# Continuum model for dislocation structures of semicoherent interfaces

Luchan Zhang, Xiaoxue Qin  Yang Xiang  Department of Mathematics, Hong Kong
University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
<EMAIL_ADDRESS>
###### Abstract

In order to relieve the misfitting elastic energy, the hetero-interfaces
become semicoherent by forming networks of dislocations. These microscopic
structures strongly influence the materials properties associated with the
development of advanced materials. We develop a continuum model for the
dislocation structures of semicoherent interfaces. The classical Frank-Bilby
equation that governs the dislocation structures on semicoherent interfaces is
not able to determine a unique solution. The available methods in the
literature either use further information from atomistic simulations or
consider only special cases (dislocations with no more than two Burgers
vectors) where the Frank-Bilby equation has a unique solution. In our
continuum model, the dislocation structure of a semicoherent interface is
obtained by minimizing the energy of the equilibrium dislocation network with
respect to all the possible Burgers vectors, subject to the constraint of the
Frank-Bilby equation. The continuum model is validated by comparisons with
atomistic simulation results.

###### keywords:

Semicoherent interfaces, Dislocations, Frank-Bilby equation, Energy
minimization

##  1 Introduction

The interfaces between different materials or different phases commonly form
semicoherent structures that consist of discrete dislocation networks to
accommodate the lattice misfit between the two materials [1, 2, 3]. Such
semicoherent interfaces play essential roles in the mechanical, electronic and
plasticity properties that are associated with the development of novel
composite materials and alloys [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15]. These properties strongly depend on the characteristics of the
dislocation networks of the semicoherent interfaces.

A semicoherent interface is composed of a network of misfitting dislocations
and coherent regions separated by these dislocations. The equilibrium
dislocation structure on a semicoherent interface is governed by the Frank-
Bilby equation [16, 17], which determines the net Burgers vector content
$\mathbf{B}(\mathbf{p})$ crossing a probe vector $\mathbf{p}$ on the
interface. The Frank-Bilby equation strongly depends on the reference state
and the possible Burgers vectors defined associated with it, on which there
have been some in-depth discussions in the literature, e.g., [6, 7, 8, 9, 10,
11, 12, 13].

However, even though the reference state and all the associated possible
Burgers vectors on a semicoherent interface are determined, the Frank-Bilby
equation is still not able to give a unique dislocation structure. See the
example discussed at the end of Sec. 2. Wang et al. [7, 9] have developed an
atomically informed Frank-Bilby theory by combining the classical Frank-Bilby
theory and atomistic simulations to determine the reference lattice and
interfacial dislocation structure of a heterogeneous interface. The
dislocation line directions and their Burgers vectors in the dislocation
structure in the Frank-Bilby equation are informed by atomistic simulation
results. Vattre and Demkowicz, Abdolrahim and Demkowicz [8, 10, 11] have
formulated approaches to determine the reference state for the interfacial
misfitting dislocation arrays, linking the Frank–Bilby equation and
anisotropic elasticity theory under the condition of vanishing far-field
stresses. They considered two sets of misfitting dislocations (i.e.,
dislocations with two possible Burgers vectors) in their theory, for which the
Frank–Bilby equation is able to give a unique solution. Generalization of this
approach has been proposed by Vattre [12] to incorporate hexagonal misfitting
dislocation networks with new dislocation segments with the third Burgers
vector formed by dislocation reaction, from the lozenge dislocation network of
two sets of dislocations solved using the method in Ref. [8, 10]. Further
generalizations have been made by Vattre and Pan [13] for interaction and
movements of various dislocations in anisotropic bicrystals with semicoherent
interfaces. There are also simulations for the dislocation/disconnection
structures on semicoherent interfaces of precipitates with prescribed Burgers
vectors [4, 5].

In this paper, we present a continuum model to obtain the dislocation
structure of a semicoherent interface, given the reference state and all
possible Burgers vectors. In the continuum model, the energy of the
equilibrium misfitting dislocation network is minimized with respect to all
possible Burgers vectors subject to the constraint of the Frank-Bilby
equation. The continuum model is based on the orientation-dependent
dislocation densities of the dislocation structure. Since the Frank-Bilby
equation holds, the long-range elastic energy vanishes, and the energy of the
heterogeneous interface consists of only the local energy of the equilibrium
dislocation network, for which the continuum formulation for the local energy
of dislocation arrays [18] is used. When the dislocation network consists of
straight dislocations, our continuum model gives the exact solution (i.e.,
exact dislocation line directions and inter-dislocation distances) of the
dislocation network. We also develop an identification method based on
dislocation reactions to recover the exact dislocation network (e.g., the
hexagonal network) from the orientation-dependent dislocation densities
obtained in the continuum model. This model is a generalization of the method
proposed in Ref. [19] for finding dislocation structures of low angle grain
boundaries.

Numerically, the constrained minimization problem in our continuum model is
solved by the penalty method. We use our continuum model to study the fcc/bcc
semicoherent interfaces with the Nishiyama-Wassermann (NW) and Kurdjumov-Sachs
(KS) orientation relations. Comparisons with atomistic simulations show that
our continuum model can provide excellent predictions of the dislocation
structures of semicoherent interfaces.

This paper is organized as follows. In Sec. 2, we review the semicoherent
interfaces and the Frank-Bilby equation. In Sec. 3, we present our continuum
model for the dislocation structure. The reference lattices and possible
Burgers vectors of some semicoherent interfaces are reviewed in Sec. 4. In
Sec. 5, we apply our continuum model to obtain the dislocation structures of
Cu/Nb semicoherent interfaces, and compare the results with those of atomistic
simulations in Ref. [7, 9]. Conclusion and discussion are made in Sec. 6.

