$4.25^{\circ}<\theta<5.25^{\circ}$, where $\gamma_{\mathrm{e}}$ is lowest for
case 2. $\gamma_{\mathrm{e}}$ and $\gamma$ are in qualitative agreement over
the entire twist angle range and in quantitative agreement for
$\theta>5^{\circ}$. As described in the Supplementary Note from Ref. [252], it
is found comparable agreement between the elasticity-based model and atomistic
interface energies for the remaining two compositions. Agreement between
$\gamma_{\mathrm{e}}$ and $\gamma$ is not sufficient to validate the present
formalism. For that, it must be determined whether the lowest energy cases
predicted by the elasticity-based model match the misfit dislocation patterns
in atomistic simulations. Each of the three Frank-Bilby solutions predicts a
different misfit dislocation pattern and therefore also a different
disregistry. The present goal is to compare the disregistries of all three
cases with that found in atomistic simulations. The model is validated if the
case with lowest $\gamma_{\mathrm{e}}$ has the best match with the atomistic
disregistry. As shown in Figs. (3.18a) and (b), and detailed in Ref. [252],
the disregistry analysis is in agreement with the elastic predictions for all
Cu/Nb and Ag/V interfaces (circle filled with light grey) except Cu/Nb at
$\theta=0^{\circ}$. The disagreement is attributed to the reconstruction of
the misfit dislocation network that is known to occur at that interface [281],
which can be treated by further extensions from section 3.6. One further case
of disagreement where dislocation network reconstruction occurs is found for
Cu/Mo at $\theta=0^{\circ}$ (see Supplementary Note). However, the agreement
between the elasticity-based model and the atomistic models is excellent,
overall.

The general approach may be compared with several ad hoc parameters proposed
previously to determine which of the cases predicted by the Frank-Bilby
equation is likeliest. Bollmann suggested that the likeliest case minimizes
[31]

(3.58) $P=\sum_{i}\frac{b_{i}^{2}}{d_{i}^{2}}\,,$

which is analogous to the Frank rule for predicting dislocation reactions
[122]. Similarly, Ecob and Ralph propose two parameters [80] to distinguish
between cases, defined by $Q$ and $R$, as follows

(3.59)
$Q=\sum_{i}\sum_{j}\frac{b_{i}b_{j}}{d_{i}d_{j}}\,,~{}~{}\mbox{and},~{}~{}R=\sum_{i}\sum_{j}\sqrt{\frac{b_{i}b_{j}}{d_{i}d_{j}}}\,,$

using geometrical arguments for the energy of semicoherent interfaces. Figures
(3.18c) and (d) plot these parameters for Cu/Nb and Ag/V interfaces. Comparing
with Figs. (3.18a) and (b), none of them predicts the misfit dislocation
patterns seen in atomistic models. For example, for Cu/Nb, all three
parameters favor case 2, while the true interface structure is case 3. The
elasticity-based model is therefore viewed as superior to these parameters and
as validated for the purpose of computational design of patterned interfaces.

### Application to the sink strength of semicoherent interfaces

Clean, safe, and economical nuclear energy requires new materials capable of
withstanding severe radiation damage. One way of removing radiation-induced
defects is to provide a high density of sinks, such as GBs or heterophase
interfaces [231] that continually absorb defects as they are created. This
motivation underlies ongoing exploration of the radiation response of
nanocomposite materials [72, 56], due to the large total interface area per
unit volume they contain. These investigations have demonstrated wide
variations in sink behavior of different interfaces. Some easily absorb
defects, preventing damage in neighbouring material, but become damaged
themselves [112]. Others are poor sinks for isolated defects, but excellent
sinks for defect clusters [73]. The sink behavior of yet others changes with
radiation dose [14, 15]. This wide variety of radiation responses prompts the
physicists to ask:

  * $\ast$

Are some specific interfaces best suited to mitigate radiation damage?

  * $\ast$

Is it possible to identify them without resorting to resource-intensive
irradiation experiments?

Here it is demonstrated that elastic interactions between point defects and
semicoherent interfaces lead to a marked enhancement in interface sink
strength. The conclusions stem from simulations that integrate first
principles, object kinetic Monte Carlo, and anisotropic elasticity
calculations. Surprisingly, the enhancement in sink strength is not due
primarily to increased thermodynamic driving forces [148, 137], but rather to
reduced defect migration barriers, which induce a preferential drift of
defects towards interfaces. The sink strength enhancement is highly sensitive
to the detailed character of interfacial stresses, suggesting that "super-
sink" interfaces may be designed by optimizing interface stress fields. These
findings motivate a computational search for "super-sink" interfaces: ones
that optimally attract, absorb, and annihilate radiation-induced defects.

####  3.5.1 Computational multi-model strategy

To answer the aforementioned questions, an improved computational method for
rapidly assessing the vacancy and interstitial sink strength of semicoherent
interfaces is proposed. This method builds on the interfacial dislocation-
based model for elastic fields of heterophase bicrystals, previously
described. Such interfaces are of particular interest because many of them
contain a high density of defect trapping sites [70, 224]. Moreover,
semicoherent interfaces generate elastic fields that interact directly with
radiation-induced defects [258]. These elastic fields have an unexpectedly
large influence on interface sink strength, as quantified by the following
computational multi-model approach.

##### Elastic dipole tensor calculation

Defect P-tensors are calculated using VASP [153], a plane wave-based, first
principles density functional theory code. A fcc supercell containing $256\pm
1$ atoms ($+1$ and $-1$ for interstitial and vacancy, respectively) is used.
Calculations are also performed LAMMPS [209] classical potential simulations
using embedded atom method potentials for Ag [87] and Cu [190] to study the
convergence of the elastic dipole tensors up to supercell sizes of $2048$
atoms. The discrepancy in the elastic P-tensor components between the
$256$-atom supercell and that of $2048$-atom supercell is found lower than
$4\%$. This supercell size ensures the convergence of defect formation
energies to within few meV, as detailed in the Supplementary Note from Ref.
[258]. The $256$-atom density functional theory simulations is therefore
viewed as well converged with respect to model size. A $3\times 3\times 3$
shifted Monkhorst-Pack $K$-point grid mesh, a Hermite-Gaussian broadening of
$0.25$ eV [188], and a plane wave cutoff energy of $400$ eV are used. The
change of the elastic dipole tensors is less than $0.5\%$ compared to tighter
settings. The Perdew-Burke-Ernzerhof [208] exchange-correlation functional is
conveniently used within the projector-augmented-wave approach [154]. The
structures are internally relaxed with a force convergence criterion of
$10^{-3}~{}\mbox{eV}$/Å. The climbing image nudged elastic band method [118]
is employed to find the saddle points for defect migration.

##### Object kinetic Monte Carlo algorithm

The defect diffusion is investigated by using an object kinetic Monte Carlo
code with a residence time algorithm to advance the simulation clock [39,
102]. At time $t$, the time step is chosen according to $\Delta
t=-(\mathrm{ln}\,r_{1})/w_{\mbox{\scriptsize tot}}$, where $r_{1}$ is a random
number with $r_{1}\in]0,1]$ and $w_{\mbox{\scriptsize tot}}$ is the sum of
frequencies of all events that may occur at $t$, i.e. $w_{\mbox{\scriptsize
tot}}=\sum_{i}^{N}w_{i}$. The chosen event $j$ is such that
$\sum_{i}^{j-1}w_{i}<r_{2}w_{\mbox{\scriptsize tot}}\leq\sum_{i}^{j}w_{i}$,
where $r_{2}$ is another random number with $r_{2}\in]0,1]$.

Three kinds of events are considered in the simulations: the jump of a point
defect from one stable point to a neighbouring one, the absorption of a defect
by an interface, and the creation of a new point defect through irradiation.
Jump frequencies are given by $w_{i}=\nu\,\mathrm{exp}(-\Delta E_{i}/(kT))$,
where $\nu$ is an attempt frequency and $\Delta E_{i}=E_{i}^{\mbox{\scriptsize
sad}}-E_{i}^{\mbox{\scriptsize sta}}$ is the energy difference between the
saddle position and the initial stable position of the jump considered. The
stable point energy is

(3.60) $E_{i}^{\mbox{\scriptsize sta}}=-\sum_{k,l}P^{\mbox{\scriptsize
sta}}_{kl,i}\;\varepsilon_{kl}^{\mbox{\scriptsize
int}}(\textbf{{r}}_{i}^{\mbox{\scriptsize sta}})\,,$

while the saddle point energy is

(3.61) $E_{i}^{\mbox{\scriptsize sad}}=E^{\mbox{\scriptsize
m}}-\sum_{k,l}P^{\mbox{\scriptsize
sad}}_{kl,i}\;\varepsilon_{kl}^{\mbox{\scriptsize
int}}(\textbf{{r}}_{i}^{\mbox{\scriptsize sad}})\,,$

with $E^{\mbox{\scriptsize m}}$ the migration energy in the absence of elastic
interactions. Here, $\textbf{P}^{\mbox{\scriptsize sta}}$ and
$\textbf{P}^{\mbox{\scriptsize sad}}$ are the defect P-tensors in the ground
state and saddle point configurations, respectively. For simplicity, the
position of the saddle point $\textbf{{r}}_{i}^{\mbox{\scriptsize sad}}$ is
taken mid-way between the two stable points explored by the jump [240].

The defect is considered to have been absorbed by an interface if it reaches
the nearest atomic row to the interface. It is then simply removed from the
simulation. This absorption condition is used to obtain a first estimate of
sink strength, without taking into account the diffusion of point defects
along interfaces or their possible reemission. The irradiation rate is fixed
at the beginning of each simulation to keep the average number of point
defects equal to $200$ in the material where the measurements are performed,
if no elastic interactions are considered. The actual number of point defects
in the system, averaged over the simulation time when steady state is reached,
constitutes the basis for the sink strength calculation.

The concentration of defects is recorded every $10^{4}$ iterations, after the
concentration has become stationary. At the end of the simulation, an estimate
of the average defect concentration $\mkern
1.0mu\overline{\mkern-0.4muC\mkern-0.4mu}\mkern 0.4mu$ is computed by
averaging over the values $C_{j}$, with $j=1,\dots,n$, as follows

(3.62) $\mkern 1.0mu\overline{\mkern-0.4muC\mkern-0.4mu}\mkern
0.4mu_{n}=\frac{1}{n}\sum_{j=1}^{n}C_{j}\,.$

The final time is adjusted to obtain sufficient accuracy on $\mkern
1.0mu\overline{\mkern-0.4muC\mkern-0.4mu}\mkern 0.4mu$ and thus on the
associated sink strength $k^{2}$ in accordance with the mean field rate theory
formalism [306]. For this purpose, the estimation of the error on the
concentration is given by the standard error of the mean value, i.e.

(3.63) $\delta\mkern 1.0mu\overline{\mkern-0.4muC\mkern-0.4mu}\mkern
0.4mu_{n}=\frac{\sigma_{n}}{\sqrt{n}}\,,$

where

(3.64) $\sigma_{n}^{2}=\frac{1}{n-1}\sum_{j=1}^{n}\left(C_{j}-\mkern
1.0mu\overline{\mkern-0.4muC\mkern-0.4mu}\mkern 0.4mu_{n}\right)^{2}\,.$

The final time for each system is chosen so that the relative error on $\mkern
1.0mu\overline{\mkern-0.4muC\mkern-0.4mu}\mkern 0.4mu$ and $k^{2}$ is less
than 0.5%.

####  3.5.2 Kinetic Monte Carlo simulations with elastic interactions

Modelling the removal of radiation-induced point defects at sinks is a
challenging task: on one hand, the variety and complexity of defect behaviors
call for the flexibility of atomistic modelling. On the other, the relatively
slow, thermally activated mechanisms of defect motion require longer
simulation times than may be reached using conventional atomistic techniques,
such as molecular dynamics. The object kinetic Monte Carlo (OKMC) method [39,
102, 55, 140] is employed, which is well suited to modeling long-time,
thermally activated processes yet is also able to account for nuances of
defect behavior uncovered through atomistic modeling.

Figure (3.19) illustrates the setup of the simulations containing two
crystalline layers$-$A and B$-$separated by semicoherent interfaces. Periodic
boundary conditions are applied in all directions, so each model contains two
A-B interfaces. Due to their inherent internal structure, the interfaces
create characteristic stress fields in the neighbouring crystalline layers.
These stress fields interact with radiation-induced point defects, modifying
their diffusion.

Figure 3.19: Schematic illustration of the diffusion of radiation-induced
point defects (illustrated by ovals) to interfaces under the influence of
interface elastic fields. In general, materials A and B may be any two
crystalline solids. In the present work, they are chosen to be either Cu or
Ag.

The interface stress fields is computed by the approach discussed in section
3.2. For illustration, two specific interfaces are treated in the present
work: a low-angle twist GB on a (001) plane in Ag and a pure misfit (zero
misorientation) heterophase interface between (001) planes of Ag and Cu.
Figure (3.20a) shows a plan view of the Ag twist GB, where the adjacent GB
planes have been rotated by $\pm\theta/2$ ($\theta$: twist angle). The
boundary plane contains two sets of parallel, pure screw dislocations: one
aligned with the $\textbf{{x}}=[110]$ direction and the other with the
$\textbf{{y}}=[\mkern 1.0mu\overline{\mkern-0.4mu1\mkern-0.4mu}\mkern
0.4mu10]$ direction. For a relative twist angle of $\theta=7.5^{\circ}$, the
spacing between dislocations within each set is $\sim 2.2$ nm. Figure (3.20b)
shows the interface plane of the Ag/Cu pure misfit interface. Similar to the
twist boundary in Fig. (3.20a), this interface also contains two sets of
parallel dislocations aligned with the $\textbf{{x}}=[110]$ and
$\textbf{{y}}=[\mkern 1.0mu\overline{\mkern-0.4mu1\mkern-0.4mu}\mkern
0.4mu10]$ directions. Furthermore, the spacing between dislocations in the
Ag/Cu interface is the same as in the twist boundary of Fig. (3.20a): $\sim
2.2$ nm. However, unlike in the twist boundary, both sets of dislocations in
the misfit interface are of pure edge type.

Figure 3.20: Planar semicoherent interfaces with identical misfit dislocation
arrangements in (a) Ag twist GB with pure screw dislocations and (b) a Ag/Cu
misfit interface with pure edge dislocations.

The two interfaces in Fig. (3.20) have identical dislocation arrangements, but
different dislocation characters. Thus, they contain identical dislocation
densities, but have differing stress fields. For instance, all normal stress
components for the twist GB are zero throughout the entire bicrystal. This
stress field is therefore purely deviatoric. By contrast, due to symmetry, the
shear stress $\sigma_{12}$ is everywhere zero for the Ag/Cu interface, but all
of its other stress components are in general non-zero. In particular, this
interface generates significant hydrostatic stresses. These differences have
important implications for interface-defect interactions and defect migration
pathways.

The force dipole moment approximation is used to compute elastic interaction
energies between point defects and interfaces, $E^{\mbox{\scriptsize PD/int}}$
[145, 230, 69]:

(3.65) $E^{\mbox{\scriptsize
PD/int}}=-P_{ij}\,\varepsilon^{\,\mbox{\scriptsize
int}}_{ij}\left(\textit{x},\textit{y},\textit{z}\right)\,.$

Here, $\varepsilon^{\,\mbox{\scriptsize
int}}_{ij}\left(\textit{x},\textit{y},\textit{z}\right)=E_{ij}\left(\textit{x},\textit{y},\textit{z}\right)$
are the short-range components of the previously calculated interface strain
field, given by eq. (3.12a). On the other hand, $P_{ij}$ are the components of
the elastic dipole tensor (the "P-tensor"), which describes the elastic fields
generated by a point defect. $E^{\mbox{\scriptsize PD/int}}$ values are used
to compute stress-dependent energy barriers for defect migration at each
location in the simulation cell. A similar approach has been adopted in
previous OKMC studies to describe point defect interactions with dislocations
[232, 240].

The density functional theory is used to calculate P-tensors for two types of
point defects in Ag and Cu: vacancies and self-interstitials of lowest
formation energy, namely $\left\langle 100\right\rangle$-split dumbbells [82].
The P-tensor values for these defects are obtained in their ground states as
well as at their saddle point configurations during migration (found using the
climbing image nudged elastic band method [118]). Starting from a simulation
cell containing a perfect, stress-free crystal, the point defect of interest
is inserted in the desired location and relax the atom positions while keeping
the simulation cell shape fixed. The point defect induces stresses,
$\sigma_{ij}$, in the simulation cell. They are related to the defect P-tensor
through

(3.66) $P_{ij}=V\,\sigma_{ij}=P_{ij}^{\,\mbox{\scriptsize
d}}+p^{\,\mbox{\scriptsize h}}\,\delta_{ij}\,,$

where $V$ is the simulation cell volume. $P_{ij}^{\,\mbox{\scriptsize d}}$ and
$p^{\,\mbox{\scriptsize h}}$ are the deviatoric and hydrostatic (isotropic)
P-tensor components, respectively. The former is associated with a pure shear
(no volume change) while the latter is related to isotropic tension
(interstitials) or compression (vacancies), which leads to a volume change.

Table 3.6 lists the P-tensors used in the present study. All of them are
expressed in the Nye frame, where the X-, Y-, and Z-axes are aligned with the
$[100]$, $[010]$, and $[001]$ Miller index directions, respectively. The form
of the P-tensor reflects the symmetry of the corresponding defect. Thus, the
P-tensor for a vacancy in its ground state is isotropic while that of an
interstitial is tetragonal. P-tensors for defect orientations other than those
given in Table 3.6 may be calculated using coordinate system rotations. The
P-tensors for $\left\langle 100\right\rangle$-split dumbbell self-
interstitials and vacancies in Cu agree with experimental data [115, 82, 290].
Furthermore, the present calculations of relaxation volumes of a vacancy in Ag
and Cu are in very good agreement with recent ab-initio predictions [195].

Element | Interstitial | Vacancy
---|---|---
Ground state | Saddle point | Ground state | Saddle point
Ag | $\scriptsize\begin{bmatrix}26.80&0&0\\\ 0&26.86&0\\\ 0&0&26.86\end{bmatrix}$ | $\scriptsize\begin{bmatrix}26.69&2.59&0\\\ 2.59&26.69&0\\\ 0&0&27.74\end{bmatrix}$ | $\scriptsize\begin{bmatrix}-3.04&0&0\\\ 0&-3.04&0\\\ 0&0&-3.04\end{bmatrix}$ | $\scriptsize\begin{bmatrix}-2.64&-0.39&0\\\ -0.39&-2.64&0\\\ 0&0&2.15\end{bmatrix}$
Cu | $\scriptsize\begin{bmatrix}17.46&0&0\\\ 0&17.66&0\\\ 0&0&17.66\end{bmatrix}$ | $\scriptsize\begin{bmatrix}18.01&1.78&0\\\ 1.78&18.01&0\\\ 0&0&18.46\end{bmatrix}$ | $\scriptsize\begin{bmatrix}-3.19&0&0\\\ 0&-3.19&0\\\ 0&0&-3.19\end{bmatrix}$ | $\scriptsize\begin{bmatrix}-3.61&-0.37&0\\\ -0.37&-3.61&0\\\ 0&0&2.12\end{bmatrix}$
Table 3.6: Elastic dipole tensors P-tensors (in eV) of point defects from
first principles for a $\left\langle 100\right\rangle$-split dumbbell self-
interstitial and a vacancy in Ag and Cu at both the ground state and saddle
point configurations. The ground state interstitial is oriented in the $[100]$
direction. Its saddle point configuration is for a $[100]$-to-$[010]$
migration path. The vacancy saddle point is for migration along the $[110]$
direction. Figure 3.21: Elastic interaction energy between (a) an interstitial
with the Ag twist GB ($E^{\mbox{\scriptsize PD/int}}<-0.002$ eV in the blue
isovolume), and between the Ag/Cu misfit interface with (b) an interstitial
and (c) a vacancy ($E^{\mbox{\scriptsize PD/int}}<-0.06$ eV in the blue
isovolume; $E^{\mbox{\scriptsize PD/int}}>0.06$ eV in the red; gray contours
are locations with zero interaction energy).

