# Machine learning potential for the Cu-W system

Manura Liyanage<EMAIL_ADDRESS>Vladyslav Turlo  Laboratory for
Advanced Materials Processing, Empa - Swiss Federal Laboratories for Materials
Science and Technology, Thun, Switzerland  W. A. Curtin  National Centre for
Computational Design and Discovery of Novel Materials MARVEL, École
Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland  School of
Engineering, Brown University, Providence, RI 02906 USA

###### Abstract

Combining the excellent thermal and electrical properties of Cu with the high
abrasion resistance and thermal stability of W, Cu-W nanoparticle-reinforced
metal matrix composites and nano-multilayers (NMLs) are finding applications
as brazing fillers and shielding material for plasma and radiation. Due to the
large lattice mismatch between fcc Cu and bcc W, these systems have complex
interfaces that are beyond the scales suitable for ab initio methods, thus
motivating the development of chemically accurate interatomic potentials.
Here, a neural network potential (NNP) for Cu-W is developed within the
Behler-Parrinello framework using a curated training dataset that captures
metallurgically-relevant local atomic environments. The Cu-W NNP accurately
predicts (i) the metallurgical properties (elasticity, stacking faults,
dislocations, thermodynamic behavior) in elemental Cu and W, (ii) energies and
structures of Cu-W intermetallics and solid solutions, and (iii) a range of
fcc Cu/bcc W interfaces, and exhibits physically-reasonable behavior for solid
W/liquid Cu systems. As will be demonstrated in forthcoming work, this near-ab
initio-accurate NNP can be applied to understand complex phenomena involving
interface-driven processes and properties in Cu-W composites.

neural network potentials; copper-tungsten; nano-multilayers; molecular
dynamics

##  1 Introduction

The alloying of Cu, having high thermal and electrical conductivity, and W,
having low thermal expansion coefficient, high elastic moduli, high hardness,
good thermal stability, and abrasion resistance [1, 2, 3, 4, 5] leads to Cu-W
composite materials with outstanding properties [6, 7]. A highly positive heat
of formation between Cu and W makes them immiscible under equilibrium
conditions [1, 8, 9, 10, 4, 11], although direct alloying of Cu-W has been
realized through non-equilibrium processing techniques such as mechanical
alloying, ion beam mixing, and direct diffusion methods, all used to improve
the bonding strength of the Cu-W interfaces [3, 11]. Cu-W composites with
$\mathrm{\mu m}$-sized grains of Cu and W can be used in electric resistance
welding, electrical contacts, and electro-discharge machining electrodes [5,
12], heat sinks in fusion reactors [13, 14], as well as power packaging in
microelectronics and optoelectronics [15].

Nano-multilayers (NMLs) are a special kind of composite that can be
manufactured with physical vapor deposition methods and are characterized by a
large density of interfaces. Experimental and computational work on fcc-bcc
multilayers has shown the potential of NML by acting as obstacles to slip and
sinks for radiation-induced defects [16, 17, 18]. Cu-W NMLs specifically, have
strong potential to be used as brazing fillers and shielding material for
radiation and plasma [19, 20, 21, 22, 23, 24, 25, 7]. For brazing
applications, Cu-W NMLs take advantage of microstructural evolution to a
nanocomposite structure of W particles in a Cu matrix occurring at
$\sim$700-800 ∘C [20, 21, 22, 23, 24, 25, 26, 27] \- well below the melting
points of Cu and W, while for shielding applications the NML interfaces act as
sinks for radiation-generated defects [28, 29, 30, 31].

Experimental observations have confirmed a strong bias toward forming
interfaces with $\\{111\\}_{\mathrm{Cu}}||\\{110\\}_{\mathrm{W}}$ orientation
relationship as expected for NMLs made of fcc (face-centered cubic) and bcc
(body-centered cubic) metals with a strong fiber texture [32, 33, 34, 20, 21,
23, 35, 36]. These interfaces are expected to be highly incoherent due to the
large lattice mismatch between Cu and W. However, there are conflicting views
regarding the in-plane orientation relationship with the Nishiyama-Wasserman
(NW) [32, 37] and Kurdjumov-Sachs (KS) [29, 23, 25] misorientations, common
for bcc/fcc NMLs, reported as possible candidates. Ma  _et al._ [31] state
that the KS orientation relationship possesses the lowest possible interface
energy, which contradicts ab initio simulations in Bodlos  _et al._ [38] and a
previous machine learning potential (MLP) [39]. Intermixing and alloying at
the interface [40, 32, 41, 11] and the adsorption of bcc Cu on W surfaces (up
to 3 layers) [42] have also been observed in experiments; these highlight the
complex nature of Cu-W interfaces, with structural and chemical features
strongly affecting Cu-W NML properties.