##  2 Semicoherent interfaces and Frank-Bilby equation

We first review the semicoherent interfaces and the Frank-Bilby equation. The
geometry of a bicrystal hetero-interface is illustrated schematically in Fig.
1(a). Two materials (or two phases) $\alpha$ and $\beta$ with different
lattice structures are joined, and a hetero-interface is formed between them.
The interface plane is set as the $xy$ plane. When the lattice structures of
the adjacent crystals are similar, and the lattice spacing difference between
the unstrained adjacent crystals are relatively small, the interface usually
becomes semicoherent by forming a network of misfit dislocations on the
interface, and atoms in two adjacent lattices are adjusted by additional
strains or rotations. Figure 1(b) demonstrates a natural dichromatic pattern
of the semicoherent interface between fcc$(110)$/bcc$(001)$.

Figure 1: (a) Illustration of the bicrystal hetero-interface. (b) The natural
dichromatic pattern of the interface between fcc$(110)$/bcc$(001)$.

Frank and Bilby proposed a theory that provides an geometry constraint of the
equilibrium dislocation structure on a semicoherent interface [16, 17]. Given
the reference lattice, the Frank-Bilby theory determines the net Burgers
vector $\mathbf{B}(\mathbf{p})$ crossing an interfacial probe vector
$\mathbf{p}$ as
$\mathbf{B}(\mathbf{p})=(\mathbf{S}_{\beta}^{-1}-\mathbf{S}_{\alpha}^{-1})\mathbf{p}$,
where $\mathbf{S}_{\alpha}^{-1},\mathbf{S}_{\beta}^{-1}$ respectively are
inverse matrices of the distortion transformation matrices
$\mathbf{S}_{\alpha},\mathbf{S}_{\beta}$ that map the lattice vectors from the
natural unstrained lattices $\alpha$, $\beta$ to the reference lattice.

Once the reference lattice is determined, there will be finite number of
possible Burgers vectors associated with it, which are the lattice vectors in
the reference lattice. We denote these Burgers vectors by $\mathbf{b}_{j}$,
$j=1,2,\cdots,J$. The net Burgers vector $\mathbf{B}(\mathbf{p})$ can be
expressed in the reference lattice as
$\mathbf{B}(\mathbf{p})=\sum_{j=1}^{J}(\mathbf{N}_{j}\cdot\mathbf{p})\mathbf{b}_{j}$.
Recall that in the classical dislocation model of grain boundaries and
interfaces [3, 20], the dislocation structure on a boundary/interface is
described by the reciprocal vector $\mathbf{N}$ lying in the
boundary/interface plane that is perpendicular to the dislocation and has
magnitude $N=1/D$, where $D$ is the inter-dislocation distance. The local
dislocation line direction is $\bm{\xi}=(\mathbf{N}/N)\times\mathbf{n}$, where
$\mathbf{n}$ is the unit normal vector of the interface (which is in the $+z$
direction here). The dislocation density is $N$. For multiple arrays of
dislocations on the interface, the dislocation array with Burgers vectors
$\mathbf{b}_{j}$ are represented by $\mathbf{N}_{j}$, $j=1,2,\cdots,J$.
Accordingly, the Frank-Bilby equation can be written as

$\displaystyle\sum_{j=1}^{J}\left(\mathbf{N}_{j}\cdot\mathbf{p}\right)\mathbf{b}_{j}=(\mathbf{S}_{\beta}^{-1}-\mathbf{S}_{\alpha}^{-1})\mathbf{p},$
(1)

for any interfacial probe vector $\mathbf{p}$.

The Frank-Bilby equation in general is not able to uniquely determine the
dislocation structure. For example, on the fcc$(110)$/bcc$(001)$ interface
with $[001]_{\rm fcc}\|[010]_{\rm bcc}$ and $[1\bar{1}0]_{\rm fcc}\|[100]_{\rm
bcc}$, there are four possible Burgers vectors [7], leading to $8$ unknowns of
the four vectors $\mathbf{N}_{j}=(N_{jx},N_{jy})$ for the orientations and
inter-dislocation distances of the four sets of Burgers vectors; whereas there
are only $4$ equations in Frank-Bilby theory, which are not enough to
determine the $8$ unknown quantities. Only when there are no more than two
possible Burgers vectors in the semicoherent interface, the Frank-Bilby
equation can uniquely determine a dislocation structure.

Our continuum model will be based on the given reference state. In this paper,
we simply adopt the median lattice (average of the two lattices) with
isotropic elasticity. In practice, the median lattice or one of the adjacent
lattices have often been used as the reference lattice [16, 21, 3].
Especially, the median lattice [16] is an excellent approximation of the
reference lattice for symmetric and isotropic interfaces, leading to equal
partition of the elastic fields of the two crystals. Recently, methods of
determining the reference lattice in general cases such as anisotropic or
unsymmetrical interfaces have also been developed [6, 7, 8, 9, 10, 11].

The reference lattices and possible Burgers vectors of some hetero-interfaces
will be reviewed in Sec. 4.

##  3 Continuum model

Now we present a continuum model to obtain the dislocation structure of the
semicoherent hetero-interface with the given reference state (or equivalently,
the distortion transformation matrices $\mathbf{S}_{\alpha}$ and
$\mathbf{S}_{\beta}$) and all possible Burgers vectors ($\mathbf{b}_{j}$,
$j=1,2,\cdots,J$). Since Frank-Bilby equation is not sufficient to uniquely
determined the dislocation structure on the interface, we identify the
equilibrium dislocation structure by minimizing the local energy associated
with the constituent dislocations of the interface subject to the constraint
of the Frank-Bilby equation. That is, we solve the following constrained
energy minimization problem for the dislocation structure:

minimize $\displaystyle E=\int_{S}\gamma dS,$ (2) $\displaystyle{\rm with}$
$\displaystyle\gamma=\sum_{j=1}^{J}\frac{\mu
b_{j}^{2}}{4\pi(1-\nu)}\\!\left(1-\nu\frac{(\mathbf{N}_{j}\times\mathbf{n}\cdot\mathbf{b}_{j})^{2}}{b_{j}^{2}N_{j}^{2}}\right)\\!N_{j}\log\\!\frac{1}{r_{g}\sqrt{N_{j}^{2}+\epsilon}},$
(3) subject to
$\displaystyle\mathbf{h}=\sum_{j=1}^{J}\left(\mathbf{N}_{j}\cdot\mathbf{p}\right)\mathbf{b}_{j}-(\mathbf{S}_{\beta}^{-1}-\mathbf{S}_{\alpha}^{-1})\mathbf{p}=\mathbf{0}.$
(4)