Figure (3.21) shows the distribution of ground state interstitial and vacancy
interaction energies with the Ag twist GB and the Ag/Cu misfit interface. A
$\left\langle 100\right\rangle$-split dumbbell interstitial may take on three
different orientations. Figure (3.21) uses the orientation with lowest
$E^{\mbox{\scriptsize PD/int}}$. For the Ag twist GB, interstitial interaction
energies are negative at all locations, as shown in Fig. (3.21a). Thus, all
interstitials in the vicinity of this GB experience a thermodynamic driving
force to migrate towards the boundary. The interstitials, however, have nearly
isotropic P-tensors (see Table 3.6), so their interaction energies with the Ag
twist GB are very small. The interaction energy of vacancies with the Ag twist
GB is everywhere zero due to the absence of hydrostatic stresses near this
interface. However, the anisotropy of the vacancy saddle point configuration
leads to non-zero interaction energies of migrating vacancies with the GB.

Interstitial interaction energies near the Ag/Cu misfit interface, shown in
Fig. (3.21b), may be attractive or repulsive, depending on the location of the
defect. Thus, interstitials in Ag are expected to migrate towards the center
of the dislocation pattern while those in Cu are expected to migrate to
dislocation cores. Figure (3.21c) shows the interaction energy between
vacancies and the Ag/Cu misfit interface. The spatial variation of this
interaction energy is similar to that of the interstitials, but with opposite
sign.

The OKMC simulations assume a constant, uniform defect creation rate, $G$.
Defects diffuse until they are absorbed by an interface. Only individual
interstitials or vacancies are tracked in the simulations: defect reactions,
such as clustering or recombination, are not considered. After a certain
simulation time, defect distributions reach a steady state, whereupon the
defect concentration is computed as a function of position along the
z-direction (normal to the interface plane) based on the time spent by each
defect on a given atomic site.

####  3.5.3 Effect of elastic interactions on interface sink strength

Figure (3.22) shows steady-state vacancy and interstitial concentrations for
the two types of interfaces described above for models with $10$ nm-thick Ag
and Cu layers. In the absence of elastic interactions between defects and
interfaces, steady-state defect concentrations may be computed analytically,
which are successfully compared with the simulation results.

Figure 3.22: Steady-state point defect concentrations as a function of
location normal to interface planes. The black vertical lines represent the
interface planes, while the continuous gray lines denote the reference case
with no elastic interactions, computed analytically. OKMC results for both
isotropic (orange) and anisotropic (blue) saddle point configurations are
shown. (a) Vacancy and (b) interstitial profiles near Ag/Cu pure misfit
interfaces. (c) Vacancy and (d) interstitial profiles near Ag twist GBs.
Concentrations are normalized by the average concentration $\mkern
1.0mu\overline{\mkern-0.4mu\textit{C}\mkern-0.4mu}\mkern 0.4mu$ obtained when
no elastic interactions are taken into account.

Elastic interactions have a dramatic effect on defect concentration profiles.
In all cases shown in Fig. (3.22) except vacancies near Ag/Cu interfaces,
there are nearly no defects within $\sim 2$ nm-wide zones adjacent to the
interfaces. By contrast, without elastic interactions, defect concentrations
are zero only at the interfaces themselves. Moreover, even though defect-
interface elastic interaction energies are negligible beyond $\sim 2$ nm, the
zones depleted of defects near the interfaces have a pronounced effect on
defect concentrations throughout the entire layer, markedly reducing the
average defect concentration. For the simulations in Fig. (3.22), elastic
interactions reduce defect concentrations by about a factor of two even in the
middle of the layers. This effect is even more pronounced for thinner layers.
For vacancies in Ag/Cu, local traps are responsible for the sharp increase in
concentration near the interface.

The simulations account for numerous aspects of defect-interface elastic
interactions, such as defect anisotropy or differences in defect ground state
and saddle point properties. To discover which ones are primarily responsible
for the defect concentrations shown in Fig. (3.22), some of these
characteristics are artificially "switched off" and repeated the OKMC
simulations to see whether doing so changes the steady-state defect
concentrations. These calculations demonstrate that the anisotropy of the
P-tensor in the saddle point configurations is primarily responsible for the
reduced defect concentrations in Figs. (3.22a) and (3.22b).

The saddle point anisotropy is "switched off" by replacing the saddle point
P-tensor with $\textbf{P}^{\mbox{\scriptsize sad}}=p^{\,\mbox{\scriptsize
h}}_{\mbox{\scriptsize sad}}\,\textbf{I}$, where I is the identity matrix and
$p^{\,\mbox{\scriptsize h}}_{\mbox{\scriptsize sad}}$ is one third of the
trace of the true saddle point P-tensor. This assumption is tantamount to
modelling defects at saddle points as misfitting spherical inclusions in
isotropic media. Concentration profiles obtained with this approximation are
markedly different from the anisotropic case, as shown in Fig. (3.22). In the
case of the Ag twist GB (Figs. (3.22c) and (3.22d)), isotropic saddle points
yield the same defect concentrations as when there are no defect-interface
interactions at all. Indeed, since the twist interface generates no
hydrostatic strain field, only the deviatoric components of defect P-tensors
may interact with these interfaces. Ground state vacancies have zero
deviatoric P-tensor components, so the interaction energy with the Ag twist GB
vanishes, similar to ground state interstitials with nearly isotropic
P-tensors (Table 3.6). The same conclusions hold at saddle positions if saddle
point anisotropy is "switched off", as describe above. Elastic interactions
then do not affect migration energies, explaining why defect concentrations
are identical to the case without elastic interactions.

For the Ag/Cu interface, concentration profiles computed without saddle point
anisotropy lie between the non-interacting and fully anisotropic cases, as
shown in Figs. (3.22a) and (3.22b). Vacancy concentrations are only marginally
lower than the non-interacting case (Fig. (3.22a)), demonstrating the
overriding importance of saddle point anisotropy in their behavior.
Interstitial concentrations obtained without saddle anisotropy lie
approximately mid-way between the fully anisotropic and non-interacting cases
(Fig. (3.22b)), demonstrating that saddle point anisotropy is at least as
important to their behavior as are $p\,\Delta V$ interactions, which are more
commonly investigated.

Figure 3.23: Preferential migration paths and local concentrations of (a)
vacancies and (b) interstitials on the Ag side of the Ag/Cu interface and of
(c) vacancies and (d) interstitials in the Ag twist GB. Migration paths are
shown as gray lines originating from $1$ nm away from the interface. The
square grid of black lines represents interface dislocations. Concentrations
are plotted in a plane located two atomic distances away from the interface.
The concentrations are normalized by $\mkern
1.0mu\overline{\mkern-0.4mu\textit{C}\mkern-0.4mu}\mkern 0.4mu$: the average
concentration when no interactions are considered. Any normalized
concentration values higher than $0.015$ are shown as equal to $0.015$.

Figure (3.23) gives a more detailed view of defect concentrations at different
locations in the Ag layer of the Ag/Cu interface and in the Ag twist GB. Close
to these interfaces, concentrations vary as a function of location parallel to
the interface plane, following the strain field pattern created by the
interfaces. Indeed, the strain field creates preferential paths for defect
migration, as shown by the gray trajectories in Fig. (3.23). These paths are
in general different for interstitials and vacancies. For both the Ag/Cu
interface and Ag twist GB, vacancies preferentially migrate to the dislocation
lines, while interstitials are mostly absorbed between dislocations. This
preferential, non-random walk drift of point defects to specific locations is
responsible for the enhanced interface sink strengths. Knowing the steady-
state defect concentrations obtained by OKMC, sink strengths are derived for
the two interfaces considered above. In the mean field rate theory formalism
[41], "sink strengths" quantify the ability of sinks, such as interfaces, to
absorb defects. Within this formalism, the evolution equation for the average
defect concentration, $\mkern
1.0mu\overline{\mkern-0.4mu\textit{C}\mkern-0.4mu}\mkern 0.4mu$, follows

(3.67) $\dfrac{d\mkern
1.0mu\overline{\mkern-0.4mu\textit{C}\mkern-0.4mu}\mkern
0.4mu}{dt}=G-k^{2}D\mkern
1.0mu\overline{\mkern-0.4mu\textit{C}\mkern-0.4mu}\mkern 0.4mu\,,$

where $G$ is the defect creation rate and $D$ is bulk defect diffusivity. The
second term on the right hand side is related to the loss of defects at sinks
with associated sink strength, $k^{2}$. At steady state, the sink strength may
be computed from the average concentration:

(3.68) $k^{2}=\dfrac{G}{D\mkern
1.0mu\overline{\mkern-0.4mu\textit{C}\mkern-0.4mu}\mkern 0.4mu}\,.$

Using the average of the concentration profile computed for defect removal at
interfaces in the absence of elastic interactions, the interface sink strength
is analytically found to be $k^{2}=12/d^{2}$ [47]. When interactions between
interfaces and defects are present, the sink strength is numerically
determined through eq. (3.68), by using the average steady-state concentration
obtained by OKMC simulations and the diffusion coefficient without elastic
interactions. The resulting vacancy and interstitial sink strengths for both
interfaces are shown in Fig. (3.24a$-$f) as a function of layer thickness.

Figure 3.24: Enhancement in sink strength of Ag/Cu interfaces and Ag twist GBs
for (a$-$c) vacancies ($k^{2}_{v}$) and (d$-$f) interstitials ($k^{2}_{i}$) in
a given layer (Ag or Cu), as a function of layer thickness, $d$. (g$-$i) Bias
factors of Ag/Cu interface and Ag twist GB. The gray line corresponds to the
analytical solution when no interaction is present ($k^{2}=12/d^{2}$). Orange
and blue lines correspond to OKMC calculations without saddle point anisotropy
and with the fully anisotropic interaction model, respectively.

In all cases, the sink strength increases significantly when elastic
interactions are taken into account. This effect is especially pronounced for
thinner layers, as defects undergo elastic interactions with interfaces over a
larger fraction of the layer. It is particularly strong for interstitials,
whatever the interface type, and for vacancies for the twist interface. These
results also confirm the importance of saddle point anisotropy: by comparing
with OKMC simulations that use isotropic saddle-point P-tensors, it yields
order-of-magnitude increases in sink strength, in some cases.

Another quantity of interest for radiation response is the bias factor, $B$,
which expresses the propensity of a given sink to absorb more interstitials
than vacancies. It is defined as

(3.69) $B=\dfrac{k^{2}_{i}-k^{2}_{v}}{k^{2}_{i}}\;,$

where $k^{2}_{v}$ and $k^{2}_{i}$ are the sink strengths for vacancies and
interstitials, respectively. For example, small interstitial clusters and
dislocations exhibit positive bias factors (typically between $0.01$ and $0.3$
[45, 117]) and thus absorb more interstitials than vacancies. The preferential
absorption of interstitials by biased sinks leads to an excess of remaining
vacancies, which cluster and eventually aggregate into voids [45, 182].

Bias factors for the semicoherent interfaces are shown in Fig. (3.24g$-$i).
Values larger than $0.2$ are obtained for the fully anisotropic interaction
model in the case of the Ag/Cu interface. Such interfaces would compete for
interstitials with dislocations. The presence of two sinks of differing bias
magnitude has been given as a possible cause for void swelling suppression in
ferritic steels [175]. Interestingly, for the Ag twist GB the bias factor is
negative, meaning that these interfaces tend to absorb more vacancies than
interstitials. Similar observations have been made in Ref. [233], where the
bias factor for single screw dislocations is negative when using anisotropic
elasticity theory and zero in the isotropic approximation. Such GBs may
therefore deplete excess vacancy concentrations sufficiently to inhibit void
nucleation.

### Elastic strain relaxation in interfacial dislocation patterns

The interfacial dislocation-based model described in section 3.2 has been
extended to investigate the equilibrium relaxed dislocation microstructures
with specified constraints on semicoherent interfaces [260, 261]. The present
parametric energy-based framework includes surface/interface stress and
elasticity effects as additional constitutive relations, which are viewed as
infinitely thin membranes in contact with each individual material, give rise
to non-classical boundary conditions. The elastic field solutions are used to
compute the corresponding strain energy landscapes for planar hexagonal-shaped
configurations containing three sets of misfit dislocations with unextended
three-fold nodes.

####  3.6.1 General considerations on hexagonal-shaped dislocation patterns

The mechanical dislocation-based problem for determining the elastic strain
relaxation of interfacial patterns formed by joining two linear anisotropic
elastic materials A and B is described by adopting specific notations and
conventions in Fig. (3.25). In the global coordinate system
$(\mbox{O},\,\textbf{{x}}^{\mbox{\scriptsize
or}}_{1},\,\textbf{{x}}^{\mbox{\scriptsize
or}}_{2},\,\textbf{{x}}^{\mbox{\scriptsize or}}_{3})$, corresponding to the
orientation relations along fixed crystal directions of the system of
interest, the semicoherent interface is located at the coordinate
$\textit{x}^{\mbox{\scriptsize or}}_{2}=0$, with
$\textit{x}^{\mbox{\scriptsize or}}_{2}>0$ for material A, and
$\textit{x}^{\mbox{\scriptsize or}}_{2}<0$ for material B. Such directions are
not necessary related to high symmetry directions, so that the anisotropic
elastic constants may be displayed in the most general form. In the present
work, the unit vector normal to the interface is
$\textbf{{n}}\parallel\textbf{{x}}^{\mbox{\scriptsize or}}_{2}$, and a
coplanar free surface to the semicoherent interface is potentially introduced
at $\textit{x}^{\mbox{\scriptsize or}}_{2}=h_{\mbox{\scriptsize A}}$, whereas
B is always a semi-infinite linear elastic crystal.

The crystallography of all interfaces is completely specified between close-
packed planes of neighboring materials, so that both orientation relations of
crystals A and B with relative misorientations (tilt and twist) and differing
lattice parameters (misfit) are described using the previous concept of
reference/natural states, as defined in section 3.2. As an example, the
$2.5^{\circ}$ Ta (tantalum) twist boundary is illustrated in Fig. (3.25a). In
the reference state, the interface is coherent, but the interface is not
coherent in the natural state, and the atomic structures of interfaces lead to
the formation of periodic networks of misfit dislocations that may undergo
local relaxations or reconstructions [95].

Figure 3.25: Geometry of a hexagonal-shaped dislocation pattern containing
three sets of interface dislocations with the associated individual Burgers
vectors. (a) The orientation relationships between the adjacent linear
materials are defined with respect to the global coordinate system
$(\mbox{O},\,\textbf{{x}}^{\mbox{\ssmall
or}}_{1},\,\textbf{{x}}^{\mbox{\ssmall or}}_{2},\,\textbf{{x}}^{\mbox{\ssmall
or}}_{3})$, within which the semicoherent interface is located at
$\textit{x}^{\mbox{\ssmall or}}_{2}=0$. For illustration, the current
intrinsic dislocation structure is associated with a planar
$\\{011\\}\parallel\textbf{{n}}$ twist GB between two bcc crystals with a
$2.5^{\circ}$ rotation angle. (b) Anisotropic elasticity calculations are
performed in the non-orthogonal
$(\mbox{O},\,\textbf{{x}}_{1}^{\prime}\parallel\textbf{{p}}_{1}^{\mbox{\scriptsize
o}},\,\textbf{{x}}_{2}\parallel\textbf{{n}},\,\textbf{{x}}_{3}^{\prime}\parallel\textbf{{p}}_{2}^{\mbox{\scriptsize
o}})$ frame with fixed basis vectors, where
$\textbf{{p}}_{1}^{\mbox{\scriptsize o}}$ and
$\textbf{{p}}_{2}^{\mbox{\scriptsize
o}}\neq\textbf{{p}}_{1}^{\mbox{\scriptsize o}}$ are the O-lattice vectors that
describe the periodicity of the dislocation structures. The fixed red and blue
points characterize the initial lozenge-shaped unit cell and the pivot points
for elastic strain relaxations, respectively. The grey points are related to
the O-lattice points, separated by the networks of interfacial dislocations
with three-fold dislocation junction nodes where the conservation law of
Burgers vectors is satisfied, e.g. at the specific orange node J1 that is
parametrized by the dimensionless coordinates $(\eta_{1},\eta_{2})$. For
convex hexagonal-shaped dislocation configurations, J1 may move within the
shaded triangular domain $\mathcal{T}_{\mbox{\ssmall ABC}}$ in dark grey.

The closely related quantized Frank-Bilby equation [94, 29, 30] and the
O-lattice theory [31] are crystallographic approaches used to describe
intrinsic dislocation structures at semicoherent interfaces, which provide the
interfacial dislocation geometries in terms of line directions and spacings
for one, two, or three independent, planar, and uniformly spaced parallel sets
of infinitely long straight dislocations. As illustrated in the previous
sections, however, such purely geometrical approaches are not able to
characterize local reactions of crossing dislocations to form dislocation
segments with different Burgers vectors in mesh networks that are
energetically favorable.

The extended formalism for predicting the interface dislocations arrays
linking the quantized Frank-Bilby equation and anisotropic elasticity theory
under the condition of vanishing the far-field stresses is used to identify
the periodicity of the structures with two sets of dislocations from the pre-
determined O-lattice vectors $\textbf{{p}}_{1}^{\mbox{\footnotesize o}}$ and
$\textbf{{p}}_{2}^{\mbox{\footnotesize
o}}\neq\textbf{{p}}_{1}^{\mbox{\footnotesize o}}$, as illustrated in Fig.
(3.25b). These two vectors characterize the initial lozenge-shaped unit cell
of crossing dislocation sets (red points), for which the translations of the
unit cell by the basis vectors $\textbf{{p}}_{1}^{\mbox{\footnotesize o}}$ and
$\textbf{{p}}_{2}^{\mbox{\footnotesize o}}$ tessellate the entire interface
plane. In the following, the superscript ${}^{\mbox{\footnotesize un}}$ will
be used to indicate quantities related to the unrelaxed dislocation
configurations, e.g. $\boldsymbol{\xi}^{\mbox{\scriptsize
un}}_{1}\parallel\textbf{{p}}_{2}^{\mbox{\footnotesize o}}$ and
$\boldsymbol{\xi}^{\mbox{\scriptsize
un}}_{2}\parallel\textbf{{p}}_{1}^{\mbox{\footnotesize o}}$ correspond to the
initial dislocation directions of the two sets that consist of the lozenge-
shaped patterns, with Burgers vectors $\textbf{{b}}_{1}$ and
$\textbf{{b}}_{2}$, respectively, as stated in section 3.2. Planar
energetically favorable interactions may lead to the formation of dislocation
junctions with coplanar Burgers vector $\textbf{{b}}_{3}$, i.e.

