A\to\sf PM}$, the $\sf PM$ layer on the GB region abruptly
disappears, and the original austenitic bicrystal is recovered. Below $0.1$ K
of $\theta_{\sf PM\to\sf A}$, the $\sf PM$ layer was, however, present. We see
from Fig. 4 that the difference between $\theta_{\sf A\to\sf PM}$ and
$\theta_{\sf PM\to\sf A}$ increases as $\delta_{12}^{gb}$ (and so
$\bar{\Delta}_{12}$) increases, implying a monotonic increase in the
temperature hysteresis, which is measured as the difference in the
corresponding temperatures, i.e. $\theta_{\sf PM\to\sf A}-\theta_{\sf A\to\sf
PM}$ (see, e.g. [59, 58] in case of surface-induced melting). The hysteresis
is caused by a jump-like discontinuous evolution of the order parameter
$\eta_{0}$ during these two transformations. The hysteresis decreases as the
diffused GB approaches a sharp one, i.e. $\bar{\Delta}_{12}\to 0$. The
difference in the temperatures of transformations between $\theta_{\sf A\to\sf
PM}$ and $\theta_{\sf PM\to\sf M}$ is going down as the GB width increases,
similar to other interface-induced PTs [60, 61, 62, 58, 59, 55]. The
transformation temperatures $\theta_{\sf A\to\sf PM}$ is $59$ K to $101.2$ K
higher, and $\theta_{\sf PM\to\sf M}$ is $54$ K to $85.8$ K higher than the
critical temperature $\theta_{\sf A\to\sf M}^{c}=240$ K over the entire range
of $\bar{\Delta}_{12}$ shown in Fig. 4.

Although we did not find any relevant experimental or atomistic data for GB-
induced MTs in Ni-rich NiAl alloy in the literature, we found the following
limited data for a ferrous alloy and the NiTi alloy, which we would use for
comparing our numerical results qualitatively. The martensitic start
temperature was observed to be $66$ K higher in bicrystal with a planar tilt
GB than that of the single crystal of a ferrous alloy [30]. Ueda et al.
considered a Fe-Ni alloy and studied MTs in two different bicrystals with
identical (crystallographically) tilt GBs in [28] and [29], where both the
samples were obtained using an identical procedure. The martensitic start
temperatures for these two bicrystals were reported as $234$ K [28] and $210$
K [29], i.e. there is a difference of $24$ K. Since the GBs in these two
bicrystals are crystallographically identical, their energies are also the
same. However, a plausible reason for that nucleation temperature difference
could be the difference in GB widths. Since the GB widths were not reported in
these two references, we cannot confirm that. The nucleation temperatures for
those bicrystals (with tilt GB) were reported as $54$ K [28] and $68$ K [29]
higher than that of the single crystals. The atomistic simulations for NiTi
alloy in [50] showed that a tilt GB promoting MTs nucleates $\sf M$ at $16$ K
higher than its single crystal counterpart. The temperature ranges in our
simulations mentioned in the previous paragraph are close to the ranges
reported in these experimental and atomistic studies. Furthermore, the
nucleation of the product phase ($\sf PM$ or $\sf M$) on the GB in a jump-like
manner was also observed in experiments [28, 29], which has been referred to
as ‘the burst transformation’ therein. The experiments and atomistic
simulations in polycrystalline solids have shown that not all the tilt GBs
promote $\sf M$ nucleation [30, 31, 50], the reason for which was not clearly
mentioned. Therefore, it would be interesting to conduct systematic
experiments and molecular simulations to confirm the effects of GB width on
the temperatures of transformations between $\sf A$, $\sf PM$, and $\sf M$ to
compare the present PF results shown in Fig. 4.

Note that very rich transformation temperature plots for MTs with the
variation of the width of the corresponding external surface [60, 61] or a
matrix-precipitate interface [62] were developed, which were shown to agree
with the experiments. Similar plots were also developed for the
transformations between solid, melt, and premelt induced by external surface
in nanoparticles [58] and nanovoids [59], and also induced by GBs in
polycrystals [55], which explain the different temperature ranges for premelt
nucleation to complete melting reported in various experimental and atomistic
studies.

(a) $\eta_{0}$ plots during forward MT (b) $\eta_{eq}$ plots during forward MT
(c) $\eta_{0}$ plots during reverse MT (d) $\eta_{eq}$ plots during reverse MT

Figure 6: Microstructure evolution during (a,b) forward, and (c,d) reverse MTs
in a $30$ nm $\times$ $30$ nm bicrystal having a symmetric planar GB with
$\theta$ variation. We used $\Delta\gamma_{12}^{gb}=-0.5$ N/m,
$\delta_{12}^{gb}=1$ nm, and $\vartheta_{12}=40^{\circ}$. No strain applied.

Figure 7: Plots for the (a) Cauchy elastic stresses $\sigma_{(e)ij}$, (b)
Cauchy structural stresses $\sigma_{(st)ij}$, (c) $\rho_{0}\psi$ \- total
energy per unit volume of $\Omega_{0}$, and (d) driving force $X_{12}^{\eta}$
at $\theta=65$ K within the martensitic sample shown in Fig. 6(a,b) obtained
by forward MT.

(a) $\eta_{0}$ plots during forward MT (b) $\eta_{eq}$ plots during forward MT
(c) $\eta_{0}$ plots during reverse MT (d) $\eta_{eq}$ plots during reverse MT

Figure 8: Microstructure evolution during (a,b) forward, and (c,d) reverse MTs
in a $30$ nm $\times$ $30$ nm bicrystal having a symmetric planar GB with
$\theta$ variation. We used $\Delta\gamma_{12}^{gb}=-0.5$ N/m,
$\delta_{12}^{gb}=1.5$ nm, and $\vartheta_{12}=40^{\circ}$. No strain applied.

Figure 9: Plots for the (a) Cauchy elastic stresses $\sigma_{(e)ij}$, (b)
Cauchy structural stresses $\sigma_{(st)ij}$, (c) $\rho_{0}\psi$ \- total
energy per unit volume of $\Omega_{0}$, and (d) driving force $X_{12}^{\eta}$
at $\theta=100$ K within the martensitic sample shown in Fig. 8(a,b) obtained
by forward MT.

Microstructures evolution:

The microstructure evolution at different $\theta$ during the forward and
reverse transformations is shown in Figs. 6 and 8 for $\delta_{12}^{gb}=1$ nm
($\bar{\Delta}_{12}=0.667$) and $\delta_{12}^{gb}=1.5$ nm
$(\bar{\Delta}_{12}=1$), respectively, while all other parameters are
identical to that used for plotting Fig. 4. We have introduced an equivalent
order parameter defined as $\eta_{eq}=2\eta_{0}(\eta_{1}-0.5)$ for plotting
the evolution of the overall austenite-martensite microstructures in Figs.
6(b,d) and 8(b,d). We note that $\eta_{eq}=0$ in $\sf A$, and it appears as
green, $\eta_{eq}=1$ in ${\sf M}_{1}$, and it appears as dark red, and
$\eta_{eq}=-1$ in ${\sf M}_{2}$, and it appears as dark blue in the plots. The
ICs for $\eta_{0}$ and $\eta_{1}$ would be indicated separately for each case.
The subfigures (a) and (b) of Figs. 6 and 8 show the evolution of $\eta_{0}$
and $\eta_{eq}$, respectively, during the forward MTs when $\theta$ is
decreased in steps. The subfigures (c) and (d) show the the evolution of
$\eta_{0}$ and $\eta_{eq}$, respectively, during the reverse MTs when $\theta$
is increased.