Here $S$ is a periodic cell on the interface plane, $\gamma$ is the interface
energy density, $\mu$ is the shear modulus, $\nu$ is the Poisson ratio,
$b_{j}$ is the length of the $j$-th Burgers vector, $r_{g}$ is a parameter
associated with the dislocation core size, $\epsilon$ is some small positive
regularization parameter to avoid the numerical singularity when $N_{j}=0$.
The constraint in Eq. (4) is the Frank-Bilby equation, which holds for any
vector $\mathbf{p}$ on the interface.

The interface energy $\gamma$ in Eq. (3) is based on the local energy of
dislocation arrays in terms of dislocation densities [18]. Since Frank-Bilby
equation is equivalent to cancellation of the long-range elastic field [16,
17, 20, 3], there is only local energy of the constituent dislocations on the
interface when Frank-Bilby equation in Eq. (4) holds, as that for the grain
boundaries in homogeneous materials [20, 3]. The elastic constants in $\gamma$
in Eq. (3) can be chosen as the averages of those of the two materials: e.g.,
$\mu=(\mu_{\alpha}+\mu_{\beta})/2$, $\nu=(\nu_{\alpha}+\nu_{\beta})/2$. More
accurate values can be adopted if necessary.

Note that the energy $\gamma$ in Eq. (3) is based on densities of
dislocations. When all the constituent dislocations on the interface are
straight, the obtained vectors $\mathbf{N}_{j}$’s give the exact dislocation
structure. When the dislocation network consists of disconnected dislocation
segments, e.g., the hexagonal network, our continuum model gives the line
directions and densities of these dislocations (line direction
$\mathbf{N}_{j}/N_{j}$ and density $N_{j}$); and in this case, we will present
a method to recover the exact hexagonal network from the obtained dislocation
densities and line directions; see the end of this section.

The constraint of the Frank-Bilby formula in Eq. (4) holds for any probe
vector $\mathbf{p}$ if and only if it holds for the two basis vectors of the
$xy$ plane: $\mathbf{p}=\mathbf{p_{1}}=(1,0)$ and
$\mathbf{p}=\mathbf{p_{2}}=(0,1)$, i.e.,
$\tilde{\mathbf{h}}=(\mathbf{h}_{1},\mathbf{h}_{2})^{T}=\mathbf{0}$ with
$\mathbf{h}_{1}$ and $\mathbf{h}_{2}$ being the Frank-Bilby formula in Eq. (4)
when the probe vector $\mathbf{p}$ is set to be $\mathbf{p}_{1}$ and
$\mathbf{p}_{2}$, respectively.

In addition to the misfit, the Frank-Bilby equation in Eq. (4) may also
include further twist and/or tilt of the two crystals $\alpha$ and $\beta$
through the transformation matrices $\mathbf{S}_{\alpha}$ and
$\mathbf{S}_{\beta}$ [6, 7, 8, 9, 10, 11, 12, 13]. When there are rotations
around an axis perpendicular to the interface (i.e., twist), assuming that the
rotation angles of the natural lattices of the two materials and the reference
lattice are $\theta_{\alpha}$, $\theta_{\beta}$ and $\theta$, respectively,
the Burgers vectors in the rotated reference lattice can be calculated using
the rotation matrix $\mathbf{R}_{\theta}$ as $\mathbf{b}_{i}^{\rm
R}=\mathbf{R}_{\theta}\mathbf{b}_{i}$, and the distortion transformation
matrices are $\mathbf{S}_{\alpha}^{\rm
R}=\mathbf{R}_{\theta}\mathbf{S}_{\alpha}\mathbf{R}_{\alpha}^{\rm T}$ and
$\mathbf{S}_{\beta}^{\rm
R}=\mathbf{R}_{\theta}\mathbf{S}_{\beta}\mathbf{R}_{\beta}^{\rm T}$, where
$\mathbf{b}_{i}$, $\mathbf{S}_{\alpha}$ and $\mathbf{S}_{\beta}$ are those in
the un-rotated state.

Remarks:

1\. The interface energy formula in Eq. (3) can also be generalized to include
elastic anisotropy based on the energy of straight dislocations with
appropriate pre-logarithmic energy coefficients [20].

2\. We adopt the local energy of the constituent dislocations of the interface
for a simple form and efficient calculation of the continuum model. In
principle, the local energy in Eq. (3) can be replaced by the full elastic
energy with elastic anisotropy and/or different elastic constants in the two
materials [22, 8, 10, 11, 12, 13] for more accurate results.

Numerically, the constrained minimization problem can be solved by the penalty
method [23], in which it is approximated by the following unconstrained
minimization problem:

$\text{minimize
}Q=\int_{S}\left(\gamma+\frac{\alpha_{p}}{2}\|\tilde{\mathbf{h}}\|^{2}\right)dS,$
(5)

where $\alpha_{p}>0$ with large value is the penalty parameter. It has been
shown that as the penalty parameter $\alpha_{p}\rightarrow+\infty$, the
solution of this unconstrained minimization problem converges to the solution
of the constrained minimization problem [23]. (Other method such as the
augmented Lagrangian method can also be used to solve this constrained
minimization problem [23].)

This unconstrained problem is still very challenging to solve due to the
nonconvexity of the interface energy. We make a further simplification by
considering uniform distributions of straight dislocations on the interface.
In this case, each $\mathbf{N}_{j}=(N_{jx},N_{jy})$ is a constant vector, and
the problem is reduced to minimize
$q=\gamma+\alpha_{p}\|\tilde{\mathbf{h}}\|^{2}/2$.