(3.70) $\textbf{{b}}_{1}+\textbf{{b}}_{2}\to\textbf{{b}}_{3}\,,$

such that the current semicoherent interfaces contain infinite, planar, and
periodic dislocation structures with three sets of misfit dislocations. As
illustrated in Fig. (3.25b), the third newly formed set (in black) is
associated with the junction formation due to the local rearrangements between
two initial crossing dislocation arrays, shown by the blue and red dashed
lines. The current directions of the three sets of misfit dislocations are
denoted by $\boldsymbol{\xi}_{1}$, $\boldsymbol{\xi}_{2}$, and
$\boldsymbol{\xi}_{3}$ for which the latter is associated with the direction
of the in-plane dislocation junctions.

The present reactions yield to hexagonal-shaped patterns with three-fold
dislocation nodes, where the centers of the parent dislocation segments from
the lozenge-shaped unit cells consist of pinning pivot points (blue points)
for glissile planar dislocations. An useful triangular domain
$\mathcal{T}_{\mbox{\ssmall ABC}}$ for performing parametric energy-based
analyses, is represented by two blue pivot points (B and C) and the red
intersection point A in which dislocation reactions occur, as shaded in dark
grey in Fig. (3.25b). On the other hand, the newly formed representative
hexagonal-shaped unit cell (light grey domain), which contains six vertices
(dislocation nodes), indexed and ordered by $\mbox{J}_{1}$, $\mbox{J}_{2}$,
$\mbox{J}_{3}$, $\mbox{J}_{4}$, $\mbox{J}_{5}$, and $\mbox{J}_{6}$, is denoted
by $\mathcal{H}_{\mbox{\,\ssmall J}_{1}\\!\mbox{\,\ssmall
J}_{2}\\!\mbox{\,\ssmall J}_{3}\\!\mbox{\,\ssmall J}_{4}\\!\mbox{\,\ssmall
J}_{5}\\!\mbox{\,\ssmall J}_{6}}$. The determination of such infinitely
repeated dislocation nodes with the type of rearrangement defined by eq.
(3.70) produces neither orientation nor magnitude changes in the O-lattice
vectors. Thus, the two-dimensional periodicity of the dislocation networks
containing three families of straight parallel dislocation segments in the
local Cartesian frame
$(\mbox{O},\,\textbf{{x}}_{1},\,\textbf{{x}}_{2},\,\textbf{{x}}_{3})$ with
$\textbf{{x}}_{2}\parallel\textbf{{x}}^{\mbox{\scriptsize
or}}_{2}\parallel\textbf{{n}}$, remains unchanged during the elastic strain
relaxation processes.

In the previous non-orthogonal (oblique and fixed) frame with basis vectors
$(\mbox{O},\,\textbf{{x}}_{1}^{\prime},\,\textbf{{x}}_{2},\,\textbf{{x}}_{3}^{\prime})$,
where $\textbf{{x}}_{1}^{\prime}\parallel\textbf{{p}}_{1}^{\mbox{\footnotesize
o}}\parallel\boldsymbol{\xi}^{\mbox{\scriptsize un}}_{2}$ and
$\textbf{{x}}_{3}^{\prime}\parallel\textbf{{x}}_{3}\parallel\textbf{{p}}_{2}^{\mbox{\footnotesize
o}}\parallel\boldsymbol{\xi}^{\mbox{\scriptsize un}}_{1}$, the oriented angle
between $\boldsymbol{\xi}^{\mbox{\scriptsize un}}_{2}$ and
$\boldsymbol{\xi}^{\mbox{\scriptsize un}}_{1}$ is denoted by
$\phi^{\mbox{\scriptsize un}}$, so that
$\textit{x}_{1}^{\prime}=\textit{x}_{1}\,\mathrm{csc}\,\phi^{\mbox{\scriptsize
un}}$ and
$\textit{x}_{3}^{\prime}=\textit{x}_{3}-\textit{x}_{1}\,\mathrm{ctg}\,\phi^{\mbox{\scriptsize
un}}$. Thus, any position vector in this non-orthogonal frame may be expressed
as:
$\textbf{{r}}=\textit{x}_{1}^{\prime}\,\textbf{{p}}_{1}^{\mbox{\footnotesize
o}}+\textit{x}_{3}^{\prime}\,\textbf{{p}}_{2}^{\mbox{\footnotesize
o}}=(\textit{x}_{1}\,\mathrm{csc}\,\phi^{\mbox{\scriptsize
un}})\,\textbf{{p}}_{1}^{\mbox{\footnotesize
o}}+(\textit{x}_{3}-\textit{x}_{1}\,\mathrm{ctg}\,\phi^{\mbox{\scriptsize
un}})\,\textbf{{p}}_{2}^{\mbox{\footnotesize o}}$. In particular, the mobile
dislocation three-fold node of interest $\mbox{J}_{1}$, which is parametrized
by the dimensionless coordinates $(\eta_{1},\eta_{2})$ in the first quadrant
of the
$(\mbox{O},\,\textbf{{x}}_{1}^{\prime},\,\textbf{{x}}_{2},\,\textbf{{x}}_{3}^{\prime})$
frame, is also defined by:
$\textbf{{j}}_{1}=\eta_{1}\,\textbf{{p}}_{1}^{\mbox{\footnotesize
o}}+\eta_{2}\,\textbf{{p}}_{2}^{\mbox{\footnotesize o}}$, with
$(\eta_{1},\eta_{2})\in\left]0,\,1/2\right[\,\\!^{2}$, excluding $0$ and $1/2$
to describe convex hexagonal-shaped patterns with six distinct dislocation
edges. For example, the limiting case of equilibrium arrays with two sets of
orthogonal misfit dislocations is given by: $\phi^{\mbox{\scriptsize
eq}}=\pi/2$, $\eta_{1}^{\mbox{\scriptsize eq}}\to 1/2$, and
$\eta_{2}^{\mbox{\scriptsize eq}}\to 1/2$, so that
$\mbox{J}_{2}\simeq\mbox{J}_{3}$ and $\mbox{J}_{5}\simeq\mbox{J}_{6}$, as the
$(010)$ twist GBs in fcc materials. On the other hand, the regular equilibrium
hexagonal network corresponds to the particular case where:
$\phi^{\mbox{\scriptsize eq}}=\pi/3$, and $\eta_{1}^{\mbox{\scriptsize
eq}}=\eta_{2}^{\mbox{\scriptsize eq}}=1/3$, as the $(111)$ twist GBs in fcc
crystals.

####  3.6.2 Solution methodology for strain-relaxed rearrangements

During the non-random elastic strain relaxations without externally applied
stresses, misfit dislocations are rearranged into hexagonal-shaped networks
due to local reactions that lower the elastic strain energy at semicoherent
interfaces [95, 122]. Such strain-relaxed rearrangements of interfacial
dislocation patterns also involve the mechanical problem of finding the
minimum-energy paths from a given initial non-equilibrium lozenge-shaped
microstructure with two sets of parent misfit dislocations to new unique or
multiple (with the same strain energy) stable equilibrium hexagonal-shaped
dislocation patterns of lowest energies with possible metastable
configurations.

Without changing the interface crystallographic characters upon the relaxation
processes, the prescribed displacement jumps for each periodic hexagonal unit
cell are also assumed to vary linearly with the (algebraic) directed distance
between the O-lattice points (displayed by the grey points in Fig. (3.25b))
and the nearest neighbor interfacial dislocation segments. At the positions of
the dislocation segments, the relative displacements are completely described
by the directions and constant magnitudes of the associated individual Burgers
vectors. Furthermore, the non-classical boundary conditions due to the free
surface excess stress and the semicoherent interface excess stress
contributions are therefore applied at: $\textit{x}^{\mbox{\scriptsize
or}}_{2}=h_{\mbox{\scriptsize A}}$ and $\textit{x}^{\mbox{\scriptsize
or}}_{2}=0$, respectively. Thus, the minimum-energy paths are entirely
obtained by measuring the removal of the short-range elastic strain energy
with respect to the coordinates $(\eta_{1},\eta_{2})$ of $\mbox{J}_{1}$, along
which the long-range elastic strain-free state is not altered by spurious non-
zero far-field strains.

For a given crystallographic orientation relationship between materials A and
B, the methodology for determining the equilibrium dislocation configurations
for elastic strain relaxation processes along minimum-energy paths is
described below. The two first items summarize the strategy procedure for
computing the Burgers vectors of interface dislocations using anisotropic
elasticity theory, which have been introduced in section 3.2.

  1. 1.

The geometries in terms of dislocation spacings and line directions, i.e.
$\boldsymbol{\xi}^{\mbox{\scriptsize un}}_{1}$ and
$\boldsymbol{\xi}^{\mbox{\scriptsize un}}_{2}$, related to the initial
lozenge-shaped patterns are found by using the quantized Frank-Bilby equation.
For such networks containing two sets of straight, parallel, and infinite
misfit dislocations, the periodicity of the dislocation structures is also
obtained by mapping the O-lattice points at the interfaces. The corresponding
computed O-lattice vectors $\textbf{{p}}_{1}^{\mbox{\footnotesize o}}$ and
$\textbf{{p}}_{2}^{\mbox{\footnotesize
o}}\neq\textbf{{p}}_{1}^{\mbox{\footnotesize o}}$ are conveniently associated
with the fixed and non-orthogonal basis vectors of the
$(\mbox{O},\,\textbf{{x}}_{1}^{\prime},\,\textbf{{x}}_{2},\,\textbf{{x}}_{3}^{\prime})$
frame for elasticity analyses, where
$\textbf{{x}}_{1}^{\prime}\parallel\textbf{{p}}_{1}^{\mbox{\footnotesize
o}}\parallel\boldsymbol{\xi}^{\mbox{\scriptsize un}}_{2}$,
$\textbf{{x}}_{2}\parallel\textbf{{n}}$, and
$\textbf{{x}}_{3}^{\prime}\parallel\textbf{{p}}_{2}^{\mbox{\footnotesize
o}}\parallel\boldsymbol{\xi}^{\mbox{\scriptsize un}}_{1}$.

  2. 2.

The reference state, within which the individual Burgers vectors of both
dislocation sets are defined, i.e. $\textbf{{b}}_{1}$ and $\textbf{{b}}_{2}$,
is determined by combining the Frank-Bilby equation with anisotropic
elasticity theory that meets the constraints of interface crystallographic
character and zero long-range strains (or stresses) for infinite bicrystals.
Because the latter far-field condition is still fulfilled during the elastic
strain relaxation processes, the third Burgers vector $\textbf{{b}}_{3}$ for
the newly formed dislocation junctions is also obtained from the conservation
eq. (3.70) of the Burgers vector content at the three-fold node
$\mbox{J}_{1}$. In the limiting case where a coplanar free surface is located
in material A, the reference state (and therefore also, the three Burgers
vectors) is fully associated with material B, e.g. the case of a thin film on
a semi-infinite substrate.

  3. 3.

The specific triangular region $\mathcal{T}_{\mbox{\ssmall ABC}}$ in the
representative lozenge-shaped unit cell, formed by the three fixed points A,
B, and C in Fig. (3.25b), is discretized into four-node quadrilateral elements
with respect to the $i^{\mbox{\footnotesize th}}$ nodal points with
coordinates $(\eta_{1}^{i},\eta_{2}^{i})$, such that
$\\{\eta_{1}^{i},\eta_{2}^{i}\\}\in\left]0,\,1/2\right[\,\\!^{2}$. This
discretization allows to represent any convex hexagonal-shaped dislocation
patterns in the non-orthogonal
$(\mbox{O},\,\textbf{{x}}_{1}^{\prime},\,\textbf{{x}}_{2},\,\textbf{{x}}_{3}^{\prime})$
frame for mechanics-based calculations of elastic field solutions, e.g.
displacements, stresses, traction forces, etc.

  4. 4.

The elastic strain energy stored at semicoherent interfaces is computed at any
mesh point $(\eta_{1}^{i},\eta_{2}^{i})$, by using the persistent short-range
stress and strain field solutions for convex and irregular hexagonal-shaped
dislocation configurations. Furthermore, the complete elastic energy landscape
$\gamma_{\mathrm{e}}(\eta_{1},\eta_{2})$ is interpolated for any
$(\eta_{1},\eta_{2})\in\left]0,\,1/2\right[\,\\!^{2}$ with the aid of standard
finite element bilinear shape functions for four-node elements.

  5. 5.

For energetically favorable reactions, the minimum-energy dislocation
configurations are numerically obtained by using the conjugate gradient
algorithm on the pre-computed energy landscapes with a given prescribed
tolerance. Then, the nudged elastic band method [118, 226] is used to provide
access to the minimum-energy paths between the initial (non-equilibrium)
lozenge-shaped structures and the determined elastically relaxed dislocation
patterns with the aid of the elastic forces:
$\textbf{{f}}_{\mathrm{\\!e}}=-\boldsymbol{\nabla}\gamma_{\mathrm{e}}(\eta_{1},\eta_{2})$.
In practice, all elastic field solutions are recomputed along the curvilinear
reaction coordinates of the minimum-energy paths.

####  3.6.3 Parametric energy-based framework

This section is concerned with the complete expressions of elastic fields for
hexagonal-shaped dislocation patterns located at heterophase interfaces
between two dissimilar anisotropic materials. The Stroh sextic formalism of
anisotropic linear elasticity combined with the surface/interface treatment in
Ref. [108] and a Fourier series-based solution technique is therefore used to
compute the elastic fields outside the cores of dislocations. In the general
case, all surfaces of interest (i.e. semicoherent interfaces and free
surfaces) are distinctly considered as infinitely thin membranes with
different, separate, and appropriate constitutive equations than the relations
for both (bulk) materials A and B.

Again, the pre-subscripts A and B in the elastic properties and also the field
expressions will be omitted for clarity in the following if no distinction
between materials is required.

##### Elastic field equations and solutions in bulk materials

In the fixed Cartesian coordinate system
$\left(\mbox{O},\,\textbf{{x}}_{1},\,\textbf{{x}}_{2},\,\textbf{{x}}_{3}\right)$,
the three-dimensional stress field
$\boldsymbol{\sigma}(\textbf{{x}})=\sigma_{ij}(\textit{x}_{1},\textit{x}_{2},\textit{x}_{3})$
and the displacement field
$\textbf{{u}}(\textbf{{x}})=u_{i}(\textit{x}_{1},\textit{x}_{2},\textit{x}_{3})$
in both crystals A and B are related by the Hooke’s law in index form from Eq.
(3.12b), as follows

(3.71)
$\displaystyle{\sigma_{ij}}(\textit{x}_{1},\textit{x}_{2},\textit{x}_{3})$
$\displaystyle=c_{ijkl}\;{\textit{u}_{k,l}}(\textit{x}_{1},\textit{x}_{2},\textit{x}_{3})\,,$

where a comma stands for differentiation, with repeated indices denoting
summation convention ranging from 1 to 3, unless stipulated otherwise. The
anisotropic elastic constants of the fourth-order stiffness tensor
$\mathbb{C}$ are fully symmetric, i.e. $c_{ijkl}=c_{jikl}=c_{ijlk}=c_{klij}$,
and the classical partial differential eq. (3.8) of mechanical equilibrium
that is fulfilled in both crystals in terms of the displacement fields is
given by

(3.72)
$\displaystyle{\sigma_{ij,j}}(\textit{x}_{1},\textit{x}_{2},\textit{x}_{3})$
$\displaystyle=c_{ijkl}\;\textit{u}_{k,jl}(\textit{x}_{1},\textit{x}_{2},\textit{x}_{3})=0\,.$

According to eq. (3.7), the complete displacement field is expressed as the
superposition of the linear displacement contribution from the proper
selection of reference states for constrained interfaces and the total
displacement fields produced by the arrays of interfacial Volterra
dislocations. The latter dislocation displacement fields are also given as a
biperiodic Fourier series, i.e.