Forward MT. The $\sf PM$ nucleates in the GB regions in a jump-like manner at
$\theta_{n}^{f}=315.5$ K and $\theta_{n}^{f}=302.4$ K, as shown in Figs.
6(a,b) and 8(a,b), when $\delta_{12}^{gb}=1$ nm (or $\bar{\Delta}_{12}=0.667$)
and $1.5$ nm (or $\bar{\Delta}_{12}=1$), respectively. The reason for
introducing a new symbol $\theta_{n}^{f}$ for denoting the nucleation
temperature during the forward MT (instead of simply using $\theta_{\sf
A\to\sf PM}$ introduced above) is that the austenitic GB might, depending on
the material parameters, directly nucleate $\sf M$ instead of transforming via
$\sf PM$. These two temperature values were also reported on the $\sf A\to\sf
PM$ curve for the respective $\bar{\Delta}_{12}$ in Fig. 4. In both the layers
of $\sf PM$ nucleated in the GB region, $max(\eta_{0})=0.1$, and the
$\eta_{eq}$ ranges are $-0.09\leq\eta_{eq}\leq 0$ and $-0.11\leq\eta_{eq}\leq
0$, respectively. Noticing that only a single variant was observed to nucleate
in atomistic studies [39] (also see [111]), we have considered an IC on
$\eta_{1}$, which is biased towards one of the variants, say, $\sf M_{2}$;
this resulted in the nucleation of a layer of pre-$\sf M_{2}$. On choosing an
unbiased initial $\eta_{1}$, say, $\eta_{1}=0.5$ in the whole sample, we see
that the $\sf PM$ nucleates at $\theta_{n}^{f}=310.2$ K for the sample with
$\delta_{12}^{gb}=1$ nm, which is just $5.3$ K lower than that with the biased
IC described above. Such a $\sf PM$ consists of both pre-$\sf M_{1}$ and
pre-$\sf M_{2}$, and there are clear interfaces between them, which penalizes
some energy resulting in the said drop in $\theta_{n}^{f}$. However, this
paper will consider the IC $0\leq\eta_{1}(\boldsymbol{r}_{0},0)\leq 0.1$
distributed randomly for all the simulations related to the $\sf A\to\sf
PM$/$\sf M$ transformations. Our results clearly show that the higher the GB
width, the lower the $\theta_{n}^{f}$ (also see Fig. 4 for other values of
$\delta_{12}^{gb}$). The samples shown in Figs. 6(a,b) and 8(a,b) however
remained in $\sf A$ phase with $\eta_{0}=0$ everywhere at $315.6$ K and
$302.5$ K, respectively. The jump-like nucleation of $\sf PM$ or $\sf M$, also
called ‘the burst transformation’, has been observed experimentally on the GBs
within the temperature regime of $\sf A$ stability [28, 29].

As the temperatures of the samples are further reduced from the respective
$\theta_{n}^{f}$, the respective single-variant pre-$\sf M_{2}$ layers grow
and transform to complete $\sf M$ consisting of two different-scaled and
mutually orthogonal plates of two variants in each of the grains, which is
driven by a reduction in the strain energy. The twins in these two grains are
disoriented by approximately equal to the misorientation angle $40^{\circ}$.
The post-nucleation microstructures in Fig. 6(a)-(b) are shown at $\theta=300$
K, $200$ K, and $65$ K, where at $65$ K, the entire bicrystal is martensitic.
In Fig. 8(a)-(b), the post-nucleation microstructures are shown at
$\theta=280$ K, $200$ K, and $100$ K, where the sample is entirely in $\sf M$
at $100$ K. At $\theta=300$ K and $\theta=280$ K in the individual cases, we
see that the GB region has predominantly transformed to $\sf M_{1}$ and a much
smaller volume fraction of $\sf M_{2}$. By noticing the orientation of the
transformation strains (see Figs. 2, 3), we see by the appearance of a large
volume fraction of $\sf M_{1}$, the system tries to maximize the
transformation strain. However, as the temperature is further reduced, the
volume fraction of $\sf M_{2}$ also increases, and both the variants evolve
with apparently the same volume fraction. In both the samples, two mutually
orthogonal twins have developed, where the twin boundaries far from the GB
region are making approximately $45^{\circ}$ angles with the
$\boldsymbol{c}_{1}$ and $\boldsymbol{c}_{2}$-axes in the respective grains,
which is in accordance with the crystallographic solutions for twins in large
single grains undergoing cubic$\to$tetragonal MTs; see, e.g. [3] and Chapter 5
of [1]. The variant plates meeting in the GB region from both the grains are
coherent, which we can explain using the fact that the variants $\sf M_{1}$
and $\sf M_{2}$ are compatible about a symmetric tilt GB (see Appendix D for
the proof). The plates about the GB towards the middle of the samples have a
larger thickness than the other variant plates. Similar self-accommodating
twins arrangements evolution (shown in Fig. 6(a)-(b) and Fig. 8(a)-(b)) about
the tilt GBs have also been observed in experiments [28, 29] and atomistic
simulations [50]. Finer twins within the thick $\sf M$ plates in the sample
shown in Fig. 8(a)-(b) at $100$ K are observed, which is driven by a reduction
in the elastic energy of the system. Near the top and bottom boundaries of the
samples, irregularities in microstructures are observed due to the boundary
effect. Such random nucleation of twin plates away from the GB region was
observed in experiments also [28, 29]. Although the GBs are symmetric in both
samples, due to the biased IC on $\eta_{1}$, we get asymmetries in the
microstructures about the GBs. However, with an unbiased IC, say,
$\eta_{1}=0.5$ all over the samples, we verified that fully symmetric
microstructure evolves in the two grains about the GB at any
$\theta\leq\theta_{n}^{f}$.