This unconstrained problem can be solved by gradient minimization with respect
to variables $N_{jx}$, $N_{jy}$, $j=1,2,\cdots,J$, which leads to the
following evolution equations with an artificial time:

$\displaystyle(N_{jx})_{t}=-\left(\frac{\partial\gamma}{\partial
N_{jx}}+\alpha_{p}\frac{\partial c}{\partial N_{jx}}\right),\ \ \
(N_{jy})_{t}=-\left(\frac{\partial\gamma}{\partial
N_{jy}}+\alpha_{p}\frac{\partial c}{\partial N_{jy}}\right),$ (6)

for $j=1,2,\cdots,J$, where $c=\|\tilde{\mathbf{h}}\|^{2}/2$, and
$\tilde{\mathbf{h}}=(h_{1},h_{2},h_{3},h_{4})^{T}$ with
$h_{1}=\sum_{j=1}^{J}b_{jx}N_{jx}-(\mathbf{S}_{\beta}^{-1}[1,1]-\mathbf{S}_{{\alpha}}^{-1}[1,1])$,
$h_{2}=\sum_{j=1}^{J}b_{jy}N_{jx}-(\mathbf{S}_{\beta}^{-1}[2,1]-\mathbf{S}_{{\alpha}}^{-1}[2,1])$,
$h_{3}=\sum_{j=1}^{J}b_{jx}N_{jy}-(\mathbf{S}_{\beta}^{-1}[1,2]-\mathbf{S}_{{\alpha}}^{-1}[1,2])$,
$h_{4}=\sum_{j=1}^{J}b_{jy}N_{jy}-(\mathbf{S}_{\beta}^{-1}[2,2]-\mathbf{S}_{{\alpha}}^{-1}[2,2])$.
Note that when tilt of the two crystals $\alpha$ and $\beta$ is also
considered, in the Frank-Bilby equation in Eq. (4), $\mathbf{S}_{\alpha}$ and
$\mathbf{S}_{\beta}$ will be $3\times 3$ matrices, $\mathbf{b}_{j}$,
$\mathbf{p}$ and $\mathbf{h}$ will be vectors in three dimensions, and
$\tilde{\mathbf{h}}$ will be a $6\times 1$ vector.

Identification of dislocation structure from dislocation densities. Now we
present a method that recovers the exact dislocation structure based on the
densities and orientations of the constituent dislocations obtained in our
continuum model. As mentioned above, when all the constituent dislocations on
the interface are straight, the obtained vectors $\mathbf{N}_{j}$’s give the
exact dislocation structure directly. In a general dislocation structure, due
to dislocation reactions, the dislocations may not necessarily be continuous
straight lines, and they may form hexagons (not necessarily regular) with
disconnected dislocation segments; see Fig. 5(b) for an example. In the
identification method, we calculate the exact length and orientation of each
dislocation segment in the hexagonal network based on the dislocation
densities and orientations obtained by the continuum model.

Figure 2: A hexagonal dislocation structure that consists of dislocations with
Burgers vectors $\mathbf{b}_{1}$, $\mathbf{b}_{2}$ and $\mathbf{b}_{3}$, whose
line directions are $\boldsymbol{\xi}_{1}$, $\boldsymbol{\xi}_{2}$, and
$\boldsymbol{\xi}_{3}$, respectively. Vectors $\mathbf{a}_{1}$ and
$\mathbf{a}_{2}$ are the two sides of the periodic parallelogram unit cell.
The area of a unit cell is $A=\|\mathbf{a}_{1}\times\mathbf{a}_{2}\|$.

Consider a hexagonal network with dislocations of three Burgers vectors
$\mathbf{b}_{1}$, $\mathbf{b}_{2}$, and $\mathbf{b}_{3}$, in which
dislocations may have reactions, e.g,
$\mathbf{b}_{3}=\mathbf{b}_{1}+\mathbf{b}_{2}$ and
$b_{2}^{2}<b_{1}^{2}+b_{2}^{2}$; see Fig. 2. In this hexagonal network, as we
discussed before, the direction of dislocation generated by $\mathbf{b}_{j}$
is $\boldsymbol{\xi}_{j}=(\mathbf{N}_{j}/N_{j})\times\mathbf{n}$, and the
density of these dislocations is $N_{j}$. Consider a periodic parallelogram
cell as shown in Fig. 2. It can be calculated that the $\mathbf{b}_{1}$-,
$\mathbf{b}_{2}$-, and $\mathbf{b}_{3}$-dislocation segments in the
parallelogram, written in the vector form, are
$\mathbf{l}_{1}=A\mathbf{N}_{1}\times\mathbf{n}$,
$\mathbf{l}_{2}=A\mathbf{N}_{2}\times\mathbf{n}$,
$\mathbf{l}_{3}=A\mathbf{N}_{3}\times\mathbf{n}$, respectively, where
$\mathbf{n}$ is the normal vector of the interface, and $A$ is the area of the
periodic parallelogram cell. Using these results, it can be solved that in a
periodic parallelogram cell, the length of each dislocation segment is

$l_{j}=\frac{N_{j}}{\|\mathbf{N}_{1}\times\mathbf{N}_{2}\|+\|\mathbf{N}_{2}\times\mathbf{N}_{3}\|+\|\mathbf{N}_{3}\times\mathbf{N}_{1}\|},\
\ j=1,2,3,$ (7)

Using this formula of length of each dislocation segment $l_{j}$ and its
direction
$\boldsymbol{\xi}_{j}=\left(\mathbf{N}_{j}/N_{j}\right)\times\mathbf{n}$, we
can draw the exact hexagonal network structure based on the dislocation
densities and orientations represented by $\\{\mathbf{N}_{j}\\}$ in the
continuum model.