(3.73) $\displaystyle\textit{u}_{k}^{\mbox{\scriptsize
dis}}(\textit{x}_{1},\textit{x}_{2},\textit{x}_{3})=\mathrm{Re}\sum_{\textbf{{k}}\,\neq\,\mathbb{0}}\mathrm{e}^{i2\pi\textbf{{k}}\,\cdot\,\textbf{{r}}}\;\textit{u}_{k}^{\textbf{{k}}}(\textit{x}_{2})=2\,\mathrm{Re}\sum_{\mathrm{\textit{D}}}\mathrm{e}^{i2\pi\textbf{{k}}\,\cdot\,\textbf{{r}}}\;\textit{u}_{k}^{\textbf{{k}}}(\textit{x}_{2})\,,$

where the Fourier series expansion involves the harmonics $(n,\,m)$ that
belong to the upper two-dimensional half-plane domain, defined by
$\mathrm{\textit{D}}=\\{\\{n\in\mathbb{N}^{*}\\}\cup\\{m\in\mathbb{Z}^{*},\,n=0\\}\\}$.
For clarity, the subscript ${}_{\mbox{\scriptsize dis}}$ in eq. (3.7) has been
changed to superscript in eq. (3.73). The components $\textit{k}_{1}(n,m)$ and
$\textit{k}_{3}(m)$ of the wavevectors k are given by eq. (3.6) as follows

(3.74)
$\textbf{{k}}\,\cdot\,\textbf{{r}}=\dfrac{n}{\textit{p}_{1}^{\mbox{\footnotesize
o}}}\,\textit{x}_{1}^{\prime}+\dfrac{m}{\textit{p}_{2}^{\mbox{\footnotesize
o}}}\;\textit{x}_{3}^{\prime}=\left(\dfrac{n\;\mathrm{csc}\,\phi^{\mbox{\scriptsize
un}}}{\textit{p}_{1}^{\mbox{\footnotesize
o}}}-\dfrac{m\;\mathrm{ctg}\,\phi^{\mbox{\scriptsize
un}}}{\textit{p}_{2}^{\mbox{\footnotesize
o}}}\right)\textit{x}_{1}+\dfrac{m}{\textit{p}_{2}^{\mbox{\footnotesize
o}}}\;\textit{x}_{3}=\textit{k}_{1}(n,m)\;\textit{x}_{1}+\textit{k}_{3}(m)\;\textit{x}_{3}\,,$

with $\textit{p}_{1}^{\mbox{\footnotesize
o}}=\lvert\,\textbf{{p}}_{1}^{\mbox{\footnotesize o}}\rvert$ and
$\textit{p}_{2}^{\mbox{\footnotesize
o}}=\lvert\,\textbf{{p}}_{2}^{\mbox{\footnotesize o}}\rvert$. On the other
hand, the far-field components are computed for two dislocation sets to
determine the correct reference state [251], within which the Burgers vectors
$\textbf{{b}}_{1}$ and $\textbf{{b}}_{2}$ (and also $\textbf{{b}}_{3}$, by
virtue of eq. (3.70)) are defined. Because the elastic (short-range) strain
relaxations do not alter the long-range strain state during the junction
formation of the third dislocation sets, the removal of the far-field strains
(or stresses) in the natural state is fulfilled by solving the tensorial far-
field eqs. (3.1), exhibiting non-zero and heterogeneous short-range elastic
fields for interfacial dislocation patterns, only. Thus, substituting the
displacement field eq. (3.73) into eq. (3.71), the second-order differential
equation applied to both materials is obtained in index form as follows

(3.75)
$-4\pi^{2}\,\mbox{W}_{1_{ik}}\;\tilde{\textit{u}}^{\textbf{\,{k}}}_{k}(\textit{x}_{2})+i2\pi\left(\mbox{W}_{2_{ik}}+\mbox{W}_{2_{ki}}\right)\tilde{\textit{u}}^{\textbf{\,{k}}}_{k,2}(\textit{x}_{2})+\mbox{W}_{3_{ik}}\;\tilde{\textit{u}}^{\textbf{\,{k}}}_{k,22}(\textit{x}_{2})=0\,,$

where $\textbf{W}_{1}$, $\textbf{W}_{2}$, and $\textbf{W}_{3}$ are $3\times 3$
real matrices defined in eqs. (3.10). In eq. (3.75), the superimposed tilde to
any quantities will be used to indicate that the corresponding field solutions
are consistent with the Frank-Bilby equation under the condition of vanishing
far-field strains (or stresses) for any dislocation patterns. For non-zero
wavevectors k, the standard solutions satisfying eq. (3.75) can be written in
the following form [62]

(3.76)
$\tilde{\textit{u}}^{\textbf{\,{k}}}_{k}(\textit{x}_{2})=\mathrm{e}^{i2\pi\textit{p}^{\textbf{{k}}}\textit{x}_{2}}\;\textit{a}^{\textbf{\,{k}}}_{k}\,,$

where $\textit{p}^{\textbf{{k}}}=\textit{p}$ and
$\textbf{{a}}^{\textbf{{k}}}=\textit{a}_{k}$ become the complex scalar and
vectorial unknowns of the boundary value problems, respectively, for which the
superscripts k are omitted, for clarity. Introducing eq. (3.76) into eq.
(3.75), the vector a is found to satisfy the homogeneous linear system

(3.77)
$\left[\mbox{W}_{1_{ik}}+\textit{p}\,(\mbox{W}_{2_{ik}}+\mbox{W}_{2_{ki}})+\textit{p}^{2}\,\mbox{W}_{3_{ik}}\right]\textit{a}_{k}=\Pi_{ik}\,\textit{a}_{k}=0\,,$

which corresponds to the standard eigenvalue problem in anisotropic elasticity
theory [238, 244]. A non-zero (non-trivial) solution can be found only if the
determinant of $\boldsymbol{\Pi}$ is zero, i.e.

(3.78) $\mathrm{det}\;\Pi_{ik}=0\,,$

leading to a sextic equation for p. As mentioned in section 3.6.3, the
solutions of eq. (3.78) have six imaginary roots, which are arranged such that
the three first eigenvalue solutions $\textit{p}^{\alpha}$ have positive
imaginary parts, indexed by superscripts $\alpha=1,\,2,\,3$. The remaining
three solutions have negative imaginary parts, so that
$\textit{p}^{\alpha+3}=\textit{p}_{*}^{\alpha}$. The corresponding
eigenvectors $\textbf{{a}}^{\alpha}=\textit{a}^{\alpha}_{k}$ are also complex
conjugates with
$\textbf{{a}}^{\alpha+3}=\textbf{{a}}_{*}^{\alpha}=\textit{a}_{k_{\,*}}^{\alpha}$,
so that the general solution may be rewritten as a linear combination of the
three eigenfunctions, i.e.

(3.79) $\displaystyle\tilde{\textit{u}}_{k}^{\mbox{\scriptsize
dis}}(\textit{x}_{1},\textit{x}_{2},\textit{x}_{3})$
$\displaystyle=2\,\mathrm{Re}\sum_{\mathrm{\textit{D}}}\mathrm{e}^{i2\pi\textbf{{k}}\,\cdot\,\textbf{{r}}}\sum_{\alpha\,=\,1}^{3}\lambda^{\alpha}\mathrm{e}^{i2\pi\textit{p}^{\alpha}\textit{x}_{2}}\;\textit{a}^{\alpha}_{k}+\zeta^{\alpha}\mathrm{e}^{i2\pi\textit{p}_{*}^{\alpha}\textit{x}_{2}}\;\textit{a}_{k_{\,*}}^{\alpha}\,,$

which differs from eq. (3.11) by a multiplicative $i2\pi$ term, without loss
of generality. It also follows from eq. (3.71) that

(3.80) $\displaystyle\tilde{\sigma}_{ij}^{\mbox{\scriptsize
dis}}(\textit{x}_{1},\textit{x}_{2},\textit{x}_{3})$
$\displaystyle=4\pi\,\mathrm{Re}\sum_{\mathrm{\textit{D}}}i\,\mathrm{e}^{i2\pi\textbf{{k}}\,\cdot\,\textbf{{r}}}\sum_{\alpha\,=\,1}^{3}\lambda^{\alpha}\mathrm{e}^{i2\pi\textit{p}^{\alpha}\textit{x}_{2}}\;\mbox{H}^{\alpha}_{ij}+\zeta^{\alpha}\mathrm{e}^{i2\pi\textit{p}_{*}^{\alpha}\textit{x}_{2}}\;\mbox{H}^{\alpha}_{ij_{\,*}}\,,$

where the $3\times 3$ complex matrices $\textbf{H}^{\alpha}$ are related to
the eigenvectors $\textbf{{a}}^{\alpha}$ by

(3.81)
$\displaystyle\mbox{H}^{\alpha}_{ij}=\left(\textit{k}_{1}\,c_{ijk1}+\textit{k}_{3}\,c_{ijk3}+\textit{p}^{\alpha}c_{ijk2}\right)\textit{a}_{k}^{\alpha}\,,$

from selected elastic constants of materials A and B. In particular, the
surface tractions at the semicoherent interfaces, i.e. $\textit{x}_{2}=0$, are
reduced to

(3.82) $\displaystyle\tilde{\textit{t}}_{k}^{\mbox{\,\scriptsize
int}}(\textit{x}_{1},\textit{x}_{3})=\tilde{\sigma}_{ki}^{\mbox{\scriptsize
dis}}(\textit{x}_{1},0,\textit{x}_{3})\,\textit{n}_{i}$
$\displaystyle=4\pi\,\mathrm{Re}\sum_{\mathrm{\textit{D}}}i\,\mathrm{e}^{i2\pi\textbf{{k}}\,\cdot\,\textbf{{r}}}\sum_{\alpha\,=\,1}^{3}\lambda^{\alpha}\,\mbox{H}^{\alpha}_{k2}+\zeta^{\alpha}\,\mbox{H}^{\alpha}_{k2_{\,*}}\,,$

as well as the tractions at the free surface, i.e.
$\textit{x}_{2}=h_{\mbox{\scriptsize A}}$, to

(3.83) $\displaystyle\tilde{\textit{t}}_{k}^{\mbox{\,\scriptsize
fs}}(\textit{x}_{1},\textit{x}_{3})=\tilde{\sigma}_{ki}^{\mbox{\scriptsize
dis}}(\textit{x}_{1},h_{\mbox{\scriptsize A}},\textit{x}_{3})\,\textit{n}_{i}$
$\displaystyle=4\pi\,\mathrm{Re}\sum_{\mathrm{\textit{D}}}i\,\mathrm{e}^{i2\pi\textbf{{k}}\,\cdot\,\textbf{{r}}}\sum_{\alpha\,=\,1}^{3}\lambda^{\alpha}\mathrm{e}^{i2\pi\textit{p}^{\alpha}h_{\mbox{A}}}\;\mbox{H}^{\alpha}_{k2}+\zeta^{\alpha}\mathrm{e}^{i2\pi\textit{p}_{*}^{\alpha}h_{\mbox{A}}}\;\mbox{H}^{\alpha}_{k2_{\,*}}\,.$

##### Free surface and semicoherent interface elasticity contributions

Combined with the surface tractions in eqs. (3.82) and (3.83), the additional
surface/interface stress contributions, due to the work required by applying
in-plane forces to elastically stretch the pre-existing free surfaces and
interfaces neighboring both materials A and B into the correct reference
states, are introduced as follows

(3.84) $\displaystyle\tau_{\chi\varphi}(\textit{x}_{1},\textit{x}_{3})$
$\displaystyle=\gamma\,\delta_{\chi\varphi}+\frac{\partial\,\gamma}{\partial\epsilon^{\,\textit{s}}_{\chi\varphi}(\textit{x}_{1},\textit{x}_{3})}\,,$

where $\tau_{\chi\varphi}(\textit{x}_{1},\textit{x}_{3})$ and
$\epsilon_{\chi\varphi}^{\,\textit{s}}(\textit{x}_{1},\textit{x}_{3})$ are the
$2\times 2$ surface stress and strain tensors, and $\gamma$ is the surface
free energy [228, 51]. Because eq. (3.84) is derived for the plane stresses
acting in the surface area, the stress and strain fields have only in-plane
components, and Greek indices take values 1 and 3, only. In order to solve the
elasticity problems with appropriate constitutive relations between the
surface stress and strain components, a linear constitutive equation analogous
to eq. (3.71) is used [225], i.e.

(3.85) $\displaystyle\tau_{\chi\varphi}(\textit{x}_{1},\textit{x}_{3})$
$\displaystyle=\tau_{\chi\varphi}^{\scalebox{0.65}{0}}+d_{\chi\varphi\gamma\eta}\;\epsilon^{\,\textit{s}}_{\gamma\eta}(\textit{x}_{1},\textit{x}_{3})\,,$

where $\tau_{\chi\varphi}^{\scalebox{0.65}{0}}$ is the surface/interface
residual stress tensor and $d_{\chi\varphi\gamma\eta}$ is the fourth-order
stiffness tensor of surface/interface elastic constants. When the
surface/interface entities are considered as elastic isotropic media, the
elasticity tensor contains also two independent constants, known as
surface/interface Lamé constants [51].

In the case of realistic semicoherent interfaces, the atomic structures are
not exactly like those generated by the linear mappings from a reference
state, as idealized and illustrated in Fig. (3.25a) with no atomic
relaxations. Indeed, electron microscopy and atomistic calculations have
revealed that such boundaries consist of coherent patches separated by
networks of interfacial dislocations. The coherency and bounding conditions
between such boundaries and the adjacent bulk materials yield therefore to the
expression for the surface stresses in terms of the derivatives of the bulk
displacement fields, i.e.

(3.86) $\displaystyle
2\,\epsilon^{\,\textit{s}}_{\gamma\eta}(\textit{x}_{1},\textit{x}_{3})=2\,\tilde{\epsilon}_{\gamma\eta}^{\mbox{\,\scriptsize
dis}}(\textit{x}_{1},0,\textit{x}_{3})=\tilde{\textit{u}}_{\gamma,\eta}^{\mbox{\scriptsize
dis}}(\textit{x}_{1},0,\textit{x}_{3})+\tilde{\textit{u}}_{\eta,\gamma}^{\mbox{\scriptsize
dis}}(\textit{x}_{1},0,\textit{x}_{3})\,,$

for all in-plane strain components, at $\textit{x}_{2}=0$. Similarly to the
model for interface stresses with application to misfit dislocations in Ref.
[52], this interface/bulk conversion of strain fields in eq. (3.86) depends
strongly on the presence of the misfit dislocations (and, therefore, on the
non-arbitrary coherent reference states), which in the present case, gives
rise to unequally partitioned distortion states, in terms of strains as well
as tilt and twist rotations.

The elastic field solutions of the displacements, stresses and tractions of
eqs. (3.79), (3.80), and (3.82$-$3.83) respectively, with respect to
surface/interface effects defined by eqs. (3.85$-$3.86), are also written as
linear combinations of the eigenfunctions, within which
$\\{\lambda^{\alpha},\zeta^{\alpha}\\}$ are complex unknown quantities that
are to be determined by the boundary conditions. For hexagonal-shaped
dislocation patterns, these specific required conditions are expressed in
terms of the discontinuities of displacement and stress components across the
semicoherent interfaces in bimaterials in presence (if any) of a free surface
in the upper material.

####  3.6.4 Boundary conditions with surface/interface constitutive relations

In what follows in section 3.6.4, expressions of displacements, strains, and
stresses, and also all related quantities that are needed to compute these
field solutions (e.g., the elastic constants, Burgers vectors
$\textbf{{b}}_{1}$, $\textbf{{b}}_{2}$, and $\textbf{{b}}_{3}$, eigenvectors
$\textbf{{a}}^{\alpha}$, etc.), are expressed in the local oblique and fixed
$(\mbox{O},\,\textbf{{x}}_{1}^{\prime},\,\textbf{{x}}_{2},\,\textbf{{x}}_{3}^{\prime})$
frame. In particular, the boundary conditions are written with respect to the
geometry of the dislocation patterns, i.e. to the canonical coordinates
$(\eta_{1},\eta_{2})$ of the three-fold dislocation node J1, as well as the
magnitudes and directions of the individual Burgers vectors for three sets of
dislocations.

##### Convergence of the elastic field solutions

For all constrained interfaces that are consistent with the Frank-Bilby
equation, i.e. when eqs. (3.1) are fulfilled with respect to the correct
reference state, the corresponding semi-infinite linear crystal (here,
material B) is also necessary free of all far-field stress components. The
elastic stress solution in eq. (3.80) is therefore required to converge to
zero at long range, i.e. when $\textit{x}_{2}\to-\infty$. Hence,
${{}_{\mbox{\scriptsize B}}\lambda^{\alpha}}=0$, independently of interfacial
boundary conditions. For infinite bicrystals of interest, the convergence
conditions in both materials A and B yield to: ${{}_{\mbox{\scriptsize
B}}\lambda^{\alpha}}={{}_{\mbox{\scriptsize A}}\zeta^{\alpha}}=0$, when
$\textit{x}_{2}\to\pm\infty$, as already defined in section 3.2.5.

##### Relative displacement due to the interfacial dislocation patterns

In accordance with eq. (3.15), the prescribed relative displacement field
$\textbf{{u}}^{\,\textit{p}}(\textit{x}_{1},\textit{x}_{3})$ for any
(irregular) hexagonal-shaped dislocation patterns at the interface, i.e.
$x_{2}=0$, is obtained by superposing the contributions of both Burgers
vectors $\textbf{{b}}_{1}$ and $\textbf{{b}}_{2}$, i.e.

(3.87)
$\displaystyle\textbf{{u}}^{\,\textit{p}}(\textit{x}_{1},\textit{x}_{3})=\frac{\textit{x}_{1}\,\mathrm{csc}\,\phi^{\mbox{\scriptsize
un}}}{\textit{p}_{1}^{\mbox{\footnotesize
o}}}\;\textbf{{b}}_{1}+\frac{\textit{x}_{3}-\textit{x}_{1}\,\mathrm{ctg}\,\phi^{\mbox{\scriptsize
un}}}{\textit{p}_{2}^{\mbox{\footnotesize
o}}}\;\textbf{{b}}_{2}=\textit{z}_{1}(\textit{x}_{1})\;\textbf{{b}}_{1}+\textit{z}_{2}(\textit{x}_{1},\textit{x}_{3})\;\textbf{{b}}_{2}\,,$

where $\textit{z}_{1}=\textit{z}_{1}(\textit{x}_{1})$ and
$\textit{z}_{2}=\textit{z}_{2}(\textit{x}_{1},\textit{x}_{3})$ are
dimensionless linear functions. Assuming that the displacement jumps are zero
at all positions of the O-lattice points, e.g. at O in the representative unit
cell in Fig. (3.25b), the prescribed displacement field is also an odd
function with respect to r in the oblique
$(\mbox{O},\,\textbf{{x}}_{1}^{\prime},\,\textbf{{x}}_{2},\,\textbf{{x}}_{3}^{\prime})$
frame. According to the linear elasticity theory, these displacement jumps
produced by each hexagonal-shaped dislocation cell can therefore be formally
expressed as double Fourier series for any dislocation configurations with
respect to $(\eta_{1},\eta_{2})$, i.e.

(3.88)
$\displaystyle\textbf{{u}}^{\,\textit{p}}(\textit{x}_{1},\textit{x}_{3})=\mathrm{Im}\sum_{\textbf{{k}}\,\neq\,\mathbb{0}}\mathrm{e}^{i2\pi\textbf{{k}}\,\cdot\,\textbf{{r}}}\;\hat{\textbf{{u}}}^{\,\textit{p}}(\eta_{1},\eta_{2})=-\mathrm{Re}\,i\sum_{\textbf{{k}}\,\neq\,\mathbb{0}}\mathrm{e}^{i2\pi\textbf{{k}}\,\cdot\,\textbf{{r}}}\left(\hat{\textbf{{u}}}^{\,\textit{p}}_{1}(\eta_{1},\eta_{2})+\hat{\textbf{{u}}}^{\,\textit{p}}_{2}(\eta_{1},\eta_{2})\right)\,,$

where all real-valued expansion coefficients
$\hat{\textbf{{u}}}^{\,\textit{p}}(\eta_{1},\eta_{2})$ in eq. (3.88) are
additionally decomposed into the individual contributions
$\hat{\textbf{{u}}}^{\,\textit{p}}_{1}(\eta_{1},\eta_{2})$ and
$\hat{\textbf{{u}}}^{\,\textit{p}}_{2}(\eta_{1},\eta_{2})$, associated with
$\textbf{{b}}_{1}$ and $\textbf{{b}}_{2}$, respectively. In particular, the
vector quantity $\hat{\textbf{{u}}}^{\,\textit{p}}_{1}(\eta_{1},\eta_{2})$ for
$\textbf{{b}}_{1}$ is deduced by solving the double integral with respect to
$\textit{z}_{1}$ and $\textit{z}_{2}$, as follows

(3.89)
$\hat{\textbf{{u}}}^{\,\textit{p}}_{1}(\eta_{1},\eta_{2})=\mathrm{Re}\left[\,i\int_{\bar{\textit{z}}_{1}(\eta_{1})}^{\bar{\bar{\textit{z}}}_{1}(\eta_{1})}\left(\textit{z}_{1}\int_{\bar{\textit{z}}_{2}(\textit{z}_{1},\eta_{1},\eta_{2})}^{\bar{\bar{\textit{z}}}_{2}(\textit{z}_{1},\eta_{1},\eta_{2})}\,e^{-i2\pi(n\textit{z}_{1}+m\textit{z}_{2})}\,d\textit{z}_{2}\right)d\textit{z}_{1}\right]\textbf{{b}}_{1}\,,$

for any $(\eta_{1},\eta_{2})\in\mathcal{H}_{\mbox{\,\ssmall
J}_{1}\\!\mbox{\,\ssmall J}_{2}\\!\mbox{\,\ssmall J}_{3}\\!\mbox{\,\ssmall
J}_{4}\\!\mbox{\,\ssmall J}_{5}\\!\mbox{\,\ssmall J}_{6}}$. Moreover, eq.
(3.89) may be integrated over three separate unit domains, e.g. the
parallelogram $\mathcal{P}_{\mbox{\,\ssmall J}_{1}\\!\mbox{\,\ssmall
J}_{3}\\!\mbox{\,\ssmall J}_{4}\\!\mbox{\,\ssmall J}_{6}}$ and both triangles
$\mathcal{T}_{\mbox{\,\ssmall J}_{1}\\!\mbox{\,\ssmall
J}_{2}\\!\mbox{\,\ssmall J}_{3}}$ and $\mathcal{T}_{\mbox{\,\ssmall
J}_{4}\\!\mbox{\,\ssmall J}_{5}\\!\mbox{\,\ssmall J}_{6}}$, i.e.