The components of the Cauchy elastic and structural stresses in the fully $\sf
M$ samples are shown in Figs. 7(a) and 7(b), respectively, when
$\delta_{12}^{gb}=1$ nm, and in Figs. 9(a) and 9(b), respectively, they are
shown for $\delta_{12}^{gb}=1.5$ nm. The pockets of high stresses are mainly
concentrated across the variant-variant interfaces and the GB regions in both
the samples, similar to that observed in experiments; see, e.g. [33]. The
elastic stresses are much higher than the structural stresses, and in some
places, the difference may be of two orders of magnitude. A detailed study
with the analytical solution for the elastic and structural stresses across
the variant-variant interfaces is given in [134], and the reason for such a
large difference in these two stresses for the present kinematic model (Eq.
(2.10)) can be seen therein and also in [87, 63]. In the realistic solids,
such high stresses would relax by nucleation of defects such as dislocations,
and cracks, as observed in, say, e.g. [33]. However, our model does not
account for such coupled phenomena. The total free energy per unit volume
given by Eq. (2.13), and the driving force $X_{12}^{\eta}$ for variant-variant
transformation given by Eq. (C.24) are shown across the fully $\sf M$ samples
in Figs. 7(c) and 7(d), respectively, when $\delta_{12}^{gb}=1$ nm, and in
Figs. 9(c) and 9(d), respectively, when $\delta_{12}^{gb}=1.5$ nm. Since
$X_{0M}=0$ in fully $\sf M$ samples, we do not show its plot. Although the
large values of the driving force $X_{12}^{\eta}$ are concentrated across the
interfaces, its average over the area of the respective samples is vanishing
(note the sign changes), and hence the microstructure does not evolve further.
We note that the highest energy densities are concentrated at the right upper
corner of the sample with $\delta^{gb}_{12}=1$ nm (see Fig. 7(c)) and in a
region near the GB and close to the bottom surface of the left grain for the
sample with $\delta^{gb}_{12}=1.5$ nm (see Fig. 9(c)). This information we
will use it in the next paragraph to study the nucleation of $\sf PM$ or $\sf
A$ during the reverse MT.

Reverse MT. We now study the reverse transformations, i.e. $\sf M\to\sf PM$
and $\sf A$ transformation, from the fully martensitic samples. It is to be
clarified that in Fig. 4, the reverse transformation of the $\sf PM$/$\sf M$
layer nucleated at $\theta^{f}_{n}$ in the GB region and its small
neighbourhood only was studied. However, we consider the fully martensitic
samples obtained at $65$ K and $100$ K shown in Figs. 6(a,b) and 8(a,b) as the
reference bodies ($V_{0}$) when $\delta_{12}^{gb}=1$ nm and $1.5$ nm,
respectively. We assume an IC for $\eta_{0}(\boldsymbol{r}_{0},t=0)=0.98$
distributed uniformly in both the samples, whereas the same $\eta_{1}$
distribution in the complete $\sf M$ sample obtained during the forward
transformation is used. The same IC for $\eta_{0}$ and the similar IC for
$\eta_{1}$ (i.e. as it is in the $\sf M$ sample) for studying the reverse MTs
in fully martensitic samples are used for all the examples related to the
bicrystals and tricrystals. The microstructures evolution during the reverse
transformations for these two samples is shown in Figs. 6(c,d) and 8(c,d). The
temperatures at which the $\sf PM$, i.e. incomplete $\sf A$ nucleated are
$\theta_{n}^{r}=82.2$ K and $\theta_{n}^{r}=133.5$ K, respectively, which are
obviously in the regime of stability of $\sf M$. In Fig. 6(c,d), the $\sf PM$
nucleated at the right upper corner of the sample where the total energy
density was the highest in the fully $\sf M$ sample, as shown in 7(c) (see the
above paragraph). When $\delta_{12}^{gb}=1.5$ nm, the $\sf PM$ nucleated at a
region in the lower part of the left grain, which is a region with high total
energy density, as shown in Fig. 9(c). Such high energetic regions provide the
driving force required for nucleation of $\sf PM$ and $\sf A$ on heating of
the martensitic samples. Post $\sf PM$ nucleation, the temperature of both the
samples is further increased, and the microstructures at different $\theta$
are shown. A significant portion of both the samples is transformed to $\sf A$
at $200$ K. The third of the plots in Figs. 6(c,d) and 8(c,d) depict the
microstructures at the respective $\theta_{n}^{f}$, the temperature at which
the $\sf PM$/$\sf M$ nucleated during the forward MT. The microstructures of
Fig. 6(a,b) and that of Fig. 6(c,d) at $\theta_{n}^{f}=315.5$ K are
significantly different. At that $\theta$, the GB region contains a thin layer
of pre-$\sf M_{2}$ during the forward MT. In contrast, the GB region contains
a thick M layer containing both variants, and a significant volume fraction of
M is present in the interior of the grains during the reverse MT. A similar
conclusion is drawn for the other sample as well. The last plots in Figs.
6(c,d) and 8(c,d) show the microstructures with thin layers of $\sf PM$
containing both pre-$\sf M_{1}$ and pre-$\sf M_{2}$ only on the GB regions at
temperatures indicated therein. The samples fully transform to $\sf A$ (i.e.
$\eta_{0}=0$) at $0.1$ K above the indicated temperatures. The experimental
study of [29] in Fe-Ni bicrystal with a tilt GB also showed that the reverse
transformation starts at high energetic regions within the grains, usually far
from the GB. It was also reported that the variant plates far from the GBs
transform to A first on heating the sample further, and the twins about the GB
finally disappear to transform the sample back to austenite. Our simulation
results are hence in good agreement with the experimental observations of
[29].

(a) $\eta_{0}$ plots during forward MT (b) $\eta_{eq}$ plots during forward MT
(c) $\eta_{0}$ plots during reverse MT (d) $\eta_{eq}$ plots during reverse MT

Figure 10: Microstructure evolution during (a,b) forward, and (c,d) reverse
MTs in a $30$ nm $\times$ $30$ nm bicrystal having a symmetric planar GB with
$\theta$ variation. We took $\Delta\gamma_{12}^{gb}=-0.7$ N/m,
$\delta_{12}^{gb}=1$ nm, and $\vartheta_{12}=40^{\circ}$. No strain applied.

####  4.1.2 Effect of change in GB energy during MTs

We now study the effect of change in GB energy $\Delta\gamma_{12}^{gb}$
considering the same austenitic bicrystal with a symmetric planar tilt GB
considered in Sec 4.1.1 for $\Delta\gamma_{12}^{gb}=-0.7$ N/m, while all other
parameters are identical, i.e. $\delta_{12}^{gb}=1$ nm,
$\vartheta_{1}=-\vartheta_{2}=20^{\circ}$ ($\vartheta_{12}=40^{\circ}$), and
no strains applied. The results for $\Delta\gamma_{12}^{gb}=-0.5$ N/m and
$\Delta\gamma_{12}^{gb}=-0.7$ N/m are finally compared. The martensitic GB
energies in these two cases are obviously $\gamma_{gb}^{M}=0.4$ N/m and
$\gamma_{gb}^{M}=0.2$ N/m, respectively.