##  4 Reference lattice and possible Burgers vectors

In this section, we briefly review the reference lattice and possible Burgers
vectors for the dislocation network on a semicoherent interface. The
distortion transformation matrices $\mathbf{S}_{\alpha}$ and
$\mathbf{S}_{\beta}$ are defined as the matrices mapping from the dichromatic
patterns of the two lattices ${\alpha}$ and ${\beta}$ to the reference
lattice, and the possible Burgers vectors ar the lattice vectors in the
reference lattice [3]. In practice, the median lattice (average of the two
lattices) or one of the adjacent lattices have often been used as the
reference lattice [16, 21, 3]. Especially, the median lattice [16] is an
excellent approximation of the reference lattice for symmetric and isotropic
interfaces, leading to equal partition of the elastic fields of the two
crystals. Recently, methods of determining the reference lattice in general
cases such as anisotropic or unsymmetrical interfaces have also been developed
[6, 7, 8, 9, 10, 11].

Figure 3: Geometry of the dichromatic patterns, reference states and possible
Burgers vectors of some classical semicoherent interfaces between fcc and bcc
lattices. The natural dichromatic patterns of the fcc and bcc lattices and the
reference lattice (the median lattice) are polygons with red, blue, and black
dashed lines, respectively. The possible Burgers vectors defined in the
reference lattice are shown by black arrows. (a) Interface with the
orientation relationship $(110)_{\rm fcc}\|(001)_{\rm bcc}$, $[001]_{\rm
fcc}\|[010]_{\rm bcc}$, and $[1\bar{1}0]_{\rm fcc}\|[100]_{\rm bcc}$. The
natural dichromatic patterns are rectangles. (b) Interface with the
orientation relationship of that in (a) with further rotations around an axis
perpendicular to the interface. The reference lattice and possible Burgers
vectors can all be obtained by rotations from those in (a). (c) Interface with
the NW orientation relationship: $(111)_{\rm fcc}\|(110)_{\rm bcc}$,
$[11\bar{2}]_{\rm fcc}\|[1\bar{1}0]_{\rm bcc}$, and $[1\bar{1}0]_{\rm
fcc}\|[001]_{\rm bcc}$. The natural dichromatic patterns are hexagons. (d)
Interface with the KS orientation relationship: $(111)_{\rm fcc}\|(110)_{\rm
bcc}$, $[11\bar{2}]_{\rm fcc}\|[1\bar{1}2]_{\rm bcc}$, and $[\bar{1}10]_{\rm
fcc}\|[1\bar{1}1]_{\rm bcc}$. The fcc and bcc lattices, reference lattice
patterns and Burgers vectors can be obtained by rotating the corresponding
ones in the NW orientation relationship in (c) around an axis perpendicular to
the interface.

Figure 3 shows the reference lattices and possible Burgers vectors of several
semicoherent interfaces between fcc and bcc lattices that have been studied
recently in the literature [6, 7, 8, 9, 10, 11, 12, 13, 14]. In Fig. 3(a), the
interface orientation relationships are $(110)_{\rm fcc}\|(001)_{\rm bcc}$,
$[001]_{\rm fcc}\|[010]_{\rm bcc}$ (in $x$-direction) and $[1\bar{1}0]_{\rm
fcc}\|[100]_{\rm bcc}$ (in $y$-direction). The natural dichromatic patterns of
the fcc and bcc lattices are rectangles with parallel sides, as shown by the
red and blue dashed lines, respectively, in Fig. 3(a). The coherent
dichromatic pattern of the reference lattice is also a rectangle pattern in
between the fcc and bcc natural dichromatic patterns, as shown by the black
dashed lines in Fig. 3(a). When the median lattice (the average of the fcc and
bcc natural dichromatic patterns) is adopted as the reference lattice, the
possible Burgers vectors of dislocations on the semicoherent interface are the
lattice vectors of the reference lattice:

$\displaystyle\textstyle{\mathbf{b}_{1}=\left(\frac{1}{2}a_{\rm
fcc}+\frac{1}{2}a_{\rm
bcc},0\right),\mathbf{b}_{2}=\left(0,\frac{\sqrt{2}}{4}a_{\rm
fcc}+\frac{1}{2}a_{\rm bcc}\right)},$
$\displaystyle\textstyle{\mathbf{b}_{3}=\left(\frac{1}{4}a_{\rm
fcc}+\frac{1}{4}a_{\rm bcc},\frac{\sqrt{2}}{8}a_{\rm fcc}+\frac{1}{4}a_{\rm
bcc}\right),\mathbf{b}_{4}=\left(\frac{1}{4}a_{\rm fcc}+\frac{1}{4}a_{\rm
bcc},-\left(\frac{\sqrt{2}}{8}a_{\rm fcc}+\frac{1}{4}a_{\rm
bcc}\right)\right)},$ (8)

and the distortion transformation matrices mapping from the fcc and bcc
lattices to the reference lattice are

$\displaystyle\mathbf{S}_{\alpha}=\begin{pmatrix}\frac{a_{\rm fcc}+a_{\rm
bcc}}{2a_{\rm fcc}}&0\\\ 0&\frac{\frac{\sqrt{2}}{2}a_{\rm fcc}+a_{\rm
bcc}}{\sqrt{2}a_{\rm
fcc}}\end{pmatrix},\quad\mathbf{S}_{\beta}=\begin{pmatrix}\frac{a_{\rm
fcc}+a_{\rm bcc}}{2a_{\rm bcc}}&0\\\ 0&\frac{\frac{\sqrt{2}}{2}a_{\rm
fcc}+a_{\rm bcc}}{2a_{\rm bcc}}\end{pmatrix},$ (9)

where $a_{\rm fcc}$ and $a_{\rm bcc}$ are the lattice constants of the fcc and
bcc lattices.