(3.90)
$\hat{\textbf{{u}}}^{\,\textit{p}}_{1}(\eta_{1},\eta_{2})=\hat{\textbf{{u}}}^{\,\textit{p}}_{1}(\eta_{1},\eta_{2})|_{\mathcal{H}_{\mbox{\,\ssmall
J}_{1}\\!\mbox{\,\ssmall J}_{2}\\!\mbox{\,\ssmall J}_{3}\\!\mbox{\,\ssmall
J}_{4}\\!\mbox{\,\ssmall J}_{5}\\!\mbox{\,\ssmall
J}_{6}}}=\hat{\textbf{{u}}}^{\,\textit{p}}_{1}(\eta_{1},\eta_{2})|_{\mathcal{P}_{\mbox{\,\ssmall
J}_{1}\\!\mbox{\,\ssmall J}_{3}\\!\mbox{\,\ssmall J}_{4}\\!\mbox{\,\ssmall
J}_{6}}}+\hat{\textbf{{u}}}^{\,\textit{p}}_{1}(\eta_{1},\eta_{2})|_{\mathcal{T}_{\mbox{\,\ssmall
J}_{1}\\!\mbox{\,\ssmall J}_{2}\\!\mbox{\,\ssmall
J}_{3}}}+\hat{\textbf{{u}}}^{\,\textit{p}}_{1}(\eta_{1},\eta_{2})|_{\mathcal{T}_{\mbox{\,\ssmall
J}_{4}\\!\mbox{\,\ssmall J}_{5}\\!\mbox{\,\ssmall J}_{6}}}\,,$

as illustrated by the different vertices in Fig. (3.25b). Because the
boundaries of the hexagonal-shaped unit cells are composed of straight
dislocation segments, the integral eq. (3.89) is necessarily bounded by affine
functions with respect to the coordinates $\eta_{1}$ and $\eta_{2}$. The first
quantity
$\hat{\textbf{{u}}}^{\,\textit{p}}_{1}(\eta_{1},\eta_{2})|_{\mathcal{P}_{\mbox{\,\ssmall
J}_{1}\\!\mbox{\,\ssmall J}_{3}\\!\mbox{\,\ssmall J}_{4}\\!\mbox{\,\ssmall
J}_{6}}}$ in the right-hand side of eq. (3.90) is also computed by using the
following bounds, i.e.

(3.91) $\forall\,\\{\eta_{1},\eta_{2}\\}\in\mathcal{P}_{\mbox{\,\ssmall
J}_{1}\\!\mbox{\,\ssmall J}_{3}\\!\mbox{\,\ssmall J}_{4}\\!\mbox{\,\ssmall
J}_{6}}:~{}~{}\left\\{\begin{matrix}\begin{aligned}
\bar{\textit{z}}_{1}(\eta_{1})&=-\eta_{1}\\\
\bar{\bar{\textit{z}}}_{1}(\eta_{1})&=\eta_{1}\end{aligned}\end{matrix}\right.\,,~{}~{}~{}\mbox{and}~{}~{}~{}~{}\left\\{\begin{matrix}\begin{aligned}
\bar{\textit{z}}_{2}(\textit{z}_{1},\eta_{1},\eta_{2})&=-\mfrac{1-2\eta_{2}}{2\eta_{1}}\textit{z}_{1}-\mfrac{1}{2}\\\\[3.0pt]
\bar{\bar{\textit{z}}}_{2}(\textit{z}_{1},\eta_{1},\eta_{2})&=-\mfrac{1-2\eta_{2}}{2\eta_{1}}\textit{z}_{1}+\mfrac{1}{2}\,.\end{aligned}\end{matrix}\right.$

Similarly, the two quantities
$\hat{\textbf{{u}}}^{\,\textit{p}}_{1}(\eta_{1},\eta_{2})|_{\mathcal{T}_{\mbox{\,\ssmall
J}_{1}\\!\mbox{\,\ssmall J}_{2}\\!\mbox{\,\ssmall J}_{3}}}$ and
$\hat{\textbf{{u}}}^{\,\textit{p}}_{1}(\eta_{1},\eta_{2})|_{\mathcal{T}_{\mbox{\,\ssmall
J}_{4}\\!\mbox{\,\ssmall J}_{5}\\!\mbox{\,\ssmall J}_{6}}}$ in eq. (3.90) are
determined by considering

(3.92)
$\displaystyle\forall\,\\{\eta_{1},\eta_{2}\\}\in\mathcal{T}_{\mbox{\,\ssmall
J}_{1}\\!\mbox{\,\ssmall J}_{2}\\!\mbox{\,\ssmall
J}_{3}}:~{}~{}\left\\{\begin{matrix}\begin{aligned}
\bar{\textit{z}}_{1}(\eta_{1})&=\eta_{1}\\\
\bar{\bar{\textit{z}}}_{1}(\eta_{1})&=1-\eta_{1}\end{aligned}\end{matrix}\right.\,,~{}~{}~{}\mbox{and}~{}~{}~{}~{}\left\\{\begin{matrix}\begin{aligned}
\bar{\textit{z}}_{2}(\textit{z}_{1},\eta_{1},\eta_{2})&=\mfrac{(1-2\eta_{2})\,\textit{z}_{1}-1+\eta_{2}+\eta_{1}}{1-2\eta_{1}}\\\\[3.0pt]
\bar{\bar{\textit{z}}}_{2}(\textit{z}_{1},\eta_{1},\eta_{2})&=\mfrac{2\eta_{2}\,\textit{z}_{1}-\eta_{2}}{-1+2\eta_{1}}\,,\end{aligned}\end{matrix}\right.$
$\displaystyle\forall\,\\{\eta_{1},\eta_{2}\\}\in\mathcal{T}_{\mbox{\,\ssmall
J}_{4}\\!\mbox{\,\ssmall J}_{5}\\!\mbox{\,\ssmall
J}_{6}}:~{}~{}\left\\{\begin{matrix}\begin{aligned}
\bar{\textit{z}}_{1}(\eta_{1})&=\eta_{1}-1\\\
\bar{\bar{\textit{z}}}_{1}(\eta_{1})&=-\eta_{1}\end{aligned}\end{matrix}\right.\,,~{}~{}~{}\mbox{and}~{}~{}~{}~{}\left\\{\begin{matrix}\begin{aligned}
\bar{\textit{z}}_{2}(\textit{z}_{1},\eta_{1},\eta_{2})&=\mfrac{2\eta_{2}\,\textit{z}_{1}+\eta_{2}}{-1+2\eta_{1}}\\\\[3.0pt]
\bar{\bar{\textit{z}}}_{2}(\textit{z}_{1},\eta_{1},\eta_{2})&=\mfrac{(1-2\eta_{2})\,\textit{z}_{1}+1-\eta_{2}-\eta_{1}}{1-2\eta_{1}}\,,\end{aligned}\end{matrix}\right.$

respectively. Thus, after integrating eq. (3.90) analytically with respect to
eqs. (3.91) and (3.92), it can also be found that

(3.93)
$\hat{\textbf{{u}}}^{\,\textit{p}}_{1}(\eta_{1},\eta_{2})=\mathrm{sin}\left(2\pi\left(m\,\eta_{2}+n\,\eta_{1}\right)\right)\frac{-1+2\eta_{1}}{2\pi^{2}\left(m+n-2\left(m\,\eta_{2}+n\,\eta_{1}\right)\right)\left(2m\,\eta_{2}+n\left(-1+2\eta_{1}\right)\right)}~{}\textbf{{b}}_{1}\,,$

for any given $(\eta_{1},\eta_{2})$. Analogously to eq. (3.89), the vector
quantity $\hat{\textbf{{u}}}^{\,\textit{p}}_{2}(\eta_{1},\eta_{2})$ for
$\textbf{{b}}_{2}$ is written in the form

(3.94)
$\hat{\textbf{{u}}}^{\,\textit{p}}_{2}(\eta_{1},\eta_{2})=\mathrm{Re}\left[\,i\int_{\bar{\textit{z}}_{1}(\eta_{1})}^{\bar{\bar{\textit{z}}}_{1}(\eta_{1})}\left(\,\int_{\bar{\textit{z}}_{2}(\textit{z}_{1},\eta_{1},\eta_{2})}^{\bar{\bar{\textit{z}}}_{2}(\textit{z}_{1},\eta_{1},\eta_{2})}\,\textit{z}_{2}\,e^{-i2\pi(n\textit{z}_{1}+m\textit{z}_{2})}\,d\textit{z}_{2}\right)d\textit{z}_{1}\right]\textbf{{b}}_{2}\,,$

for which the same integral bounds defined by eqs. (3.91) and (3.92) are used
to calculate eq. (3.94) over the hexagonal-shaped dislocation patterns. Hence,
it follows

(3.95)
$\hat{\textbf{{u}}}^{\,\textit{p}}_{2}(\eta_{1},\eta_{2})=\mathrm{sin}\left(2\pi\left(m\,\eta_{2}+n\,\eta_{1}\right)\right)\frac{-1+2\eta_{2}}{2\pi^{2}\left(m+n-2\left(m\,\eta_{2}+n\,\eta_{1}\right)\right)\left(m\left(-1+2\eta_{2}\right)+2n\,\eta_{1}\right)}~{}\textbf{{b}}_{2}\,,$

for any $(\eta_{1},\eta_{2})$. Combining eq. (3.93) with eq. (3.95), the
complete vectorial solution for
$\hat{\textbf{{u}}}^{\,\textit{p}}(\eta_{1},\eta_{2})$ is given by

(3.96)
$\hat{\textbf{{u}}}^{\,\textit{p}}(\eta_{1},\eta_{2})=\frac{\mathrm{sin}\left(2\pi\left(m\,\eta_{2}+n\,\eta_{1}\right)\right)}{2\pi^{2}\left(m+n-2\left(m\,\eta_{2}+n\,\eta_{1}\right)\right)}\left[\frac{-1+2\eta_{1}}{2m\,\eta_{2}+n\left(-1+2\eta_{1}\right)}~{}\textbf{{b}}_{1}+\frac{-1+2\eta_{2}}{m\left(-1+2\eta_{2}\right)+2n\,\eta_{1}}~{}\textbf{{b}}_{2}\right]\,,$

which closely corresponds to the same expression given in Ref. [38], after
minor corrections. It is worth noting that three singular values for $n$ and
$m$ give rise to null denominators in eq. (3.96), so that three cases must be
distinguished, i.e. c1: $m+n-2(m\,\eta_{2}+n\,\eta_{1})\neq 0$, c2:
$n-2(m\,\eta_{2}+n\,\eta_{1})\neq 0$, and c3:
$m-2(m\,\eta_{2}+n\,\eta_{1})\neq 0$. By defining the function
$\textit{z}(n,m)=n\,\textit{z}_{1}+m\,\textit{z}_{2}$ in the exponential terms
of both eqs. (3.89) and (3.94), all corresponding real-valued expansion
coefficients are also obtained by replacing $m$ with $m^{\ast}$ in
$\textit{z}(n,m)$ for all different cases, i.e.

(3.97)
$\displaystyle\mbox{c1:}~{}~{}m^{\ast}=-n\,\frac{1-2\eta_{1}}{1-2\eta_{2}}~{},~{}~{}~{}~{}\mbox{c2:}~{}~{}m^{\ast}=n\,\frac{1-2\eta_{1}}{2\eta_{2}}~{},~{}\mbox{and}~{}~{}~{}\mbox{c3:}~{}~{}m^{\ast}=n\,\frac{2\eta_{1}}{1-2\eta_{2}}\,,$

for which the expressions for these three cases are given in Appendix A from
Ref. [260]. Finally, to exhibit the discontinuity condition in displacement,
the prescribed jump in eq. (3.87) with the aid of the eqs. (3.96) may finally
be related to the displacement fields generated by the interface dislocation
patterns, i.e.

(3.98)
$\displaystyle\textit{u}^{\,\textit{p}}_{k}(\textit{x}_{1},\textit{x}_{3})=\llbracket\tilde{\textit{u}}^{\mbox{\scriptsize
dis}}_{k}(\textit{x}_{1},0,\textit{x}_{3})\rrbracket_{{}_{\mbox{\scriptsize
int}}}={{}_{\mbox{\scriptsize A}}\tilde{\textit{u}}_{k}^{\mbox{\scriptsize
dis}}(\textit{x}_{1},0,\textit{x}_{3})}-{{}_{\mbox{\scriptsize
B}}\tilde{\textit{u}}_{k}^{\mbox{\scriptsize
dis}}(\textit{x}_{1},0,\textit{x}_{3})}\,,$

where the complete elastic field solutions in both materials A and B are given
by eq. (3.79). The symbol
$\llbracket\textit{y}_{k}\rrbracket_{{}_{\mbox{\scriptsize
int}}}=\Delta\,\textit{y}_{k}={{}_{\mbox{\scriptsize
A}}\textit{y}_{k}}-{{}_{\mbox{\scriptsize B}}\textit{y}_{k}}$ corresponds to
the vectorial jump of the quantity y across the interface at
$\textit{x}_{2}=0$. Although all physical displacement fields in eq. (3.98)
are defined as the real quantities of complex Fourier series-based
expressions, the real part designation in eqs. (3.79) and (3.88) are
conveniently omitted to express the complex equality, as follows

(3.99)
$\displaystyle-i\,\hat{\textit{u}}^{\,\textit{p}}_{k}(\eta_{1},\eta_{2})=\sum_{\alpha\,=\,1}^{3}{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{k}}}+{{{}_{\mbox{\scriptsize
A}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{k_{\,*}}}}-{{{}_{\mbox{\scriptsize
B}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
B}}\textit{a}^{\alpha}_{k_{\,*}}}}\,,$

so that both real and imaginary parts of eq. (3.99) lead to the equivalent
homogeneous linear system $\Sigma_{1}$ of six real equations, i.e.

(3.100) $\displaystyle(\Sigma_{1})~{}~{}\forall
k\in\\{1,2,3\\}:\left\\{\begin{matrix}\begin{aligned}
0&=\mathrm{Re}\sum_{\alpha\,=\,1}^{3}{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{k}}}+{{{}_{\mbox{\scriptsize
A}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{k_{\,*}}}}-{{{}_{\mbox{\scriptsize
B}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
B}}\textit{a}^{\alpha}_{k_{\,*}}}}\\\
-\hat{\textit{u}}^{\,\textit{p}}_{k}(\eta_{1},\eta_{2})&=\mathrm{Im}\sum_{\alpha\,=\,1}^{3}{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{k}}}+{{{}_{\mbox{\scriptsize
A}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{k_{\,*}}}}-{{{}_{\mbox{\scriptsize
B}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
B}}\textit{a}^{\alpha}_{k_{\,*}}}}\,,\end{aligned}\end{matrix}\right.$

where $\hat{\textbf{{u}}}^{\,\textit{p}}(\eta_{1},\eta_{2})$ is defined in eq.
(3.96), for any given $(\eta_{1},\eta_{2})\in\left]0,\,1/2\right[\,\\!^{2}$
and for all $\\{n,m\\}\in\mathrm{\textit{D}}$.

##### Stress conditions at the semicoherent interfaces

Due to the presence of the interfacial excess energy close to grain and
interphase boundaries, the discontinuity of the tangential stress components
is introduced using the generalized Young-Laplace equation [108, 212, 77], as
an equilibrium boundary condition to solve the present boundary-value problem
with interface stress effects, i.e.

(3.101) $\displaystyle\llbracket\tilde{\sigma}_{\varphi i}^{\mbox{\scriptsize
dis}}(\textit{x}_{1},0,\textit{x}_{3})\,\textit{n}_{i}\rrbracket_{{}_{\mbox{\scriptsize
int}}}+\tau_{\varphi\chi,\chi}=0\,,$

together with the stress discontinuity in normal direction of the boundaries,
as follows

(3.102) $\displaystyle\llbracket\tilde{\sigma}_{ij}^{\mbox{\scriptsize
dis}}(\textit{x}_{1},0,\textit{x}_{3})\,\textit{n}_{j}\,\textit{n}_{i}\rrbracket_{{}_{\mbox{\scriptsize
int}}}=\tau_{\chi\varphi}\,\kappa_{\chi\varphi}\,,$

with $\kappa_{\chi\varphi}$ the curvature tensor of the solid-state interface
of interest. Substituting the linear constitutive relation of eq. (3.85) into
eqs. (3.101) and (3.102) respectively, the governing non-classical boundary
equations lead to

(3.103) $\displaystyle\left\\{\begin{matrix}\begin{aligned}
0&=\llbracket\tilde{\textit{t}}_{\varphi}^{\mbox{\,\scriptsize
int}}(\textit{x}_{1},\textit{x}_{3})\rrbracket_{{}_{\mbox{\scriptsize
int}}}+d_{\varphi\chi\gamma\eta}\;\tilde{\textit{u}}^{\mbox{\scriptsize
dis}}_{\gamma,\eta\chi}(\textit{x}_{1},0,\textit{x}_{3})\\\\[5.0pt]
0&=\llbracket\tilde{\textit{t}}_{2}^{\mbox{\,\scriptsize
int}}(\textit{x}_{1},\textit{x}_{3})\rrbracket_{{}_{\mbox{\scriptsize
int}}}-\big{(}\tau_{\chi\varphi}^{\scalebox{0.65}{0}}+d_{\chi\varphi\gamma\eta}\;\tilde{\textit{u}}^{\mbox{\scriptsize
dis}}_{\gamma,\eta}(\textit{x}_{1},0,\textit{x}_{3})\big{)}\big{(}\kappa_{\chi\varphi}^{\scalebox{0.65}{0}}+\kappa_{\chi\varphi}^{\Delta}\big{)}\,,\end{aligned}\end{matrix}\right.$

where $\kappa_{\chi\varphi}^{\scalebox{0.65}{0}}$ and
$\kappa_{\chi\varphi}^{\Delta}$ are the deformation-independent curvature and
curvature change tensors, respectively. In the classical theory of initially
flat and infinitely thin membranes with small out-of-plane deflections [40],
as the considered (and interpreted as surface stresses) elastically stretched
membranes in Refs. [108, 109], the curvature change tensor may be approximated
by

(3.104)
$\displaystyle\kappa_{\chi\varphi}^{\Delta}=-\tilde{\textit{u}}^{\mbox{\scriptsize
dis}}_{2,\chi\varphi}(\textit{x}_{1},0,\textit{x}_{3})\,,$

without internal moments. Under the treatment of such specific boundary
conditions normal to the initially flat (but, stretched) membranes, the
distortion response caused by the presence of interface dislocations may
elastically warp the semicoherent interfaces with radii defined by
$r_{\chi\varphi}=1/\kappa_{\chi\varphi}^{\Delta}$. Thus, the right-hand side
of the second equation in eqs. (3.103) is deduced by subsequently imposing no
initial curvature and neglecting the second-order effects compared with unity,
as follows

(3.105)
$\displaystyle\big{(}\tau_{\chi\varphi}^{\scalebox{0.65}{0}}+d_{\chi\varphi\gamma\eta}\;\tilde{\textit{u}}^{\mbox{\scriptsize
dis}}_{\gamma,\eta}(\textit{x}_{1},0,\textit{x}_{3})\big{)}\big{(}\kappa_{\chi\varphi}^{\scalebox{0.65}{0}}+\kappa_{\chi\varphi}^{\Delta}\big{)}\simeq-\tau_{\chi\varphi}^{\scalebox{0.65}{0}}\;\tilde{\textit{u}}^{\mbox{\scriptsize
dis}}_{2,\chi\varphi}(\textit{x}_{1},0,\textit{x}_{3})\,,$

thus, imposing $\kappa_{\chi\varphi}^{\scalebox{0.65}{0}}=0$ and
$\tilde{\textit{u}}^{\mbox{\scriptsize
dis}}_{\gamma,\eta}(\textit{x}_{1},0,\textit{x}_{3})\,\tilde{\textit{u}}^{\mbox{\scriptsize
dis}}_{2,\chi\varphi}(\textit{x}_{1},0,\textit{x}_{3})\ll 1$. According to eq.
(3.82), both discontinuous stress boundary conditions in eqs. (3.103) can also
be recast in matrix form, i.e.