Forward MT. Figures 10(a,b) show the evolution of the microstructures with a
variation of $\theta$ when $\Delta\gamma_{12}^{gb}=-0.7$ N/m during the
forward MT. Thin layers of pre-$\sf M_{2}$, similar to the sample for
$\Delta\gamma_{12}^{gb}=-0.5$ N/m, appear jump-like at $\theta_{n}=353.5$ K on
the GB. The reason for pre-$\sf M_{2}$ is already explained in Sec 4.1.1.
However, the samples were fully austenitic at $353.6$ K. A larger reduction in
the GB energy due to $\sf A\to\sf M$ transformations increases the temperature
for nucleation of $\sf PM/\sf M$. The sample temperature is further reduced,
and the subsequent stationary microstructures at $\theta=300$ K, $200$ K, and
$60$ K are shown in Fig. 10(a,b). The microstructures at $\theta=300$ K are
similar in both samples. More than half of both the samples are transformed to
$\sf M$ phase consisting of mostly the self-accommodating twins about the GB
at $200$ K. At $\theta=60$ K, the entire sample is martensitic, consisting of
both the variants with almost equal volume fraction, when
$\Delta\gamma_{12}^{gb}=-0.7$ N/m. The overall microstructures in both the
$\sf M$ samples are qualitatively similar, except that the average thickness
of the variant plates for the case with $\Delta\gamma_{12}^{gb}=-0.5$ N/m is
smaller than the case when $\Delta\gamma_{12}^{gb}=-0.7$ N/m. The elastic and
structural stresses in the fully martensitic sample are concentrated mainly
across the GB and the variant-variant interfaces in case of
$\Delta\gamma_{12}^{gb}=-0.7$ N/m, which is similar to the other sample and
hence not shown here.

Reverse MT. We have studied the reverse MT using the sample obtained at $60$ K
shown in Fig. 10(a,b) when $\Delta\gamma_{12}^{gb}=-0.7$ N/m, whereas the
result for $\Delta\gamma_{12}^{gb}=-0.5$ N/m was already discussed in Sec.
4.1.1. As the temperature of the sample is increased from $60$ K, a small
region of $\sf PM$ appears in a jump-like manner at the upper right corner at
$\theta_{n}^{r}=120.2$ K, shown in Fig. 10(c,d), where the total energy
density was the maximum in the $\sf M$ sample at $60$ K. A significant portion
of both the grains transform to $\sf A$ at $200$ K. At
$\theta=\theta_{n}^{f}=353.5$ K, only a small volume fraction of $\sf M$
remains in the GB region, which is of a slightly higher volume fraction than
that obtained during the forward MT at the same temperature. On further
increase in $\theta$, the $\sf M$ layer gets thinner and eventually transforms
to $\sf PM$. The $\sf PM$ layer on the GB region at $393.7$ K is shown in Fig.
10(c,d), which disappears jump-like at $393.8$ K to transform the sample into
austenite. Thus we see that a higher $\Delta\gamma^{gb}_{12}$ significantly
increases the austenite-finish temperature in the bicrystals (compare with
Fig. 6(c,d)).

We cannot compare our results for the effect of $\Delta\gamma_{12}^{gb}$ on
the transformation temperatures and the microstructures with any atomistic or
experimental result due to the scarcity of relevant data in the literature.
However, the overall transformation kinetics during the forward and reverse
MTs are qualitatively in agreement with the experimental and atomistic results
[28, 29, 50].

(a) $\eta_{0}$ plots during forward MT (b) $\eta_{eq}$ plots during forward MT
(c) $\eta_{0}$ plots during reverse MT (d) $\eta_{eq}$ plots during reverse MT

Figure 11: Microstructure evolution during (a,b) forward, and (c,d) reverse
MTs in a $30$ nm $\times$ $30$ nm bicrystal having a symmetric planar GB with
$\theta$ variation. We considered $\Delta\gamma_{12}^{gb}=-0.5$ N/m,
$\delta_{12}^{gb}=1$ nm, and $\vartheta_{12}=30^{\circ}$. No strain applied.

####  4.1.3 Effect of GB misorientation

We now consider another $30$ nm $\times$ $30$ nm sample with a symmetric tilt
planar GB where the grain orientations are
$\vartheta_{1}=-\vartheta_{2}=15^{\circ}$, and hence the misorientation is
$\vartheta_{12}=30^{\circ}$. The other parameters are the same as that
considered for the case shown in Fig. 6: $\delta_{12}^{gb}=1$ nm,
$\Delta\gamma_{12}^{gb}=-0.5$ N/m, and no strain is applied. The result for
this sample is shown in Fig. 11, which is compared with that of Fig. 6. The
differences in these two cases are only in the misorientation angle and the
orientation of the Bain strains ($40^{\circ}$ used in Fig. 6). Although the GB
misorientations are different for these two samples, the GB energies
$\gamma_{gb}^{A}$ and $\gamma_{gb}^{M}$ are considered to be identical. We can
justify this by noticing that isotropic GBs with different misorientations can
have the same energy; see [129] for the examples in NiAl alloy.

Forward MT. Comparing the Figs. 6 and 11, we see that a $10^{\circ}$ reduction
in the misorientation, keeping all other parameters the same, increases the
nucleation temperature just by $0.3$ K. The thin $\sf PM$ layers appeared with
$max(\eta_{0})=0.1$ and $max(\eta_{0})=0.07$ in the samples having
$\vartheta_{12}=40^{\circ}$ and $30^{\circ}$, respectively. At $300$ K, no
significant difference in the microstructures is observed. At $200$ K, the
volume fraction of the $\sf M$ phase is, however, significantly larger when
$\vartheta_{12}=30^{\circ}$ than in the other sample. The sample with
$\vartheta_{12}=30^{\circ}$ is completely martensitic at $\theta=110$ K, while
the sample having $\vartheta_{12}=40^{\circ}$ had to be cooled down to $60$ K
to transform it to fully $\sf M$ (compare between Figs. 6 and 11). The
martensitic plates are finer for the sample with smaller misorientation.

Reverse MT. The results for the reverse MT for $\vartheta_{12}=30^{\circ}$ are
shown in Fig. 11(c,d). The $\sf A$ phase nucleated barrierlessly at
$\theta_{n}^{r}=155.6$ K at the top-left corner of the left grain, as shown in
Fig. 11(c,d). The right grain, however, remains completely martensitic at that
$\theta$. The $\sf A$-$\sf M$ phase boundary then propagates with the left
grain on further increase in $\theta$ to increase the volume fraction of $\sf
A$. The right grain remains in the $\sf M$ phase even at $\theta=200$ K, as
shown in Fig. 11(c,d). The $\sf M$ phase is still observed in the GB region
and at the top and bottom surfaces of the sample at
$\theta=\theta_{n}^{f}=315.8$ K, having a larger volume fraction of $\sf M$
than that obtained during the forward MT (see Fig. 11). The entire sample
transforms to $\sf A$ barrierlessly at $\theta=367.6$ K, whereas this happened
at $\theta=365.6$ K for the other sample with $\vartheta_{12}=40^{\circ}$. The
thin layer of $\sf PM$ in the GB region observed at $367.5$ K and with
$max(\eta_{0})=0.8$ is shown in 11(c,d).

We thus conclude that a $5^{\circ}$ difference in the orientation of the
transformation strains, keeping all other parameters identical, changes the
nucleation of $\sf M/\sf PM$ and the $\sf A$-finish temperatures marginally.
However, the microstructure evolution significantly differs.

(a) $\eta_{0}$ plots during forward MT (b) $\eta_{eq}$ plots during forward MT
(c) $\eta_{0}$ plots during reverse MT (d) $\eta_{eq}$ plots during reverse MT

Figure 12: Microstructure evolution during (a,b) forward, and (c,d) reverse
MTs in a $30$ nm $\times$ $30$ nm bicrystal having a circular GB of $15$ nm
radius with $\theta$ variation. We used $\Delta\gamma_{12}^{gb}=-0.5$ N/m,
$\delta_{12}^{gb}=1$ nm, and $\vartheta_{12}=40^{\circ}$. No strain applied.