Figure 3(b) shows the orientation relationship of the fcc$(110)$/bcc$(001)$
interface in Fig. 3(a) with further rotations around an axis perpendicular to
the interface. The rotation angles of the natural fcc, bcc lattices and the
reference lattice are $\theta_{\alpha}$, $\theta_{\beta}$ and $\theta$,
respectively. The Burgers vectors in the rotated reference lattice can be
calculated by multiplying the Burgers vectors in the un-rotated reference
lattice by the rotation matrix $\mathbf{R}_{\theta}$, i.e.,

$\displaystyle\mathbf{b}_{i}^{\rm R}=\mathbf{R}_{\theta}\mathbf{b}_{i},\text{
with }\mathbf{R}_{\theta}=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}.$ (10)

The distortion transformation matrices are

$\displaystyle\mathbf{S}_{\alpha}^{\rm
R}=\mathbf{R}_{\theta}\mathbf{S}_{\alpha}\mathbf{R}_{\alpha}^{\rm
T},\quad\mathbf{S}_{\beta}^{\rm
R}=\mathbf{R}_{\theta}\mathbf{S}_{\beta}\mathbf{R}_{\beta}^{\rm T}.$ (11)

Figure 3(c) demonstrates the classical Nishiyama-Wassermann (NW) orientation
relationship: $(111)_{\rm fcc}\|(110)_{\rm bcc}$, $[11\bar{2}]_{\rm
fcc}\|[1\bar{1}0]_{\rm bcc}$ (in $x$-direction), and $[1\bar{1}0]_{\rm
fcc}\|[001]_{\rm bcc}$ (in $y$-direction). The natural dichromatic patterns of
fcc and bcc lattices are hexagonal patterns, and the coherent dichromatic
pattern of the reference lattice, which is the median lattice between the fcc
and bcc lattices, is also a hexagonal pattern. The possible Burgers vectors
defined in the reference lattice are

$\displaystyle\textstyle{\mathbf{b}_{1}=\left(-\left(\frac{\sqrt{6}}{8}a_{\rm
fcc}+\frac{\sqrt{2}}{4}a_{\rm bcc}\right),-\left(\frac{\sqrt{2}}{8}a_{\rm
fcc}+\frac{1}{4}a_{\rm bcc}\right)\right)},$
$\displaystyle\textstyle{\mathbf{b}_{2}=\left(-\left(\frac{\sqrt{6}}{8}a_{\rm
fcc}+\frac{\sqrt{2}}{4}a_{\rm bcc}\right),\frac{\sqrt{2}}{8}a_{\rm
fcc}+\frac{1}{4}a_{\rm
bcc}\right),\mathbf{b}_{3}=\left(0,\frac{\sqrt{2}}{4}a_{\rm
fcc}+\frac{1}{2}a_{\rm bcc}\right)},$ (12)

and the distortion transformation matrices are

$\displaystyle\mathbf{S}_{\alpha}=\begin{pmatrix}\frac{\frac{\sqrt{6}}{4}a_{\rm
fcc}+\frac{\sqrt{2}}{2}a_{\rm bcc}}{\frac{\sqrt{6}}{2}a_{\rm fcc}}&0\\\
0&\frac{\frac{\sqrt{2}}{2}a_{\rm fcc}+a_{\rm bcc}}{\sqrt{2}a_{\rm
fcc}}\end{pmatrix},\quad\mathbf{S}_{\beta}=\begin{pmatrix}\frac{\frac{\sqrt{6}}{4}a_{\rm
fcc}+\frac{\sqrt{2}}{2}a_{\rm bcc}}{\sqrt{2}a_{\rm bcc}}&0\\\
0&\frac{\frac{\sqrt{2}}{2}a_{\rm fcc}+a_{\rm bcc}}{2a_{\rm
fcc}}\end{pmatrix}.$ (13)

Another classical orientation relationship of the interface jointing
$(111)_{\rm fcc}\|(110)_{\rm bcc}$ is the Kurdjumov-Sachs (KS) orientation
relationship with $[11\bar{2}]_{\rm fcc}\|[1\bar{1}2]_{\rm bcc}$ (in
$x$-direction) and $[\bar{1}10]_{\rm fcc}\|[1\bar{1}1]_{\rm bcc}$ (in
$y$-direction); see Fig. 3(d). It can be obtained by rotating the fcc and bcc
lattices in NW orientation relationship around an axis perpendicular to the
interface. The possible Burgers vectors and the distortion transformation
matrices can also be obtained from those in the NW orientation relationship by
rotations using Eqs. (10) and (11).

##  5 Numerical simulation

We apply our continuum simulation model to obtain dislocation structures on
the semicoherent interfaces of Cu$(110)$/Nb$(001)$ and Cu$(111)$/Nb$(110)$,
which have been studied recently in the literature [6, 7, 8, 9, 10, 11, 12,
13, 14]. The orientation relationships, reference states (the median lattices)
and possible Burgers vectors of these interfaces are shown in Fig. 3. The
lattice constants of Cu and Nb are $a_{\rm Cu}=0.360$nm and $a_{\rm
Nb}=0.330$nm, respectively. We start from $N_{jx}=N_{jy}=0$, $j=1,2,\cdots,J$,
when performing the energy gradient minimization in Eq. (6). We choose a large
value for the penalty parameter $\alpha_{p}$ in Eq. (5) and further increases
of its value give only negligible changes in the converged dislocation
structures. We compare our results with those obtained using atomistic
simulations [7, 9]. The reference lattices of these semicoherent interfaces
obtained by atomistic simulations [7, 9] are very close to the median lattices
adopted in our simulations.

###  5.1 Cu$(110)$/Nb$(001)$ interface with rectangle pattern

We first consider the Cu$(110)$/Nb$(001)$ interface with the orientation
relationship as shown in Fig. 3(a), where the natural Cu (fcc), Nb (bcc), and
reference lattices have rectangle patterns. There are four possible Burgers
vectors $\mathbf{b}_{j}$, $j=1,2,3,4$. See Eqs. (4) and (9) for the formulas
of these Burgers vectors and the distortion matrices $\mathbf{S}_{\alpha}$ and
$\mathbf{S}_{\beta}$.