(3.106) $\displaystyle\llbracket\tilde{\textit{t}}_{k}^{\mbox{\,\scriptsize
int}}(\textit{x}_{1},\textit{x}_{3})\rrbracket_{{}_{\mbox{\scriptsize
int}}}-4\pi^{2}\,\mbox{V}_{ki}\;\tilde{\textit{u}}_{i}^{\mbox{\,\scriptsize
int}}(\textit{x}_{1},0,\textit{x}_{3})=0\,,$

where the $3\times 3$ real matrix V is expressed as

(3.107) $\displaystyle\mbox{V}_{ki}=\mbox{V}_{ik}$
$\displaystyle=\footnotesize\left[\\!\\!\begin{array}[]{c c
c}\textit{k}_{1}^{2}d_{11}+2\textit{k}_{1}\textit{k}_{3}d_{15}+\textit{k}_{3}^{2}d_{55}&0&\textit{k}_{1}^{2}d_{15}+\textit{k}_{1}\textit{k}_{3}(d_{13}+d_{55})+\textit{k}_{3}^{2}d_{35}\\\
0&\textit{k}_{1}^{2}\tau^{0}_{11}+2\textit{k}_{1}\textit{k}_{3}\tau^{0}_{13}+\textit{k}_{3}^{2}\tau^{0}_{33}&0\\\
\textit{k}_{1}^{2}d_{15}+\textit{k}_{1}\textit{k}_{3}(d_{13}+d_{55})+\textit{k}_{3}^{2}d_{35}&0&\textit{k}_{1}^{2}d_{55}+2\textit{k}_{1}\textit{k}_{3}d_{35}+\textit{k}_{3}^{2}d_{33}\end{array}\\!\\!\right]\,,$

within which the surface/interface elastic constants are indexed using
standard contracted notations. Mechanically balanced by the interface stress
effects, eq. (3.106) shows that the infinitesimal in-plane strain fields in
the membranes may influence the stresses in both bulk materials due to the
elasticity contributions at the interphase boundaries. In contrast to the
classical continuum elasticity, the tractions across the interface and the
displacement fields are related to each other by the interface elasticity
properties as well the interface geometries through the wavevector components.

Because the materials A and B are mapped separately from the reference state,
the coherent regions at the interfaces (separated by the networks of
interfacial dislocations) can also be viewed as infinitely thin membranes
separately in contact with each individual bulk material. Furthermore, the
determination of the reference states yielding (in general) to unequal
partitioning of elastic distortions, the tractions that act on each individual
upper and lower materials bonded by these coherent interfacial regions are
consequently assumed to be different in both magnitude and direction. Using
the concept of interface zone by in Ref. [150], the specific traction vector
${{}_{\mbox{\scriptsize coh}}\textbf{{t}}}^{\mbox{\,\scriptsize
int}}(\textit{x}_{1},\textit{x}_{3})$, acting on both neighboring crystals
with fictitious infinitely thin inter-layered coherent patches at
$\textit{x}_{2}=0$, is introduced to transfer traction forces from the upper
material to the adjacent lower material. The equilibrium condition between the
interface coherent regions and material A also reads

(3.108) $\displaystyle{{}_{\mbox{\scriptsize
A}}\tilde{\textit{t}}_{k}^{\mbox{\,\scriptsize
int}}}(\textit{x}_{1},\textit{x}_{3})-{{}_{\mbox{\scriptsize
coh}}\textit{t}_{k}^{\mbox{\,\scriptsize
int}}}(\textit{x}_{1},\textit{x}_{3})-4\pi^{2}\,{{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{ki}^{\mbox{\scriptsize int}}}\;{{}_{\mbox{\scriptsize
A}}\tilde{\textit{u}}_{i}^{\mbox{\scriptsize
dis}}(\textit{x}_{1},0,\textit{x}_{3})}=0\,,$

by use of the boundary condition in eq. (3.106), while the equilibrium
condition between the interface coherent regions and material B is given by

(3.109) $\displaystyle{{}_{\mbox{\scriptsize
coh}}\textit{t}_{k}^{\mbox{\,\scriptsize
int}}}(\textit{x}_{1},\textit{x}_{3})-_{\mbox{\scriptsize
B}}\tilde{\textit{t}}_{k}^{\mbox{\,\scriptsize
int}}(\textit{x}_{1},\textit{x}_{3})-4\pi^{2}\,{{}_{\mbox{\scriptsize
B}\\!\\!}\mbox{V}_{ki}^{\mbox{\scriptsize int}}}\;{{}_{\mbox{\scriptsize
B}}\tilde{\textit{u}}_{i}^{\mbox{\scriptsize
dis}}(\textit{x}_{1},0,\textit{x}_{3})}=0\,,$

where ${{}_{\mbox{\scriptsize A}\\!\\!}\textbf{V}}^{\mbox{\scriptsize int}}$
and ${{}_{\mbox{\scriptsize B}\\!\\!}\textbf{V}}^{\mbox{\scriptsize int}}$
depend on the elastic properties of the interfaces with respect to each
material A and B, respectively. Summing both eqs. (3.108) and (3.109), it also
follows that

(3.110) $\displaystyle{{}_{\mbox{\scriptsize
A}}\tilde{\textit{t}}_{k}^{\mbox{\,\scriptsize
int}}(\textit{x}_{1},\textit{x}_{3})-_{\mbox{\scriptsize
B}}\tilde{\textit{t}}_{k}^{\mbox{\,\scriptsize
int}}(\textit{x}_{1},\textit{x}_{3})}-4\pi^{2}\left({{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{ki}^{\mbox{\scriptsize int}}}\;{{}_{\mbox{\scriptsize
A}}\tilde{\textit{u}}_{i}^{\mbox{\,\scriptsize
int}}(\textit{x}_{1},0,\textit{x}_{3})}+{{}_{\mbox{\scriptsize
B}\\!\\!}\mbox{V}_{ki}^{\mbox{\scriptsize int}}}\;{{}_{\mbox{\scriptsize
B}}\tilde{\textit{u}}_{i}^{\mbox{\,\scriptsize
int}}(\textit{x}_{1},0,\textit{x}_{3})}\right)=0\,,$

which yields to the non-classical stress discontinuity conditions at the
mismatched interfaces. Using eqs. (3.79) and (3.82), eq. (3.110) gives rise to
the additional linear system $\Sigma_{2}$ of six equations, i.e.

(3.111) $\displaystyle(\Sigma_{2}):\left\\{\begin{matrix}\begin{aligned}
0=\mathrm{Re}\Big{[}\,\sum_{\alpha\,=\,1}^{3}&{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{h}^{\alpha}_{1}}}+{{{}_{\mbox{\scriptsize
A}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{h}^{\alpha}_{1_{\,*}}}}+i2\pi\big{(}{{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{11}^{\mbox{\scriptsize int}}}\big{(}{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{1}}}+{{{}_{\mbox{\scriptsize
A}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{1_{\,*}}}}\big{)}+{{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{13}^{\mbox{\scriptsize int}}}\big{(}{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{3}}}+{{{}_{\mbox{\scriptsize
A}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{3_{\,*}}}}\big{)}\big{)}\\\\[-5.0pt]
&-{{{}_{\mbox{\scriptsize B}}\zeta^{\alpha}}}\big{(}{{{}_{\mbox{\scriptsize
B}}\textit{h}^{\alpha}_{1_{\,*}}}}-i2\pi\big{(}{{}_{\mbox{\scriptsize
B}\\!\\!}\mbox{V}_{11}^{\mbox{\scriptsize int}}}\;{{{}_{\mbox{\scriptsize
B}}\textit{a}^{\alpha}_{1_{\,*}}}}+{{}_{\mbox{\scriptsize
B}\\!\\!}\mbox{V}_{13}^{\mbox{\scriptsize int}}}\;{{{}_{\mbox{\scriptsize
B}}\textit{a}^{\alpha}_{3_{\,*}}}}\big{)}\big{)}\Big{]}=\mathrm{Re}\sum_{\alpha\,=\,1}^{3}v^{\alpha}_{1}\\\
0=\mathrm{Re}\Big{[}\,\sum_{\alpha\,=\,1}^{3}&{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{h}^{\alpha}_{2}}}+{{{}_{\mbox{\scriptsize
A}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{h}^{\alpha}_{2_{\,*}}}}+i2\pi\big{(}{{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{22}^{\mbox{\scriptsize int}}}\big{(}{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{2}}}+{{{}_{\mbox{\scriptsize
A}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
B}}\textit{a}^{\alpha}_{2_{\,*}}}}\big{)}\big{)}-{{{}_{\mbox{\scriptsize
B}}\zeta^{\alpha}}}\big{(}{{{}_{\mbox{\scriptsize
B}}\textit{h}^{\alpha}_{2_{\,*}}}}-i2\pi\;{{}_{\mbox{\scriptsize
B}\\!\\!}\mbox{V}_{22}^{\mbox{\scriptsize int}}}{{{}_{\mbox{\scriptsize
B}}\textit{a}^{\alpha}_{2_{\,*}}}}\big{)}\Big{]}\\\\[-5.0pt]
&=\mathrm{Re}\sum_{\alpha\,=\,1}^{3}v^{\alpha}_{2}\\\
0=\mathrm{Re}\Big{[}\,\sum_{\alpha\,=\,1}^{3}&{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{h}^{\alpha}_{3}}}+{{{}_{\mbox{\scriptsize
A}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{h}^{\alpha}_{3_{\,*}}}}+i2\pi\big{(}{{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{13}^{\mbox{\scriptsize int}}}\big{(}{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{1}}}+{{{}_{\mbox{\scriptsize
A}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{1_{\,*}}}}\big{)}+{{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{33}^{\mbox{\scriptsize int}}}\big{(}{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{3}}}+{{{}_{\mbox{\scriptsize
A}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{3_{\,*}}}}\big{)}\big{)}\\\\[-5.0pt]
&-{{{}_{\mbox{\scriptsize B}}\zeta^{\alpha}}}\big{(}{{{}_{\mbox{\scriptsize
B}}\textit{h}^{\alpha}_{3_{\,*}}}}-i2\pi\big{(}{{}_{\mbox{\scriptsize
B}\\!\\!}\mbox{V}_{13}^{\mbox{\scriptsize int}}}\;{{{}_{\mbox{\scriptsize
B}}\textit{a}^{\alpha}_{1_{\,*}}}}+{{}_{\mbox{\scriptsize
B}\\!\\!}\mbox{V}_{33}^{\mbox{\scriptsize int}}}\;{{{}_{\mbox{\scriptsize
B}}\textit{a}^{\alpha}_{3_{\,*}}}}\big{)}\big{)}\Big{]}=\mathrm{Re}\sum_{\alpha\,=\,1}^{3}v^{\alpha}_{3}\\\
0=\mathrm{Im}\Big{[}\,\sum_{\alpha\,=\,1}^{3}&v^{\alpha}_{k}\Big{]}~{},~{}~{}\forall
k\in\\{1,2,3\\}\,,\end{aligned}\end{matrix}\right.$

with $\textit{h}^{\alpha}_{k}=\mbox{H}^{\alpha}_{k2}$, for any given
$(\eta_{1},\eta_{2})\in\left]0,\,1/2\right[\,\\!^{2}$ and for all
$\\{n,m\\}\in\mathrm{\textit{D}}$.

##### Stress conditions at the free surfaces

Similarly to the semicoherent interface treatment, the free surfaces
experience excess energy and excess energy due to different energy profiles
close to such singular membrane-like boundaries. Thus, additional non-
classical boundary conditions as eq. (3.110) are introduced on the outer free
surface, at $\textit{x}^{\mbox{\scriptsize or}}_{2}=h_{\mbox{\scriptsize A}}$,
i.e.

(3.112) $\displaystyle{{}_{\mbox{\scriptsize
A}}\tilde{\textit{t}}_{k}^{\mbox{\,\scriptsize
fs}}}(\textit{x}_{1},\textit{x}_{3})+4\pi^{2}\,{{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{ki}^{\mbox{\scriptsize fs}}}\;{{}_{\mbox{\scriptsize
A}}\tilde{\textit{u}}_{i}^{\mbox{\scriptsize
dis}}(\textit{x}_{1},h_{\mbox{\scriptsize A}},\textit{x}_{3})}=0\,,$

where ${{}_{\mbox{\scriptsize A}\\!}\textbf{V}}^{\mbox{\scriptsize fs}}$
depends on the elastic constants of the free surfaces. It also yields to the
following system $\Sigma_{3}$ of six other equations, i.e.

(3.113) $\displaystyle(\Sigma_{3}):\left\\{\begin{matrix}\begin{aligned}
0=\mathrm{Re}\Big{[}\,\sum_{\alpha\,=\,1}^{3}&{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}\mathrm{e}^{i2\pi\textit{p}^{\alpha}h_{\mbox{A}}}\big{(}{{{}_{\mbox{\scriptsize
A}}\textit{h}^{\alpha}_{1}}}-i2\pi\big{(}{{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{11}^{\mbox{\scriptsize fs}}}\;{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{1}}}+{{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{13}^{\mbox{\scriptsize fs}}}\;{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{3}}}\big{)}\big{)}\\\\[-5.0pt]
&+{{{}_{\mbox{\scriptsize
A}}\zeta^{\alpha}}}\mathrm{e}^{i2\pi\textit{p}_{*}^{\alpha}h_{\mbox{A}}}\big{(}{{{}_{\mbox{\scriptsize
A}}\textit{h}^{\alpha}_{1_{\,*}}}}-i2\pi\big{(}{{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{11}^{\mbox{\scriptsize fs}}}\;{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{1_{\,*}}}}+{{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{13}^{\mbox{\scriptsize fs}}}\;{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{3_{\,*}}}}\big{)}\big{)}\Big{]}=\mathrm{Re}\sum_{\alpha\,=\,1}^{3}w^{\alpha}_{1}\\\
0=\mathrm{Re}\Big{[}\,\sum_{\alpha\,=\,1}^{3}&{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}\mathrm{e}^{i2\pi\textit{p}^{\alpha}h_{\mbox{A}}}\big{(}{{{}_{\mbox{\scriptsize
A}}\textit{h}^{\alpha}_{2}}}-i2\pi\,{{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{22}^{\mbox{\scriptsize fs}}}\;{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{2}}}\big{)}+{{{}_{\mbox{\scriptsize
A}}\zeta^{\alpha}}}\mathrm{e}^{i2\pi\textit{p}_{*}^{\alpha}h_{\mbox{A}}}\big{(}{{{}_{\mbox{\scriptsize
A}}\textit{h}^{\alpha}_{2_{\,*}}}}-i2\pi\,{{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{22}^{\mbox{\scriptsize fs}}}\;{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{2_{\,*}}}}\big{)}\Big{]}\\\\[-5.0pt]
&=\mathrm{Re}\sum_{\alpha\,=\,1}^{3}w^{\alpha}_{2}\\\
0=\mathrm{Re}\Big{[}\,\sum_{\alpha\,=\,1}^{3}&{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}\mathrm{e}^{i2\pi\textit{p}^{\alpha}h_{\mbox{A}}}\big{(}{{{}_{\mbox{\scriptsize
A}}\textit{h}^{\alpha}_{3}}}-i2\pi\big{(}{{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{13}^{\mbox{\scriptsize fs}}}\;{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{1}}}+{{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{33}^{\mbox{\scriptsize fs}}}\;{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{3}}}\big{)}\big{)}\\\\[-5.0pt]
&+{{{}_{\mbox{\scriptsize
A}}\zeta^{\alpha}}}\mathrm{e}^{i2\pi\textit{p}_{*}^{\alpha}h_{\mbox{A}}}\big{(}{{{}_{\mbox{\scriptsize
A}}\textit{h}^{\alpha}_{3_{\,*}}}}-i2\pi\big{(}{{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{13}^{\mbox{\scriptsize fs}}}\;{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{1_{\,*}}}}+{{}_{\mbox{\scriptsize
A}\\!\\!}\mbox{V}_{33}^{\mbox{\scriptsize fs}}}\;{{{}_{\mbox{\scriptsize
A}}\textit{a}^{\alpha}_{3_{\,*}}}}\big{)}\big{)}\Big{]}=\mathrm{Re}\sum_{\alpha\,=\,1}^{3}w^{\alpha}_{3}\\\
0=\mathrm{Im}\Big{[}\,\sum_{\alpha\,=\,1}^{3}&w^{\alpha}_{k}\Big{]}~{},~{}~{}\forall
k\in\\{1,2,3\\}\,,\end{aligned}\end{matrix}\right.$

for any given $(\eta_{1},\eta_{2})\in\left]0,\,1/2\right[\,\\!^{2}$ and for
all $\\{n,m\\}\in\mathrm{\textit{D}}$.