####  4.1.4 Effect of GB curvature

We now consider a stress-free $\sf A$ bicrystal with a circular symmetric tilt
GB of $15$ nm radius embedded inside the $30$ nm $\times$ $30$ nm sample as
shown in Fig. 3(b) to study the role of GB curvature. We take
$\vartheta_{1}=-\vartheta_{2}=20^{\circ}$, i.e. $\vartheta_{12}=40^{\circ}$,
and all other parameters are identical to the bicrystal with a planar GB
considered in Fig. 6: $\Delta\gamma_{12}^{gb}=-0.5$ N/m, $\delta_{12}^{gb}=1$
nm, and no strain is applied. The microstructure evolution during the forward
and reverse MTs are shown in Fig. 12.

Forward MT. The $15$ nm radius of curvature increases the nucleation
temperature $\theta_{n}^{f}$ only by $1$ K (compare between Figs. 12(a,b) and
6(a,b)). However, the nucleated layer in the curved GB region is fully
martensitic, containing both the variants with equal volume fractions in
contrast to the sample with a planar GB where it was a pre-$\sf M_{2}$ layer
only (see Fig. 6). As $\theta$ is further reduced, the martensitic plates grow
from the GB regions into both grains. The embedded grain, however, transforms
much faster than the outer grain due to the additional driving force
contributed by the $\sf A$-$\sf M$ interface curvature, and we see that at
$200$ K itself, it is fully martensitic, whereas the outer grain is partially
transformed (see Fig. 12(a,b). On the other hand, both the grains were
partially transformed in the sample with planar GB at $200$ K. On the further
decrease in $\theta$, a fully martensitic sample is seen at $100$ K. In the
outer grain, two sets of mutually orthogonal arrangements of $\sf M_{1}$ and
$\sf M_{2}$ plates are observed away from the GB region at $65^{\circ}$ and
$25^{\circ}$ angles with respect to the $\boldsymbol{e}_{1}$-axis, which
agrees with the crystallographic theory in an isolated large grain. The
average plate thickness of the embedded grain is smaller than that of the
outer grain. A continuous change in orientation of the GB region results in a
very complex microstructure there. The GB curvature was shown to promote MTs
with nucleation of multiple variants in the PF study of Heo and Chen [111],
which is similar to our results.

Reverse MT. A small region of $\sf A$ nucleates near the top surface of the
outer grain at $\theta_{n}^{r}=140.3$ K on heating the martensitic sample. On
further increase in $\theta$, a significant volume fraction of the outer grain
transforms to $\sf A$ at $200$ K, while the inner grain is still completely
martensitic. As the temperature is raised to $\theta=\theta_{n}^{f}=316.5$ K,
almost half of the GB region transforms to $\sf A$, and only a small volume
fraction of the inner and outer grains are still martensitic. At $351$ K, two
thin layers of $\sf PM$ still remain in the GB region, which vanish in a jump-
like manner at $351.1$ K.

Although the circular GB raises the martensitic nucleation temperature
$\theta_{n}^{f}$ by $1$ K only compared to the planar GB, the austenite finish
temperature is decreased significantly, i.e. by $14.5$ K.

(a) $\eta_{0}$ plots during forward MT (b) $\eta_{eq}$ plots during forward MT

Figure 13:  Microstructure evolution during (a,b) forward MT in a $30$ nm
$\times$ $30$ nm bicrystal having a symmetric planar GB with $\theta$
variation. We used $\Delta\gamma_{12}^{gb}=-0.5$ N/m, $\delta_{12}^{gb}=1$ nm,
$\vartheta_{12}=40^{\circ}$, and applied $\varepsilon_{1}=0.04$.

####  4.1.5 Effect of applied strain

We consider the same initial austenitic bicrystal with a planar symmetric tilt
GB having $\delta_{12}^{gb}=1$ nm considered in Fig. 6 (see Sec. 4.1.1), which
is now subjected to a normal strain $\varepsilon_{1}=0.04$ on the right
vertical surface, having the outward unit normal parallel to
$-\boldsymbol{e}_{2}$-axis. All other parameters are identical, i.e.
$\Delta\gamma_{12}^{gb}=-0.5$ N/m, and $\vartheta_{12}=-40^{\circ}$. Recall
that in Fig. 6, no external strain was applied. We have only shown the results
for the forward MT in the present case. We have applied $\varepsilon_{1}=0.04$
on the mentioned surface at $t=0$ and hold it fixed for all $t>0$, while the
normal displacement to all other surfaces is zero for all the time. All the
surfaces are free of the tangential component of the traction. A complete $\sf
M$ phase nucleates at $\theta_{n}^{f}=309.9$ K in a jump-like manner on the GB
and in its neighbourhood, as shown in Fig. 13, where the $\theta_{n}^{f}$ is
$5.6$ K lower than the case when no strain was applied. In the present case,
the volume fraction of nucleated $\sf M$ is much larger than when
$\varepsilon_{1}=0$ (compare between Figs. 13 and 6). A significant drop in
the nucleation temperature of $\sf M$ from the tilt GB was also observed in
the experiment with a Fe-Ni bicrystal under tensile strain in [28]. A larger
volume fraction of variant plates was observed around the GB therein when the
strain was applied than in the case with no external strain [28]. Under the
applied strain, the volume fraction of the $\sf M$ phase increases faster for
the decrease in $\theta$, as shown in Fig. 6. Thick $\sf M_{1}$ and $\sf
M_{2}$ plates have grown near the GB region, whereas the plates away from the
GB are much thinner. The sample is transformed to $\sf M$ except for a tiny
region in the left corner of the top surface of the sample at $100$ K. The
simulation results under applied strain are qualitatively in agreement with
the experimental observations described above.

(a) $\eta_{0}$ plots during forward MT (b) $\eta_{eq}$ plots during forward MT

Figure 14: Microstructure evolution during (a,b) forward MT in a $40$ nm
$\times$ $40$ nm bicrystal having a symmetric planar GB with $\theta$
variation. We consider $\Delta\gamma_{12}^{gb}=-0.5$ N/m, $\delta_{12}^{gb}=1$
nm, and $\vartheta_{12}=40^{\circ}$. No strain applied.

####  4.1.6  Effect of sample size

We now consider a larger bicrystal of $40$ nm $\times$ $40$ nm size with a
symmetric planar tilt GB (similar to Fig. 14). The microstructure evolution
during the forward MT is presented in Fig. 14. All the ICs and other
parameters are identical to the case shown in Fig. 6:
$\Delta\gamma_{12}^{gb}=-0.5$ N/m, $\delta_{12}^{gb}=1$ nm,
$\vartheta_{1}=-\vartheta_{2}=20^{\circ}$, i.e. $\vartheta_{12}=40^{\circ}$,
and no strain is applied. The nucleation temperature $\theta_{n}^{f}$ remains
unchanged, but $\max(\eta_{0})$ is slightly less in this larger sample.
However, the martensitic microstructures evolution in these two samples
significantly differ on decreasing $\theta$. The sample is completely
martensitic at $100$ K, and the variant plates are more regularly arranged in
this larger sample than in the smaller sample shown in Fig. 6.