Figure 4: Dislocation structure of this Cu$(110)$/Nb$(001)$ interface
calculated using our continuum model. It is a rectangular network that
consists of two arrays of dislocations with Burgers vectors $\mathbf{b}_{1}$
and $\mathbf{b}_{2}$ (red vertical lines and blue horizontal lines,
respectively). Unit: nm.

The dislocation structure obtained using our continuum model is shown in Fig.
4. The dislocation structure is a rectangular network that consists of two
arrays of dislocations with Burgers vectors $\mathbf{b}_{1}$, $\mathbf{b}_{2}$
and represented by the reciprocal vectors $\mathbf{N}_{1}=(-0.2658,0)/$nm,
$\mathbf{N}_{2}=(0,0.8681)/$nm, respectively. These two arrays of dislocations
are both edge dislocations, and are in the $+y$ ($[1\bar{1}0]_{\rm
Cu}\|[100]_{\rm Nb}$) and $+x$ ($[001]_{\rm Cu}\|[010]_{\rm Nb}$) directions,
with inter-dislocation distances $D_{1}=1/{N_{1}}=3.76\rm{nm}$ and
$D_{2}=1/{N_{2}}=1.15\rm{nm}$, respectively. Dislocations with Burgers vectors
$\mathbf{b}_{3}$ and $\mathbf{b}_{4}$ do not appear in the converged
dislocation structure, i.e., $\mathbf{N}_{3},\mathbf{N}_{4}$ converge to
$\mathbf{0}$ in the simulation. These results of rectangular network,
dislocation line directions and inter-dislocation distances agree excellently
with those obtained using atomistic simulation in Ref. [7], in which the
inter-dislocation distances in the two dislocation arrays are
$D_{1}=3.80\rm{nm}$ and $D_{2}=1.13\rm{nm}$.

###  5.2 Cu$(111)$/Nb$(110)$ interface with NW orientation relationship

We then consider the Cu$(111)$/Nb$(110)$ interface with hexagonal pattern in
the classical NW orientation relationship as shown in Fig. 3(c). The natural
dichromatic patterns of Cu (fcc) and Nb (bcc) lattices and the coherent
dichromatic pattern of the reference lattice are hexagonal patterns. The three
possible Burgers vectors defined in the reference lattice and the distortion
matrices $\mathbf{S}_{\alpha}$ and $\mathbf{S}_{\beta}$ are given in Eqs. (4)
and (13).

Figure 5:  (a) Dislocation structure calculated using our continuum model. The
dislocation structure consists of a hexagonal network of three arrays of
dislocations with Burgers vectors $\mathbf{b}_{1}$, $\mathbf{b}_{2}$ and
$\mathbf{b}_{3}$, shown by red, blue and black line segments, respectively.
Unit: nm. (b) and (c) Disregistries along some cross-section lines obtained by
atomistic simulations in Ref. [9] (reproduced from their data): (b)
disregistry in the $x$ direction ($[11\bar{2}]_{\rm Cu}\|[1\bar{1}0]_{\rm
Nb}$) along a cross-section line parallel to the $x$ axis, and (c) disregistry
in the $y$ direction ($[1\bar{1}0]_{\rm Cu}\|[001]_{\rm Nb}$) along a cross-
section line parallel to the $y$ axis. The black and red lines show the actual
disregistry and the unrelaxed uniform disregistry, respectively. The $\perp$
symbols show the locations of dislocations.

In the dislocation network of this semicoherent interface obtained using our
continuum model, dislocations with all the three Burgers vectors are present;
see Fig. 5(a). The converged values obtained using our continuum model are
$\mathbf{N}_{1}=(-0.1149,-0.0660)$/nm, $\mathbf{N}_{2}=(-0.1149,0.0660)$/nm
and $\mathbf{N}_{3}=(0,0.8022)$/nm, respectively. This result means that the
dislocations with the three Burgers vectors have line directions of
$\boldsymbol{\xi}_{1}=(-0.4981,0.8671)$,
$\boldsymbol{\xi}_{2}=(0.4981,0.8671)$ and $\boldsymbol{\xi}_{3}=(1,0)$ in the
$xy$ plane, respectively, and their densities are $\rho_{1}=N_{1}=0.1325$/nm.
$\rho_{2}=N_{2}=0.1325$/nm and $\rho_{3}=N_{3}=0.8022$/nm, respectively.
Recall that the local dislocation line direction is calculated by
$\bm{\xi}_{j}=(\mathbf{N}_{j}/N_{j})\times\mathbf{n}$, where $\mathbf{n}$ is
the unit normal vector of the interface. The $\mathbf{b}_{3}$-dislocations are
edge dislocations, and the $\mathbf{b}_{1}$\- and
$\mathbf{b}_{2}$-dislocations are very close to edge, with an angle of
$93^{\circ}$ between the dislocations and their Burgers vectors.

These three arrays of dislocations form a hexagonal network, because we have
$\mathbf{b}_{3}=(-\mathbf{b}_{1})+\mathbf{b}_{2}$ with
$b_{3}^{2}<b_{1}^{2}+b_{2}^{2}$ and dislocation reaction from a
$\mathbf{b}_{1}$-dislocation and a $\mathbf{b}_{2}$-dislocation into a
$\mathbf{b}_{3}$-dislocation is energetically favorable. In this case,
dislocations are disconnected segments instead of connected straight lines.
Using the identification method presented at the end of Sec. 3 (Eq. (7)), we
can calculate the lengths of the three types of dislocation segments, which
are $0.6642$nm, $0.6642$nm and $4.0208$nm for the $\mathbf{b}_{1}$-,
$\mathbf{b}_{2}$-, and $\mathbf{b}_{3}$-dislocation segments, respectively.
Based on these orientations and lengths of these dislocation segments, we can
recover the hexagonal network as shown in Fig. 5(a).