##### Determination of the minimum-energy paths

When the linear systems in eqs. (3.100) with (3.111) and (3.113) are combined,
the set $\mathrm{E\mbox{{\footnotesize cst}}}$ of all eighteen real unknowns
(twelve and six for A and B, respectively) are also solved with respect to the
prescribed boundary conditions, i.e.

(3.114) $\mathrm{E\mbox{{\footnotesize
cst}}}=\sum_{\alpha\,=\,1}^{3}\left\\{\,\mathrm{Re}\,{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}},\,\mathrm{Im}\,{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}},\,\mathrm{Re}\,{{}_{\mbox{\scriptsize
A}}\zeta^{\alpha}},\,\mathrm{Im}\,{{}_{\mbox{\scriptsize
A}}\zeta^{\alpha}},\,\mathrm{Re}\,{{}_{\mbox{\scriptsize
B}}\zeta^{\alpha}},\,\mathrm{Im}\,{{}_{\mbox{\scriptsize
B}}\zeta^{\alpha}}\,\right\\}\,,$

completing the solutions of the elastic displacement and stress fields, given
by eqs. (3.79) and (3.80), respectively. Following the procedure described in
section 3.6.2, the upper triangular domain $\mathcal{T}_{\mbox{\ssmall ABC}}$
in the representative unit dislocation cell, denoted by ABC in Fig. (3.25b),
is discretized into four-node quadrilateral elements with respect to the
$i^{\mbox{\footnotesize th}}$ nodal point coordinates
$(\eta_{1}^{i},\eta_{2}^{i})$, such that
$\\{\eta_{1}^{i},\eta_{2}^{i}\\}\in\left]0,\,1/2\right[\,\\!^{2}$ for convex
hexagonal-shaped dislocation patterns. Thus, for any dislocation pattern that
is geometrically characterized by the given coordinates
$(\eta_{1}^{i},\eta_{2}^{i})$, the corresponding elastic strain energy can be
computed as a volume integral over the heterostructure of interest, i.e.

(3.115)
$\gamma_{\mathrm{e}}^{i}(\eta_{1}^{i},\eta_{2}^{i})=\dfrac{1}{2\,\textit{A}}\int\\!\\!\\!\\!\int\\!\\!\\!\\!\int_{V}\;\tilde{\sigma}_{ij}^{\mbox{\scriptsize
dis}}(\textit{x}_{1},\textit{x}_{2},\textit{x}_{3})\;{\tilde{\textit{u}}^{\mbox{\scriptsize
dis}}_{j,i}}(\textit{x}_{1},\textit{x}_{2},\textit{x}_{3})\;\mbox{d}\textit{V}\,,$

where all persistent short-range field solutions of the integrand depend
specifically on $(\eta_{1}^{i},\eta_{2}^{i})$ by the treatment of boundary
conditions, described in section 3.6.4. For far-field stress-free bicrystals
at equilibrium, the standard volume integral eq. (3.115) may be reduced to a
surface integral by the use of integration by parts, together with the
divergence theorem without any body forces [227, 251], as follows

(3.116)
$\gamma_{\mathrm{e}}^{i}(\eta_{1}^{i},\eta_{2}^{i})=\dfrac{1}{2\,\textit{A}}\int\\!\\!\\!\\!\int_{\\!A(r_{\mbox{0}})}\;\tilde{\textit{t}}_{k}^{\mbox{\,\scriptsize
int}}(\textit{x}_{1},\textit{x}_{3})\;\;\llbracket\tilde{\textit{u}}^{\mbox{\scriptsize
dis}}_{k}(\textit{x}_{1},0,\textit{x}_{3})\rrbracket_{{}_{\mbox{\scriptsize
int}}}\;\mbox{d}\textit{S}\,,$

where $A(r_{\scalebox{0.65}{0}})$ is the hexagonal-shaped unit cell. In eqs.
(3.115) and (3.116), the expressions of elastic strain energy are conveniently
expressed per unit area, for which
$\textit{A}=\textit{A}(r_{\scalebox{0.65}{0}}=0)$, and account for several
different contributions, i.e. interaction between Volterra-type dislocations
against the misfit strain state, self-energy induced by individual hexagonal-
shaped dislocation configurations, as well as the interaction between the
hexagonal-shaped unit cell with all infinitely repeated cells. Finally, the
complete elastic strain energy landscape
$\gamma_{\mathrm{e}}(\eta_{1},\eta_{2})$ is interpolated for any
$(\eta_{1},\eta_{2})\in\left]0,\,1/2\right[\,\\!^{2}$, as follows

(3.117)
$\gamma_{\mathrm{e}}(\eta_{1},\eta_{2})=\sum_{i=1}^{4}N_{i}(\eta_{1},\eta_{2})\,\gamma_{\mathrm{e}}^{i}(\eta_{1}^{i},\eta_{2}^{i})\,,$

where $N_{i}(\eta_{1},\eta_{2})$ are the standard finite element bilinear
shape functions for four-node elements.

For elastic strain landscapes that favor the formation of dislocation
junctions, the minimum-energy configurations are determined by computing the
conjugate gradient algorithm, while the nudged elastic band method is used to
find the corresponding minimum-energy paths. The nudged elastic band method is
a chain-of-states method in which a string of images is used to describe the
reaction pathways. These configurations are connected by spring forces to
ensure equal spacing along the paths of interest. The ensemble of the
configurations is then relax through a force projection scheme to converge to
the most energetically favorable pathways [118, 226]. To identify the minimum-
energy paths between the initial (non-equilibrium) lozenge-shaped pattern and
the final elastically relaxed configurations (previously computed by the
conjugate gradient algorithm), all images are simultaneously evolved to
equilibrium under a nudged elastic band force (on image indexed by $s_{\eta}$)
that contains two independent components on all images $s_{\eta}$, i.e.

(3.118)
$\textbf{{f}}^{\mathrm{NEB}}_{\\!s_{\eta}}=\textbf{{f}}^{\perp}_{\\!s_{\eta}}+\textbf{{f}}^{\parallel}_{\\!s_{\eta}}\,,$

where $\textbf{{f}}^{\perp}_{\\!s_{\eta}}$ is the component of the elastic
force acting normal to the tangent of the elastic landscape, as follows

(3.119)
$\textbf{{f}}^{\perp}_{\\!s_{\eta}}=-\boldsymbol{\nabla}\gamma_{\mathrm{e}}(\eta_{1},\eta_{2})+\left(\boldsymbol{\nabla}\gamma_{\mathrm{e}}(\eta_{1},\eta_{2})\cdot\hat{\boldsymbol{\tau}}_{\\!s_{\eta}}\right)\hat{\boldsymbol{\tau}}_{\\!s_{\eta}}\,,$

with $\hat{\boldsymbol{\tau}}_{\\!s_{\eta}}$ the unit tangent to the elastic
energy landscape. In addition, the spring force
$\textbf{{f}}^{\parallel}_{\\!s_{\eta}}$ in eq. (3.118), acting parallel to
the energy landscape [118, 226] is defined by

(3.120)
$\textbf{{f}}^{\parallel}_{\\!s_{\eta}}=k\big{(}|\boldsymbol{\eta}_{\\!s_{\eta}+1}-\boldsymbol{\eta}_{\\!s_{\eta}}|-|\boldsymbol{\eta}_{\\!s_{\eta}}-\boldsymbol{\eta}_{\\!s_{\eta}-1}|\big{)}\,,$

where
$\boldsymbol{\eta}_{\\!s_{\eta}}=\boldsymbol{\eta}_{\\!s_{\eta}}(\eta_{1},\eta_{2})$
is the position of the ${s_{\eta}}^{\mbox{\footnotesize th}}$ image and $k$
the spring constant. The spring interaction between adjacent images is added
to ensure continuity of the chain.

The present numerical procedure is identical to nudged elastic band
calculations recently performed to analyze the calculation of attempt
frequency for a dislocation bypassing an obstacle [235] using a nodal
dislocation dynamics simulation with non-singular treatments for isotropic
elastic fields [50].

####  3.6.5 Application to Au/Cu heterosystems

The section gives applications to two examples of the general parametric
energy-based framework. The first simple and limiting case is concerned with
two dislocation sets in pure misfit $(010)$ Au/Cu interfaces, for which the
strain energy landscape for formation of dislocation junctions is unfavorable.
The subsequent investigation of the effects of surface/interface stress and
elasticity properties with different boundary conditions in $(010)$ Au/Cu
interfaces can be found in Ref. [260]. On the other hand, the second case
deals with the minimum-energy reaction pathway of the pre-computed $(111)$
Au/Cu elastic energy landscape, where the initial and unrelaxed dislocation
pattern solution is described by the Frank-Bilby equation. The materials
properties used in these examples are listed in Table 3.7.

Symbols |  | Au (material A) |  | Cu (Material B) |  | Units | References
---|---|---|---|---|---|---|---
Lattice parameters |  |
$a$ |  | $0.4078$ |  |  | $0.3615$ |  |  | nm |  [105]
Elastic components (Voigt notation) |  |
$c_{11}$ |  | $187.0$ |  |  | $168.4$ |  |  | GPa |  [122]
$c_{12}$ |  | $157.0$ |  |  | $121.4$ |  |  | GPa |  [122]
$c_{44}$ |  | $43.6$ |  |  | $75.4$ |  |  | GPa |  [122]
Elasticity properties for the semicoherent interfaces (Voigt notation) |
$\ast$ Interface stress |
$\tau_{11}$ |  | $-0.0465$ |  |  | $0.645$ |  |  | N/m |  [150]
$\tau_{13}$ |  | $0$ |  |  | $0$ |  |  | N/m |  [150]
$\tau_{33}$ |  | $-0.0465$ |  |  | $0.645$ |  |  | N/m |  [150]
$\ast$ Interface modulus |
$d_{11}$ |  | $-6.84$ |  |  | $-5.99$ |  |  | N/m |  [150]
$d_{13}$ |  | $-3.47$ |  |  | $0.6540$ |  |  | N/m |  [150]
$d_{33}$ |  | $-6.84$ |  |  | $-5.99$ |  |  | N/m |  [150]
$d_{15}$ |  | $0.0042$ |  |  | $0.0032$ |  |  | N/m |  [150]
$d_{35}$ |  | $0.0042$ |  |  | $0.0032$ |  |  | N/m |  [150]
$d_{55}$ |  | $-1.91$ |  |  | $-3.67$ |  |  | N/m |  [150]
Elasticity properties for the free surface (Voigt notation) |
$\ast$ Surface stress |
$\tau_{11}$ |  | $1.49$ |  |  | $-$ |  |  | N/m |  [189]
$\tau_{13}$ |  | $0$ |  |  | $-$ |  |  | N/m |  [189]
$\tau_{33}$ |  | $1.49$ |  |  | $-$ |  |  | N/m |  [189]
$\ast$ Surface modulus |
$d_{11}$ |  | $-7.10$ |  |  | $-$ |  |  | N/m |  [189]
$d_{13}$ |  | $-5.67$ |  |  | $-$ |  |  | N/m |  [189]
$d_{33}$ |  | $-3.17$ |  |  | $-$ |  |  | N/m |  [189]

Table 3.7: Lattice parameters $a$ of Au and Cu crystals, material properties
$c_{ij}$ of both bulk materials, surface stress $\tau_{\chi\varphi}$ and
surface modulus $d_{\chi\varphi}$ of the semicoherent Au/Cu heterophase
interface and the $(010)$ free surface in Au.  Figure 3.26: Dependence of the
total far-field stress component $\sigma^{\mbox{\,\scriptsize
c}}_{11}+\sigma^{\pm\infty}_{11}$ on $\delta$ in the Au and Cu materials for
the $(010)$ and $(111)$ Au/Cu semicoherent interfaces. Figure 3.27: Elastic
strain energy landscapes $\gamma_{\mathrm{e}}$ in J.m-2 of the hexagonal-
shaped patterns with three-fold dislocation nodes as a function of $\eta_{1}$
and $\eta_{2}$, for the (a) $(010)$ and (b) $(111)$ Au/Cu heterophase
interface cases. The large points at $\eta_{1}=\eta_{2}=1/2$ correspond to the
initial lozenge-shaped patterns, for which the two crossing dislocation sets
are related to equilibrium and non-equilibrium dislocation configurations for
the $(010)$ and $(111)$ interface planes, respectively. The latter case gives
rise to the presence of a minimum-energy path (in black) between the initial
pattern and the fully elastically strain-relaxed dislocation structure
(magenta point) at stable equilibrium state. An intermediate state is
displayed by the orange point.

##### Case 1: The (010) Au/Cu interface with two sets of dislocations

As a limiting case, the atomically sharp $(010)$ Au/Cu misfit interface
contains two sets of orthogonal dislocations in cube-cube orientation
relationship, i.e. $\textbf{{x}}^{\mbox{\scriptsize or}}_{1}=[10\bar{1}]$,
$\textbf{{x}}^{\mbox{\scriptsize or}}_{2}=\textbf{{n}}=[010]$, and
$\textbf{{x}}^{\mbox{\scriptsize or}}_{3}=[101]$. Similar to eq. (3.37), the
net Burgers vectors are expressed by using the quantized Frank-Bilby equation
[94, 29, 30], as follows

(3.121) $\left(\dfrac{\textbf{{n}}\times\boldsymbol{\xi}^{\mbox{\scriptsize
un}}_{1}}{\textit{d}^{\,\mbox{\scriptsize
un}}_{1}}\cdot\textbf{{p}}\right)\textbf{{b}}_{1}+\left(\dfrac{\textbf{{n}}\times\boldsymbol{\xi}^{\mbox{\scriptsize
un}}_{2}}{\textit{d}^{\,\mbox{\scriptsize
un}}_{2}}\cdot\textbf{{p}}\right)\textbf{{b}}_{2}=n_{1}\,\textbf{{b}}_{1}+n_{2}\,\textbf{{b}}_{2}=\big{(}\textbf{F}^{-1}_{\mbox{\scriptsize
Au}}-\textbf{F}^{-1}_{\mbox{\scriptsize Cu}}\big{)}\,\textbf{{p}}\,,$

where $\textit{d}^{\,\mbox{\scriptsize un}}_{1}$ and
$\textit{d}^{\,\mbox{\scriptsize un}}_{2}$ are the regularly spaced inter-
dislocation spacings, and the interface Burgers vectors
$\textbf{{b}}_{1}\parallel[10\bar{1}]$ and $\textbf{{b}}_{2}\parallel[101]$
are both parallel to $\textbf{{x}}_{1}$\- and $\textbf{{x}}_{3}$-axis,
respectively. As a result of arbitrarily selecting the reference state
identical to the Au (or Cu) natural state, for which the geometry of interface
dislocations (line directions and spacings) is independent of the choice of
reference state, the line directions are defined by
$\boldsymbol{\xi}^{\mbox{\scriptsize un}}_{1}\parallel[101]$ and
$\boldsymbol{\xi}^{\mbox{\scriptsize un}}_{2}\parallel[10\bar{1}]$, and the
inter-dislocation spacings are given by $\textit{d}^{\,\mbox{\scriptsize
un}}_{1}=\textit{d}^{\,\mbox{\scriptsize
un}}_{2}=\textit{p}_{1}^{\mbox{\footnotesize
o}}=\textit{p}_{2}^{\mbox{\footnotesize o}}=2.25144~{}$nm. Thus, the Frank-
Bilby equation predicts that an orthogonal network of straight parallel
dislocations with pure edge characters is also needed to accommodate the pure
misfit $(010)$ Au/Cu interface.

The geometry of such orthogonal grid of dislocations can also be characterized
by $\eta_{1}\to 1/2$ and $\eta_{2}\to 1/2$ in the general parametric
framework, because $\phi^{\mbox{\scriptsize un}}=\pi/2$. According to the
bilinear function $\textbf{{u}}^{\,\textit{p}}(\eta_{1}\to 1/2,\eta_{2}\to
1/2)$ for the prescribed displacement field in eq. (3.87), the corresponding
real-valued expansion functions in eq. (3.88) for the individual set 1 can be
computed by imposing $m=0$, as follows

(3.122) $\displaystyle\left\\{\begin{matrix}\begin{aligned}
\lim_{\eta_{1}\,\to\,1/2}\;{{}_{\mbox{\scriptsize
c2}}\hat{\textbf{{u}}}}^{\,\textit{p}}_{1}(\eta_{1},\eta_{2})&=\lim_{\eta_{2}\,\to\,1/2}\;{{}_{\mbox{\scriptsize
c2}}\hat{\textbf{{u}}}}^{\,\textit{p}}_{1}(\eta_{1},\eta_{2})=\frac{(-1)^{n+1}}{2\pi
n}~{}\textbf{{b}}_{1}\\\
\lim_{\eta_{2}\,\to\,1/2}\;\lim_{\eta_{1}\,\to\,1/2}\;{{}_{\mbox{\scriptsize
c2}}\hat{\textbf{{u}}}}^{\,\textit{p}}_{2}(\eta_{1},\eta_{2})&=\lim_{\eta_{2}\,\to\,1/2}\frac{(-1)^{n+1}\,(-1+2\eta_{2})}{2\pi
n}~{}\textbf{{b}}_{2}=\mathbb{0}\,,\end{aligned}\end{matrix}\right.$

exhibiting that $\hat{\textbf{{u}}}^{\,\textit{p}}(\eta_{1}\to 1/2,\eta_{2}\to
1/2)=\hat{\textbf{{u}}}_{1}^{\,\textit{p}}(\eta_{1}\to 1/2,\eta_{2}\to 1/2)$
is evidently written as a function of $\textbf{{b}}_{1}$, for set 1. By
superposing the similar contribution of set 2 with $n=0$, the final prescribed
displacement field produced by an orthogonal network of dislocations in eq.
(3.88) is therefore written in the form of two distinct one-dimensional
sawtooth-shaped functions with Fourier sine series, as follows

(3.123)
$\displaystyle\textbf{{u}}^{\,\textit{p}}(\textit{x}_{1},\textit{x}_{3})=-\sum_{\begin{subarray}{c}n\,\geq\,1\\\
m\,=\,0\end{subarray}}\frac{(-1)^{n}}{\pi
n}\mathrm{sin}\left(2\pi\textit{k}_{1}(n,0)\;\textit{x}_{1}\right)~{}\textbf{{b}}_{1}-\sum_{\begin{subarray}{c}n\,=\,0\\\
m\,\geq\,1\end{subarray}}\frac{(-1)^{m}}{\pi
m}\mathrm{sin}\left(2\pi\textit{k}_{3}(m)\;\textit{x}_{3}\right)~{}\textbf{{b}}_{2}\,,$

where $\textit{k}_{1}(n,0)$ and $\textit{k}_{3}(m)$ are defined in eq. (3.74),
with $\phi^{\mbox{\scriptsize un}}=\pi/2$. Here, the sawtooth-shaped functions
in eq. (3.123) differ from eq. (3.19) by individual translations of magnitude
$\textit{d}^{\,\mbox{\scriptsize un}}_{i}/2$. Similarly to $\Sigma_{1}$ and
$\Sigma_{2}$ in eqs. (3.20) and (3.25), the simplest limiting case of
bicrystals without any surface/interface elasticity effects leads to a set of
twelve real and linear equations, i.e.