(a) $\eta_{0}$ plots during forward MT (b) $\eta_{eq}$ plots during forward MT
(c) $\eta_{0}$ plots during reverse MT (d) $\eta_{eq}$ plots during reverse MT

Figure 15: Microstructure evolution during (a,b) forward, and (c,d) reverse
MTs in a $30$ nm $\times$ $30$ nm bicrystal having a asymmetric planar GB with
$\theta$ variation. We consider $\Delta\gamma_{12}^{gb}=-0.5$ N/m,
$\gamma_{gb}^{A}=0.9$ N/m, $\delta_{12}^{gb}=1$ nm,
$\vartheta_{1}=30^{\circ}$, and $\vartheta_{2}=-10^{\circ}$. No strain
applied. Figure 16: Plots for the (a) Cauchy elastic stresses
$\sigma_{(e)ij}$, and (b) Cauchy structural stresses $\sigma_{(st)ij}$ at
$\theta=80$ K within the martensitic sample shown in Fig. 15(a,b) obtained by
forward MT.

####  4.1.7  Results for asymmetric GB

We now consider a stress-free austenitic bicrystal with an asymmetric planar
tilt GB, where the grain orientations are $\vartheta_{1}=30^{\circ}$ and
$\vartheta_{2}=-10^{\circ}$ (see Fig. 3(a)). The GB misorientation is thus
$\vartheta_{12}=40^{\circ}$. All other parameters are identical to that
considered for Fig. 6: $\delta_{12}^{gb}=1$ nm, $\Delta\gamma_{12}^{gb}=-0.5$
N/m, and no strain is applied. The stationary microstructures obtained during
the forward and reverse MTs are shown in Fig. 15.

Forward MT. As discussed in Appendix D, neither of the variants $\sf M_{1}$
and $\sf M_{2}$ is incompatible across the asymmetric GB of this sample as
$|\vartheta_{1}|\neq|\vartheta_{2}|$. We note that the nucleation temperature
$\theta_{n}^{f}$ is $1$ K higher than the sample with a symmetric GB (compared
with Fig. 6), where the misorientation is identical. Only incomplete $\sf
M_{1}$ variant plates are developed in $G_{1}$ about the GB at $300$ K, and
the bottom part of it is transformed. The grain $G_{2}$, on the other hand, is
still primarily austenitic except for a region near the bottom surface. On the
further decrease in $\theta$, the variant-variant plates grow mainly from the
GB region at $200$ K, and also few plates grow from other parts of the grains.
The sample is fully martensitic at $80$ K. The orientation of the variant-
variant interfaces within each grain, which are away from the GB, agrees with
the crystallographic solutions ([3] and Chapter 5 of [1]). The coherency of
the $\sf M$ plates of each variant from the two grains is clearly absent in
the asymmetric GB (see Fig. 15(b)), which is due to the reason that the
variants are not compatible across such a GB. However, the variant plates were
coherently structures about the symmetric planar GB shown in Fig. 6 (also in
Figs. 8, 10, 11, 13, 14).

The elastic and structural Cauchy stresses within the fully martensitic sample
are shown in Fig. 16(a) and (b), respectively. Compared to the martensitic
bicrystal with a symmetric GB, more high amplitude elastic and structural
stress pockets are observed within the GB region of the martensitic bicrystal
with an asymmetric GB (compare between Figs. 7 and 16).

Reverse MT. On heating the fully $\sf M$ sample from $80$ K, a $\sf PM$ region
nucleates in $G_{2}$ at $140.6$ K, as shown in Fig. 15(c,d). The
microstructure evolution with a further increase in $\theta$ is also shown.
The entire sample becomes austenitic at $365.1$ K. Note that the bicrystal
with the symmetric GB transformed to $\sf A$ at $365.6$ K (see Fig. 6(c,d)).

The incompatibility of the variants at the GB marginally influences the $\sf
M$ nucleation and $\sf A$ finish temperatures. The microstructures, however,
significantly differ, especially within the GB region.

(a) $\eta_{0}$ plots during forward MT (b) $\eta_{eq}$ plots during forward MT
(c) $\eta_{0}$ plots during reverse MT (d) $\eta_{eq}$ plots during reverse MT

Figure 17: Microstructure evolution during (a,b) forward, and (c,d) reverse
MTs in a $30$ nm $\times$ $30$ nm tricrystal having symmetric planar GBs with
$\theta$ variation. We used
$\Delta\gamma_{12}^{gb}=\Delta\gamma_{13}^{gb}=\Delta\gamma_{23}^{gb}=-0.5$
N/m, $\delta_{12}^{gb}=\delta_{13}^{gb}=\delta_{23}^{gb}=1$ nm, and
$\rho_{0}K_{IJ}^{gb}=\rho_{0}K_{123}=800$ GPa. No strain applied. Figure 18:
Plots for the (a) Cauchy elastic stresses $\sigma_{(e)ij}$, and (b) Cauchy
structural stresses $\sigma_{(st)ij}$ at $\theta=85$ K within the martensitic
tricrystal shown in Fig. 17(a,b) obtained by forward MT.

###  4.2 Results for tricrystal

We now present the results for the tricrystals with three planar symmetric
tilt GBs; a typical austenitic tricrystal is shown in Fig. 3(c). The grain
orientation angles are taken as $\vartheta_{1}=-15^{\circ}$,
$\vartheta_{2}=-75^{\circ}$, and $\vartheta_{3}=-45^{\circ}$. The
misorientations of the three GBs as per the definition given by Eq. (4.2) are
$\vartheta_{12}=60^{\circ}$, $\vartheta_{23}=-30^{\circ}$, and
$\vartheta_{13}=30^{\circ}$. However, noticing the cubic symmetry of the
parent $\sf A$ phase in 2D (see Fig. 2), it can be easily verified that all
the austenitic GBs are symmetric and equivalent to having $30^{\circ}$
misorientations. We thus consider identical austenitic GB energy
($\gamma^{A}_{12}=\gamma^{A}_{13}=\gamma^{A}_{23}$), and identical change in
GB energy due to MT
$\Delta\gamma^{gb}_{12}=\Delta\gamma^{gb}_{13}=\Delta\gamma^{gb}_{23}$. The
width for the GBs is also taken identical:
$\delta^{gb}_{12}=\delta^{gb}_{13}=\delta^{gb}_{23}$. We will now study the
effect of the parameters, including the GB width, change in GB energy during
MTs, triple junction energy, and sample size on nucleation of the phases and
the subsequent microstructures evolution in detail. No external strain is
applied to the surfaces. All other BCs and the ICs are already prescribed at
the beginning of the section.