We compare the dislocation structure obtained by our continuum model with the
atomistic simulation result in Ref. [9]. Disregistries along some cross-
section lines obtained by atomistic simulations in Ref. [9] are shown in Figs.
5(b) and (c). Figure 5(b) shows the disregistry in the $x$ direction
($[11\bar{2}]_{\rm Cu}\|[1\bar{1}0]_{\rm Nb}$) along a cross-section line
parallel to the $x$ axis, and this cross-section line intersects with
dislocations periodically with average distance of $4.40$nm between two
neighboring intersecting points. Figure 5(c) shows the disregistry in the $y$
direction ($[1\bar{1}0]_{\rm Cu}\|[001]_{\rm Nb}$) along a cross-section line
parallel to the $y$ axis, and this cross-section line intersects with
dislocations periodically with average distance between two neighboring
intersecting points $1.14$nm. On the other hand, in the dislocation structure
obtained by our continuum model shown in Fig. 5(a), the horizontal line
intersects with four $\mathbf{b}_{1}$\- and $\mathbf{b}_{2}$-dislocations for
$-8{\rm nm}\leq x\leq 8{\rm nm}$ and the distance between two neighboring
intersecting points is $4.35$nm, and the vertical line intersects with the
array of the $\mathbf{b}_{3}$-dislocation segments with inter-dislocation
distance of $1.15$nm. These results agree excellently with those of atomistic
simulations shown in Figs. 5(b) and (c) (see also Figs. 3-5 in Ref. [9]).

Dislocation structure on the NW semicoherent interface as well as the
reference state have also be obtained in Ref. [13] by using their continuum
framework, in which a full anisotropic elasticity problem in the bulk of
bicrystal is solved and the hexagonal network on the interface with new
dislocation segments with the third Burgers vector is formed by dislocation
reaction from the lozenge dislocation network of two sets of dislocations. The
hexagonal networks obtained by using our continuum model and that by using
their model (Fig. 6(d) in Ref. [13]) are quite similar to each other.
Quantitatively, across the same horizontal line as that in Fig. 5(a), the
distance between two neighboring intersecting points is $4.05$nm in their
result (measured from their Fig. 6(d)), and the distance between neighboring
$\mathbf{b}_{3}$-dislocation segments is $1.13$nm (calculated from their data
of the lozenge dislocation network). These agreements between the results of
the two models provide mutual validations of these models.

###  5.3 Cu$(111)$/Nb$(110)$ interface with KS orientation relationship

We also consider the Cu$(111)$/Nb$(110)$ interface with hexagonal pattern in
the KS orientation relationship as shown in Fig. 3(d), in which there is a
relative rotation between the natural Cu fcc and Nb bcc lattices. The rotation
angle of the natural Cu fcc lattice is $\theta_{\alpha}=60^{\circ}$, and that
of the natural Nb bcc lattice is $\theta_{\beta}=54.74^{\circ}$. The rotation
angle of the reference lattice is approximately chosen as the average of
$\theta_{\alpha}$ and $\theta_{\beta}$, i.e.,
$\theta=(\theta_{\alpha}+\theta_{\beta})/2$. The three possible Burgers
vectors and the distortion matrices of this interface can be calculated by
rotations from the corresponding ones in the interface with the NW orientation
relationship, using Eqs. (10), (11) and (4), (13).

Figure 6: Dislocation structure of the Cu$(111)$/Nb$(110)$ interface in KS OR
calculated using our continuum model. The dislocation structure consists of a
parallelogram network of dislocation arrays with Burgers vectors
$\mathbf{b}_{1}$ and $\mathbf{b}_{3}$, shown by red and black lines,
respectively.

Simulation result using our continuum model shows that the dislocation
structure of this Cu$(111)$/Nb$(110)$ interface with KS orientation
relationship consists of a parallelogram network of two arrays of dislocations
with Burgers vectors $\mathbf{b}_{1}$ and $\mathbf{b}_{3}$ represented by the
reciprocal vectors $\mathbf{N}_{1}=(0.2173,-0.4118)/$nm and
$\mathbf{N}_{3}=(-0.7913,-0.0032)/$nm, respectively; see Fig. 6 for the
obtained dislocation network. These results of the continuum model mean that
the two arrays of dislocations have line directions
$\boldsymbol{\xi}_{1}=(-0.8844,-0.4667)$ and
$\boldsymbol{\xi}_{2}=(-0.0040,1.0000)$ with inter-dislocation distances
$D_{1}=2.148\rm{nm}$ and $D_{3}=1.264\rm{nm}$, respectively. Accordingly, the
angles between the two arrays of dislocations and the $x$ axis are
$-152.2^{\circ}$ and $90.2^{\circ}$, respectively. We compare this dislocation
structure with that obtained by atomistic simulation in Ref. [9] (Figs. 8 and
9 in Ref. [9]), and observe excellent agreement between the two results. In
the atomistic simulation result in Ref. [9], the inter-dislocation distances
in the two arrays of dislocations are $D_{1}=2.131\rm{nm}$,
$D_{3}=1.245\rm{nm}$ and the dislocation line directions have angles
$152^{\circ}$ and $90^{\circ}$ with respect to the $x$-axis, respectively (see
Table 2 in Ref. [9]. Notice that the $+z$ direction is pointing downward
there).

##  6 Conclusions

In summary, we have developed a continuum model for the dislocation structures
of semicoherent interfaces based on constrained energy minimization. In our
model, the dislocation structure of a semicoherent interface is obtained by
minimizing the energy of the equilibrium dislocation network with respect to
all the possible Burgers vectors, subject to the constraint of the Frank-Bilby
equation. Comparisons with atomistic simulation results and results of other
available models show that our continuum model is able to give excellent
predictions of dislocation structures on semicoherent interfaces.

## Acknowledgements

This work was supported by the Hong Kong Research Grants Council General
Research Fund through grant 16302818.

## Data availability

The datasets generated in study are available upon request.

## References

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