(3.124) $\left\\{\begin{matrix}\begin{aligned}
\mathrm{Re}\sum_{\alpha\,=\,1}^{3}{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textbf{{a}}^{\alpha}}}-{{{}_{\mbox{\scriptsize
B}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
B}}\textbf{{a}}^{\alpha}_{*}}}&=\mathbb{0}\\\
\mathrm{Im}\sum_{\alpha\,=\,1}^{3}{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textbf{{a}}^{\alpha}}}-{{{}_{\mbox{\scriptsize
B}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
B}}\textbf{{a}}^{\alpha}_{*}}}&=\boldsymbol{\vartheta}\,,~{}~{}~{}\mbox{with:}~{}~{}\boldsymbol{\vartheta}=\left\\{\begin{matrix}\begin{aligned}
-&\frac{(-1)^{n}}{\pi
n}~{}\textbf{{b}}_{1}&&\mbox{~{}~{}~{}if~{}~{}}m=0~{}~{}&n\in\mathbb{N}^{*}&\\\\[3.99994pt]
-&\frac{(-1)^{m}}{\pi
m}~{}\textbf{{b}}_{2}&&\mbox{~{}~{}~{}if~{}~{}}n=0~{}~{}&m\in\mathbb{N}^{*}&\\\\[3.99994pt]
&~{}\mathbb{0}&&\mbox{~{}~{}~{}if~{}~{}}nm\neq
0~{}~{}~{}&n\in\mathbb{N}^{*},m\in\mathbb{N}^{*}&\end{aligned}\end{matrix}\right.\\\\[-10.00002pt]
\mathrm{Re}\sum_{\alpha\,=\,1}^{3}{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textbf{{h}}^{\alpha}}}-{{{}_{\mbox{\scriptsize
B}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
B}}\textbf{{h}}^{\alpha}_{*}}}&=\mathbb{0}\\\
\mathrm{Im}\sum_{\alpha\,=\,1}^{3}{{{}_{\mbox{\scriptsize
A}}\lambda^{\alpha}}}{{{}_{\mbox{\scriptsize
A}}\textbf{{h}}^{\alpha}}}-{{{}_{\mbox{\scriptsize
B}}\zeta^{\alpha}}}{{{}_{\mbox{\scriptsize
B}}\textbf{{h}}^{\alpha}_{*}}}&=\mathbb{0}\,,\end{aligned}\end{matrix}\right.\,$

with respect to the six associated complex unknown quantities, i.e.
${{}_{\mbox{\scriptsize A}}\lambda^{\alpha}}$ and ${{}_{\mbox{\scriptsize
B}}\zeta^{\alpha}}$.

Following the procedure described in section 3.3.3, the two deformation
gradients $\textbf{F}^{-1}_{\mbox{\scriptsize Au}}$ and
$\textbf{F}^{-1}_{\mbox{\scriptsize Cu}}$ in eq. (3.121) (also, the magnitudes
of both $\textbf{{b}}_{1}$ and $\textbf{{b}}_{2}$) are determined by ensuring
the condition of vanishing far-field stresses along a transformation pathway
between both materials Au and Cu. For cube-cube orientation relation, this
condition is met by continuously adjusting the reference lattice parameter
$a_{\mbox{\scriptsize ref}}$ along a specified reaction pathway coordinate
$\delta$, starting with the pure lattice parameter of Au to Cu, i.e.

(3.125) $a_{\mbox{\scriptsize
ref}}=\left(1-\delta\right)\,a_{\mbox{\scriptsize
Au}}+\delta\,a_{\mbox{\scriptsize Cu}}\,,$

where $0\leq\delta\leq 1$ is a dimensionless variable that interpolates
linearly between $a_{\mbox{\scriptsize Au}}$ and $a_{\mbox{\scriptsize Cu}}$.

According to the far-field eq. (3.1), the dependence of the total large-range
stress components ${\sigma}^{\,\mathrm{c}}_{11}+{\sigma}^{\pm\infty}_{11}$ in
Au (black line with symbols) and Cu (red line with symbols) on the
transformation pathway coordinate $\delta$ is plotted in Fig. (3.26). For the
$(010)$ misfit case, both far-field stress components vanish for
$\delta_{(010)}=0.60392$, so that the corresponding reference state is closer
to Cu than to Au, i.e. $\delta_{(010)}>0.5$, where ${{}_{\mbox{\scriptsize
Cu}}c}_{11}<{{}_{\mbox{\scriptsize Au}}c}_{11}$ and ${{}_{\mbox{\scriptsize
Cu}}c}_{12}<{{}_{\mbox{\scriptsize Au}}c}_{12}$, but ${{}_{\mbox{\scriptsize
Cu}}c}_{44}>{{}_{\mbox{\scriptsize Au}}c}_{44}$. All other elastic components
are consistent with the absence of strains in the long range and no rotations
are induced along the transformation path. Thus, it gives rise to the
reference lattice parameter $a_{\mbox{\scriptsize ref}}=0.37984$ nm, and also
the magnitudes of correct Burgers vectors, i.e.
$\textit{b}_{1}=\textit{b}_{2}=0.26859$ nm, selected by the coherent reference
state. When an incorrect reference state is arbitrary chosen, the
corresponding Burgers vectors deviate in magnitude and non-zero spurious
stress fields exist in the microstructure. For instance, a residual stress
state in Au persists with ${{}_{\mbox{\scriptsize
Au}}\sigma}^{\,\mathrm{c}}_{11}+{{}_{\mbox{\scriptsize
Au}}\sigma}^{+\infty}_{11}\simeq 6.29~{}$GPa and $\simeq-3.66~{}$GPa, for
$\delta_{(010)}=0$ and $1$, respectively. A larger residual stress field
exists in Cu as well, where ${{}_{\mbox{\scriptsize
Cu}}\sigma}^{\,\mathrm{c}}_{11}+{{}_{\mbox{\scriptsize
Cu}}\sigma}^{-\infty}_{11}\simeq-8.77~{}$GPa, for $\delta_{(010)}=0$, and
$\simeq 5.09~{}$GPa, for $\delta_{(010)}=1$.

For the following calculations in interfacial hexagonal-shaped dislocation
patterns, the upper half-plane domain $\mathrm{\textit{D}}=\\{\\{0\leq n\leq
n_{\mbox{\ssmall max}}\\}\cup\\{\lvert m\rvert\leq n_{\mbox{\ssmall
max}}\\}\\}\setminus\\{m\leq 0,\,n=0\\}\\}$ is defined by setting
$n_{\mbox{\ssmall max}}=50$, which is large enough to ensure accurate
solutions in truncated elastic stress fields with three sets of dislocations.

Figure (3.27a) shows the elastic strain energy landscape for the $(010)$ Au/Cu
misfit interface with classical boundary conditions between both neighboring
semi-infinite Au and Cu crystals, for simplicity. To determine such energy
landscape, the triangular domain $\mathcal{T}_{\mbox{\ssmall ABC}}$ is first
discretized into $121$ nodal points with coordinates
$(\eta_{1}^{i},\eta_{2}^{i})$, such that
$\\{\eta_{1}^{i},\eta_{2}^{i}\\}\in\left]0,\,1/2\right[\,\\!^{2}$, as depicted
by the gray dots in Fig. (3.27a). Using the persistent short-range elastic
fields, the finite (guaranteed by the zero far-field stresses) stored elastic
energy per unit area is computed for any $(\eta_{1}^{i},\eta_{2}^{i})$ using
eq. (3.116) with $r_{\scalebox{0.65}{0}}=\textit{b}_{1}/4$. Following the
standard interpolation procedure of eq. (3.117), the elastic strain energy for
any given $(\eta_{1},\eta_{2})\in\left]0,\,1/2\right[\,\\!^{2}$ shows a smooth
and symmetric landscape with respect to the median $(\eta_{1}=\eta_{2})$ of
the triangular domain, within which the unique strain energy minima is
obtained at $\eta_{1}\to 1/2$ and $\eta_{2}\to 1/2$, with
$\gamma_{\mathrm{e}}^{\mbox{\scriptsize min}}=\gamma_{\mathrm{e}}(\eta_{1}\to
1/2,\eta_{2}\to 1/2)\simeq 0.57344$ J.m-2. Planar dislocation reactions and
junctions for $(010)$ misfit interfaces are also shown to be energetically
unfavorable. It is therefore demonstrated that the initial orthogonal grid of
uniformly spaced edge dislocations corresponds to the equilibrium structures
for the $(010)$-type misfit interfaces, which satisfies the condition of
vanishing far-field stresses as well as the minimum-energy criterion for
predicting the most favorable dislocation structures.

Near the unreacted state of the $(010)$ Au/Cu system, the present energy
landscape shows concave slope profiles at $\eta_{1}\simeq\eta_{2}\simeq 1/2$.
For calculations with other fcc/fcc heterosystems in the $(010)$ cube-cube
orientation relationship (not shown here), the corresponding unreacted state
can exhibit convex energy profiles, which suggest different bound crossed
states of dislocation reactions for the $(010)$ twist GBs. Thus, the parent
dislocations could also exhibit strong repulsive interactions or crossed
states where local bend and twist of dislocations may locally occur at the
short-range distances, as observed in non-coplanar dislocations [180].

##### Case 2: The (111) Au/Cu interface with three sets of dislocations

In contrast to the $(010)$ Au/Cu case, the $(111)$-oriented habit interface
planes exhibit different arrangements of atoms, which yield to more complex
interface dislocation patterns and also to general elastic states where both
constituent strains and rotations are unequally partitioned between the
crystals [125].

Figure 3.28: (a) Dependence on $\eta_{1}=\eta_{2}$ of the elastic energy
$\gamma_{\mathrm{e}}$ in J.m-2, i.e. along the bisecting lines of the
admissible triangular domains $\mathcal{T}_{\mbox{ABC}}$, as displayed in
Figs. (3.27). The blue and the red curves correspond to the mismatched $(010)$
and $(111)$ Au/Cu interfaces, respectively. The latter exhibits black dots,
indexed by $s_{\eta}=1,\dots,16$, which represent the minimum-energy path from
Fig. (3.27b). The large points at $\eta_{1}=\eta_{2}=1/2$ are related to the
initial lozenge-shaped patterns with two crossing sets of dislocations,
whereas the vertical arrow shows the minimum-energy configuration associated
with the $(111)$ semicoherent interface. (b) Dependence on $s_{\eta}$ of the
dislocation characters $\phi_{i}$ for the three sets and the angle $\phi$
between the two parent dislocations for the corresponding $(111)$ Au/Cu case.
All these quantities are expressed in ∘.  Figure 3.29: Plots of initial,
intermediate, and final states along the computed minimum-energy path for the
$(111)$ Au/Cu heterophase interface. The initial periodic network of lozenge-
shaped misfit dislocations undergoes local relaxations and also leads to a
final elastically relaxed hexagonal-shaped dislocation pattern with lowest
short-range strain energy. (a) Dislocation structures. Distribution of (b) the
normal displacement component $\textit{u}_{2}$ and (c) the displacement norm
u. See text for the displacement field expressions.

The present orientation relations associated with the $(111)$ Au/Cu misfit
case are defined by $\textbf{{x}}^{\mbox{\scriptsize or}}_{1}=[11\bar{2}]$,
$\textbf{{x}}^{\mbox{\scriptsize or}}_{2}=[111]\parallel\textbf{{n}}$, and
$\textbf{{x}}^{\mbox{\scriptsize or}}_{3}=[1\bar{1}0]$, within which the fcc
$\\{111\\}$ close-packed planes contain $a_{\mbox{\scriptsize ref}}/2\langle
110\rangle$-type Burgers vectors. Similarly to the $(010)$ case, such Burgers
vectors must be defined in the proper reference state under the condition of
vanishing far-field stresses in the $(111)$ Au/Cu bicrystal. By arbitrarily
choosing $\textbf{{b}}_{1}=a_{\mbox{\scriptsize Au}}/2[10\bar{1}]$ and
$\textbf{{b}}_{2}=a_{\mbox{\scriptsize Au}}/2[0\bar{1}1]$ as the reference
Burgers vectors, the quantized Frank-Bilby eq. (3.121) gives rise to the
lozenge-shaped dislocation structure that is specifically comprised of two
arrays of parallel dislocations (with no local reactions at nodes): the
initial line directions are defined by $\boldsymbol{\xi}^{\mbox{\scriptsize
un}}_{1}\parallel[01\bar{1}]$ and $\boldsymbol{\xi}^{\mbox{\scriptsize
un}}_{2}\parallel[10\bar{1}]$, so that the individual characters are
$\phi^{\mbox{\scriptsize un}}_{1}=\phi^{\mbox{\scriptsize
un}}_{2}=60^{\circ}$, and the angle between these two unrelaxed sets of
dislocations is $\phi^{\mbox{\scriptsize un}}=60^{\circ}$. In addition,
$\textit{p}_{1}^{\mbox{\footnotesize o}}=\textit{p}_{2}^{\mbox{\footnotesize
o}}=2.25144~{}$nm, so that the inter-dislocation spacings are given here by
$\textit{d}^{\,\mbox{\scriptsize un}}_{1}=\textit{d}^{\,\mbox{\scriptsize
un}}_{2}=1.94980~{}$nm.

As illustrated in Fig. (3.26), the dependence of the total far-field stress
components in the $(111)$ system, i.e. in both Au (black line) and Cu (red
line) on $\delta$, yields to a predicted reference state for
$\delta_{(111)}=0.57962$, so that $a_{\mbox{\scriptsize ref}}=0.38096$ nm, and
also to the magnitudes of correct Burgers vectors are defined by
$\textit{b}_{1}=\textit{b}_{2}=0.26938$ nm. Moreover, Fig. (3.26) shows
stronger spurious stress values for the $(111)$ than $(010)$ system cases, by
a factor of $2.33$ ($2.15$) in Au (Cu) when $\delta=0$, i.e. when Au is
improperly selected as the reference state. The same qualitative conclusion
regarding the spurious stress state can be drawn for $\delta=1$.

Using the aforementioned Frank-Bilby solution as the initial dislocation
structure for possible elastic strain relaxation, Fig. (3.27b) shows the pre-
computed elastic landscape as function of $\eta_{1}$ and $\eta_{2}$,
associated with the $(111)$ misfit interface case. The symmetric landscape has
been computed using the same number of nodal points than in Fig. (3.27a), for
which the orientations of both plots are different for clarity. The elastic
energy per unit interface area for the unrelaxed lozenge-shaped dislocation
pattern is given by $\gamma_{\mathrm{e}}(\eta_{1}\to 1/2,\eta_{2}\to
1/2)\simeq 0.49568$ J.m-2, with $r_{\scalebox{0.65}{0}}=\textit{b}_{1}/4$,
which is slightly lower than the stored energy for the $(111)$ Au/Cu system
for $\eta_{1}\to 1/2$ and $\eta_{2}\to 1/2$. Here, the landscape for the
$(111)$ system is qualitatively and quantitatively different than the $(010)$
case, since the former gives rise to the existence of a unique minimum-energy
dislocation configuration with three sets of dislocations resulting from
junction reactions.

The energy minimization procedure that involves the conjugate gradient
algorithm is performed by using a prescribed convergence criterion in the pre-
computed energy landscape. The interface dislocation structures with the
lowest elastic energy are considered to be found when the difference between
the values of the stored elastic energy for two subsequent iterations is less
than $10^{-4}$ J.m-2. The corresponding minimum-energy path is determined by
using the nudged elastic band method between the initial non-equilibrium and
the minimum-energy states, for which the spring constant $k$ in eq. (3.120)
has been varied over several orders of amplitude without noticeable effects on
the computed path. The obtained minimum-energy path is displayed in Fig.
(3.27b) by the black curved chain with equidistantly positioned images (i.e.
intermediate states), where the final configuration state is designated as the
final elastically strain-relaxed dislocation pattern. Here, the smooth path
has no energy barrier (therefore also, no saddle point) and 15 intermediate
states, which connect the initial and final states, are constructed. The
minimum strain energy related to the relaxed dislocation pattern is given by
$\gamma_{\mathrm{e}}^{\mbox{\scriptsize min}}=\gamma_{\mathrm{e}}(\eta_{1}\to
0.31981,\eta_{2}\to 0.31981)=0.44733$ J.m-2, which corresponds to a
significant decrease in strain energy of $9.75\%$.

The variations of strain energy along the median $(\eta_{1}=\eta_{2})$ of the
two $(010)$ and $(111)$ Au/Cu landscapes, as displayed by the blue and red
dotted lines in the insets of Fig. (3.28a), start from their initial
corresponding lozenge-shaped dislocation structures at $\eta_{1}=\eta_{2}=1/2$
with different stored energy values. The red (blue) line illustrates the
(un)favorable elastic energy profile for junction formation that continuously
decreases (increases) with decreasing both values of $\eta_{1}$ and $\eta_{2}$
from $1/2$ at the $(111)$ ($(010)$) Au/Cu heterophase interface. The
intermediate states between the lozenge-shaped and the relaxed hexagonal-
shaped dislocation configurations for the $(111)$ case are indexed by
$s_{\eta}=1,\dots,15$. Such considerable saving in strain energy along
$s_{\eta}$ is related to the change in dislocation structures, e.g.
dislocation characters $\phi_{i}$ and the angle $\phi$ between
$\boldsymbol{\xi}_{2}$ and $\boldsymbol{\xi}_{1}$, which can be examined along
the determined minimum-energy path. Figure (3.28b) plots these geometrical
characteristics in terms of $\phi$ (in green), $\phi_{1}$ (blue), $\phi_{2}$
(red), and $\phi_{3}$ (black, for the newly formed set of dislocation
junction) as a function of $s_{\eta}$. It is also found that the geometrical
equilibrium configuration of the minimum-energy dislocation pattern is
characterized by $\phi^{\mbox{\scriptsize eq}}\simeq 128.4^{\circ}$,
$\phi^{\mbox{\scriptsize eq}}_{1}=\phi^{\mbox{\scriptsize eq}}_{2}\simeq
85.8^{\circ}$, and $\phi^{\mbox{\scriptsize eq}}_{3}=90^{\circ}$. Both sets 1
and 2 deviate by $4.2^{\circ}$ from pure edge characters, and the dislocation
structure deviates by $8.4^{\circ}$ from regular hexagonal-shaped
configuration. Such dislocation arrangement is in agreement with atomistic
analysis in iron, where deviations from pure screw dislocations in $(110)$ bcc
twist GBs with comparable order of dislocation spacings have been reported
using molecular statics simulations [295].

Figures (3.29) illustrate the strain-relaxed rearrangements of the interfacial
dislocations from the lozenge-shaped configurations on the $(111)$ heterophase
interface using different elastic quantities, which can, for example, be used
to analyze the likely regions for nucleating interface dislocations or
absorbing and annihilating point defects (interstitials and vacancies). All
contour plots are displayed at $\textit{x}_{2}=3\,a_{\mbox{\scriptsize Au}}$
with respect to the three dislocation configurations shown in Figs. (3.29a),