(a) $\eta_{0}$ plots during forward MT (b) $\eta_{eq}$ plots during forward MT
(c) $\eta_{0}$ plots during reverse MT (d) $\eta_{eq}$ plots during reverse MT

Figure 19: Microstructure evolution during (a,b) forward, and (c,d) reverse
MTs in a $30$ nm $\times$ $30$ nm tricrystal having symmetric planar GBs with
$\theta$ variation. We used
$\Delta\gamma_{12}^{gb}=\Delta\gamma_{13}^{gb}=\Delta\gamma_{23}^{gb}=-0.5$
N/m, $\delta_{12}^{gb}=\delta_{13}^{gb}=\delta_{23}^{gb}=1.5$ nm, and
$\rho_{0}K_{IJ}^{gb}=\rho_{0}K_{123}=800$ GPa. No strain applied. Figure 20:
Plots for the (a) Cauchy elastic stresses $\sigma_{(e)ij}$, and (b) Cauchy
structural stresses $\sigma_{(st)ij}$ at $\theta=90$ K within the martensitic
tricrystal shown in Fig. 19(a,b) obtained by forward MT.

####  4.2.1 Effect of GB width

We consider two $30$ nm $\times$ $30$ nm-sized tricrystals where the stress-
free austenitic GBs have theoretical widths of
$\delta_{12}^{gb}=\delta_{23}^{gb}=\delta_{13}^{gb}=1$ nm and
$\delta_{12}^{gb}=\delta_{23}^{gb}=\delta_{13}=1.5$ nm, respectively, but all
other parameters are identical: $\gamma_{IJ}^{A}=0.9$ N/m,
$\Delta\gamma_{IJ}^{gb}=-0.5$ N/m, and
$\rho_{0}K_{12}=\rho_{0}K_{13}=\rho_{0}K_{23}=\rho_{0}K_{123}=800$ GPa (see
Eq. (C.15)). The stress-free austenitic tricrystal shown in Fig. 3(c) is
obtained by solving Eqs. (C.26)3,4,5 and neglecting mechanics (also see [55]).
The numerical width of the GBs would differ from the theoretical widths away
from the TJ region since a fraction of nonzero $\xi_{H}$ would always be
present on the GB $\Gamma_{IJ}$ for $H\neq I,J$ (see [55] for numerical
examples). Following [80], we define the numerical width of a GB, say,
$\Gamma_{12}$, as the distance between the points across the GB where
$\xi_{1}=0.05$ (or $\xi_{2}=0.95$) and $\xi_{1}=0.95$ (or $\xi_{2}=0.05$). For
the tricrystals under consideration, we obtain the numerical widths of the GBs
as $1.07$ nm and $1.58$ nm, respectively, showing a deviation of $7$% and $5$%
from the respective theoretical widths. The microstructure evolution during
the forward and reverse MTs for these two samples is shown in Figs. 17 and 19,
respectively.

Forward MT. Thick $\sf M$ layers containing both $\sf M_{1}$ and $\sf M_{2}$
have nucleated in the junction and GB regions at $\theta_{n}^{f}=313.3$ K and
$307.7$ K in a jump-like manner within the tricrystals with
$\delta_{IJ}^{gb}=1$ nm and $1.5$ nm, respectively. However, the samples were
fully austenitic at $313.4$ K and $307.8$ K, respectively. The overall GB and
junction region contain both variants, with an equal volume fraction. We also
note that each of the grains nucleates both the variants near the GBs and TJ.
The $\sf M$ phase within the small TJ region contains incomplete $\sf M_{1}$
and $\sf M_{2}$. With a further reduction in $\theta$, the variant plates grow
and progress inside the grains. The two samples are completely martensitic at
$85$ K and $90$ K, respectively. According to the crystallographic theory of
martensite, the pair of variant plates for cubic to tetragonal MTs make
$45^{\circ}$ or $135^{\circ}$ with the $\boldsymbol{c}_{1}$-axis of the cubic
$\sf A$ unit cell in an isolated large single grain (see e.g. Chapter 5 of
[1]). This implies that the plates in the isolated $G_{1}$ would have made
$30^{\circ}$ or $60^{\circ}$ with the $\boldsymbol{e}_{1}$-axis, and they
should have made $30^{\circ}$ or $160^{\circ}$ angles in the isolated $G_{2}$,
and $90^{\circ}$ or $0^{\circ}$ in the isolated $G_{3}$ with the
$\boldsymbol{e}_{1}$-axis. However, since the grains are constrained, the
evolving microstructures in the samples are rather more complex and not so
intuitive, especially near the GBs and TJ. The orientations of some of the
variant plates in each grain away from the GB regions are close to the above
mentioned crystallographic solution in the fully martensitic samples of Figs.
17(b) and 19(b). However, the orientation of the plates deviates in the GB and
TJ regions in trying to form a compatible $\sf M_{1}$-$\sf M_{1}$ and $\sf
M_{2}$-$\sf M_{2}$ microstructure therein, as all the GBs are symmetric (see
Appendix D). Interestingly we note in Figs. 17 and 19 that the variant plates
obtained in the grain $G_{2}$ at $200$ K rearrange themselves perpendicularly
in the completely martensitic samples at $85$ K and $90$ K, respectively. The
fully $\sf M$ samples in Figs. 17(b) and 19(b) are similar except for
variations in the variant plate thickness in some regions within the grains
and near the TJs.

We can qualitatively compare the results for the tricrystal in Fig. 17 with
that of the bicrystal whown in Fig. 11 where the theoretical GB width, the
misorientation, and all other material parameters are identical. However, a
quantitative comparison between the nucleation temperatures $\theta_{n}^{f}$
in these two samples is not possible, as the numerical GB widths in the
tricrystals differ from the analytical ones (=$1$ nm). In contrast, the
bicrystals were obtained using the analytical solution for $\xi_{1}$ (Eq.
(3.4)), and the GB width therein is hence the theoretical value only (=$1$
nm). In contrast to the tricrystals, where thick $\sf M$ layers consisting of
both the variants nucleated in the GB and TJ regions, much thinner layers of
pre-$\sf M_{2}$ nucleated on the GBs of the bicrystals at $\theta_{n}^{f}$.
The TJ size and energy (controlled by the parameter $\rho_{0}K_{123}$ and
$\rho_{0}K_{IJ}^{gb}$; see Eq. (C.15)) play an important role in determining
the $\theta_{n}^{f}$ and the corresponding volume fraction of $\sf M$ at
different temperatures in the tricrystals.

We could not find out the high-resolution martensitic microstructures near the
TJs at the nanoscale in the literature to directly compare our numerical
results. However, a qualitatively similar arrangement of $\sf M$ plates has
been reported near the triple junctions of CuNiAl SMA [33] and ceramics [34]
at the micron scale. The Cauchy elastic and structural stress components in
the fully $\sf M$ tricrystals obtained at $85$ K and $90$ K are shown in Figs.
18 and 20, respectively. The stresses are mainly concentrated across the GBs,
TJ, and variant-variant interfaces, which is as per the experimental
observations in polycrystalline SMAs (see, e.g. [33]).

Reverse MT. We now consider the fully $\sf M$ samples shown in Figs. 17(a,b)
and 19(a,b) considering the ICs mentioned above to study the reverse MTs. All
the material parameters remain identical to that used during the forward MT.
The $\sf A$ and $\sf PM$ nucleates in these two samples at
$\theta_{n}^{r}=144.7$ K and $\theta_{n}^{r}=140.7$ K, respectively, as shown
in Figs. 17(c,d) and 19(c,d). Both the nucleated regions are near the upper-
right corners of the respective samples. The nucleated $\sf A$ region in the
first sample is much larger than the $\sf PM$ region nucleated in the other
sample. As $\theta$ of the samples are raised, the volume fraction of $\sf A$
of the sample increases as desired. A fraction of the grains $G_{2}$ and
$G_{3}$ in the first sample transform to $\sf A$ at $200$ K, but $G_{1}$ still
remains martensitic, as shown in Figs. 17(c,d). On the other hand, only parts
of the $G_{1}$ and $G_{3}$ of the other sample transform to $\sf A$ at that
$\theta$, where $G_{2}$ is still fully martensitic, as shown in Fig. 19(c,d).
On further increase in $\theta$, the $\sf M$ from the grains’ interiors
continuously transforms to $\sf A$, leaving a residual $\sf M$ on the GB and
junction regions only, as shown in Figs. 17(c,d) and 19(c,d). When
$\theta=\theta_{n}^{f}$, residual $\sf M$ is still observed in all the GBs and
TJ in Fig. 17(c,d); the residual $\sf M$ is mainly observed on the two GBs, in
the TJ, and on the bottom surface of the other sample shown in Fig. 19(c,d).
The residual $\sf M$ continuously transforms to $\sf PM$ on a further increase
of $\theta$. The layers of $\sf PM$ in the GB and junction regions of the two
samples are shown at $366.4$ K and $365.7$ K, respectively, which
barrierlessly disappear on the increase in the respective sample temperatures
by $0.1$ K to yield the original austenitic samples.

(a) $\eta_{0}$ plots during forward MT (b) $\eta_{eq}$ plots during forward MT
(c) $\eta_{0}$ plots during reverse MT (d) $\eta_{eq}$ plots during reverse MT

Figure 21: Microstructure evolution during (a,b) forward, and (c,d) reverse
MTs in a $30$ nm $\times$ $30$ nm tricrystal having symmetric planar GBs with
$\theta$ variation. We used
$\Delta\gamma_{12}^{gb}=\Delta\gamma_{13}^{gb}=\Delta\gamma_{23}^{gb}=-0.7$
N/m, $\delta_{12}^{gb}=\delta_{13}^{gb}=\delta_{23}^{gb}=1$ nm, and
$\rho_{0}K_{IJ}^{gb}=\rho_{0}K_{123}=800$ GPa. No strain applied.

####  4.2.2 Effect of change in GB energy $\Delta\gamma_{IJ}^{gb}$

We now consider another $30$ nm $\times$ $30$ nm-sized tricrystal and obtain
the results for
$\Delta\gamma_{12}^{gb}=\Delta\gamma_{13}^{gb}=\Delta\gamma_{23}^{gb}=-0.7$
N/m as shown in Fig. 21. All other parameters are identical to that considered
in sample shown in Fig. 17: $\delta^{gb}_{IJ}=1$ nm, and
$\rho_{0}K_{IJ}^{gb}=\rho_{0}K_{123}=800$ GPa. The numerical width of the dry
GBs in both the austenitic samples $V_{0}$ is $1.07$ nm. The results for
$\Delta\gamma_{IJ}^{gb}=-0.5$ N/m and $\Delta\gamma_{IJ}^{gb}=-0.7$ N/m are
also compared.

Forward MT. The $\sf M$ phase nucleates in a jump-like manner on the GBs and
TJ region at $\theta_{n}^{f}=356.1$ K as shown in Fig. 21(a,b), which is
$42.8$ K higher than the case when $\Delta\gamma_{IJ}^{gb}=-0.5$ N/m. The
volume fraction of nucleated $\sf M$ in the sample of Fig. 21(a,b) is much
less than that of Fig. 17(a,b). As $\theta$ is decreased, the volume fraction
of $\sf M$ increases and progresses inside the grains. The microstructures at
$300$ K, $200$ K, and $100$ K are shown in Fig. 21(a,b). Although, there is
not much difference in the volume fraction of the $\sf M$ phase at these
temperatures between the samples in Figs. 17(a,b) and 21(a,b), the evolution
of the microstructure significantly differs. The arrangement of the twin
plates is also different in $G_{1}$ and $G_{2}$ between these samples.
Orientation of the plates far from the GBs is close to the crystallographic
solutions for a large isolated grain mentioned in Sec. 4.2.1. The arrangement
of the variants in the junctions regions of the fully martensitic samples is
different.

Reverse MT. As $\theta$ is raised from $100$ K, an $\sf A$ region nucleates at
$\theta_{n}^{r}=133.4$ K, as shown in Fig. 21(c,d), which is $11.3$ K lower
than the corresponding $\theta_{n}^{r}$ for the sample with
$\Delta\gamma_{IJ}^{gb}=-0.5$ N/m (see Fig. 17(c,d)). On the further rise of
$\theta$, we observe a large area of $G_{1}$ and $G_{3}$ transform to $\sf A$,
while $G_{2}$ is still fully martensitic at $200$ K as shown in Fig. 21(c,d).
At $\theta=\theta_{n}^{f}=356.1$ K, the thin layers of residual $\sf PM$ are
observed in the GB, and junction regions similar to the $\sf PM$ appeared
during the forward MT. The GBs and TJ retained $\sf PM$ up to $\theta=393.2$
K, and the entire tricrystal transformed to austenite in a jump-like manner at
$393.3$ K.

(a) $\eta_{0}$ plots during forward MT (b) $\eta_{eq}$ plots during forward MT
(c) $\eta_{0}$ plots during reverse MT (d) $\eta_{eq}$ plots during reverse MT

Figure 22: Microstructure evolution during (a,b) forward, and (c,d) reverse
MTs in a $30$ nm $\times$ $30$ nm tricrystal having symmetric planar GBs with
$\theta$ variation. We used
$\Delta\gamma_{12}^{gb}=\Delta\gamma_{13}^{gb}=\Delta\gamma_{23}^{gb}=-0.5$
N/m, $\delta_{12}^{gb}=\delta_{13}^{gb}=\delta_{23}^{gb}=1$ nm, and
$\rho_{0}K_{IJ}^{gb}=\rho_{0}K_{123}=1200$ GPa. No strain applied.

####  4.2.3 Effect of junction energy

We recall that in Eq. (LABEL:MF2), we introduced the parameter $K_{HIJ}$, and
by varying it, we can control the energy and size of the TJ region. We take an
austenitic tricrystal of $30$ nm $\times$ $30$ nm size obtained by considering
$\rho_{0}K_{IJ}^{gb}=\rho_{0}K_{123}=1200$ GPa, while all other parameters are
identical to that used for Fig. 17: $\delta_{IJ}^{gb}=1$ nm, and
$\Delta\gamma_{IJ}^{gb}=-0.5$ N/m. The numerical width of all the GBs in the
austenitic tricrystal is obtained as $1.06$ nm, which is closer to the
analytical width than the case with $\rho_{0}K_{IJ}^{gb}=\rho_{0}K_{123}=800$