The level of detail provided by experimental work at the micro and nanoscales
is limited in revealing the atomistic mechanisms occurring at Cu-W interfaces.
Although ab inito methods such as Density Functional Theory (DFT) have been
used to study and predict some behavior of Cu-W composites [38, 28, 27, 39]
and metastable intermetallics [10, 43], results are limited to small
computationally-feasible cell sizes that cannot capture more-realistic large
incoherent interfaces and nanometric layers. Semi-empirical potentials such as
the embedded atom method [44, 45] and the extended Finnis-Sinclair
formalism[46, 47] have thus been developed to model melting and diffusion at
Cu-W interfaces [11], Cu-W solid solutions [8, 10, 48], glass forming
capability [8], and interfacial reactions [1, 4]. However, the small number of
fitting parameters based on a limited DFT database composed of metastable
intermetallic compounds prevents such models from capturing a more
comprehensive range of chemical and structural features, such as interfacial
defects, alloying at the interfaces, and interactions between the interface
and defects in Cu or W.

Machine learning methods are now allowing the creation of machine-learned
interatomic potentials (MLPs) that provide near-DFT accuracy at computational
costs approaching traditional semi-empirical methods. Following successes in
other metals and alloys (Al-Cu [49]; Mg [50]; Al-Mg-Si [51]; Al-Mg-Cu-Zn [52];
Zr [53]), here we develop and validate an MLP using the neural network
formulation of Behler and Parinello [54, 55]. This Cu-W neural network
potential (NNP) captures complex Cu-W interfaces, solid solutions,
intermetallics, and important phenomena in fcc Cu and bcc W, including lattice
parameters, elastic parameters, stacking faults, surfaces, and dislocations.
We then show a few preliminary large-scale applications to Cu-W situations
relevant to NMLs and NCs.

While beyond the scope of this paper, forthcoming work will apply the NNP to
study a wide range of application-oriented issues in NMLs. Atomistic
simulations using the NNP will be shown to help understand intermixing, phase
stability, and observed large variations in interface stresses versus
processing conditions [56]. Similarly, the NNP will be used to clarify the
effects of layer thickness, in-plane stress, macroscopic porosity,
temperature, grain boundaries, and non-equilibrium vacancies in physical
vapor-deposited Cu/W nano-multilayers on Young’s modulus to explaining
contradictory experimental measurements versus deposition conditions and
resulting microstructures [57]. More broadly, similar NNPs could be developed
for other fcc/bcc systems that are of high interest as NML structures [58, 59,
16, 17, 18, 60].

The remainder of this paper is organized as follows. Section 2 presents the
formation of the reference dataset, a summary of the Behler-Parrinello
framework, and the selection of the descriptors for the NNP. The quality of
NNP for properties of elementary Cu and W is assessed in section 3.2. In
section 3.3, the behavior of the NNP for binary Cu-W systems, including
substitutions, metastable phases, Cu-W interfaces, and solid Cu/liquid W
interactions, is investigated. Conclusions are presented in section 4 along
with future applications of the new NNP.

##  2 Methods

The following sequence of steps needs to be followed with care to develop a
precise and useful NNP:

  * •

Development of a comprehensive reference database of structures, using ab
intio methods. This database should contain energy and force descriptions for
unitary and binary systems and be complete in terms of local atomic
environments encountered in possible applications;

  * •

Selection of a set of descriptors. These descriptors play a crucial role in
discerning the local atomic environments in the reference dataset. They should
be sufficient in describing these environments without causing overfitting, a
key consideration in the NNP development process;

  * •

Training of the NNP on the reference dataset using the selected descriptors by
optimizing the weights and biases in the neural network;

  * •

Meticulous and comprehensive assessment of the quality of the final NNP. This
includes thorough checks for overfitting and complex material behavior, such
as dislocations, phase stability, and interfaces.

Special care has to be taken to identify and avoid unphysical behavior while
complying with underlying theories and models. We will discuss how the above
steps were executed in this work to create the Cu-W NNP. Since the literature
has comprehensively addressed NNP generation in metal alloys [49, 50, 51, 52,
53], we discuss the framework only briefly.

###  2.1 Reference structures

The reference first-principles DFT dataset must be constructed carefully to
include all local atomic environments expected to be encountered in future
applications of the NNP. Marchand  _et al._ [49] has created an extensive
dataset for fcc Cu behavior Marchand  _et al._ [61] that captures lattice
parameters, equation of states ($\pm 20\%$ lattice parameter), elastic
behavior (for strains $\pm 0.2\%$), stacking faults, vacancies, interstitials,
and vicinal surfaces. We have created a similar dataset for W, capturing the
linear and non-linear behavior of the bcc W. For Cu-W solid solutions,
metastable alloys with space groups Bh, B3, L10, and L11 for the CuW
stoichiometry, space group C15 for Cu2W and CuW2 stoichiometries, and space
groups A15, C15, D03, D09, D019, D022, and L12 for Cu3W and CuW3
stoichiometries, along with the intermetallic structures available in the Open
Quantum Material Database (OQMD) database [62] are included in the equilibrium
configurations. Structures for the space groups A15, B2, D09, D019, D022, and
L12, identified as more likely, are studied under strains of 0.15% for the
Voigt directions. Cu and W are substituted in bcc W and fcc Cu, respectively,
as single atoms and pairs of atoms at 1st, 2nd, and 3rd nearest neighbor
distances. Cu-W interfaces are modeled in compact cells achievable with DFT
for $\\{100\\}_{\mathrm{Cu}}||\\{100\\}_{\mathrm{W}}$ (two orientations with
45 inclination), $\\{110\\}_{\mathrm{Cu}}||\\{110\\}_{\mathrm{W}}$,
$\\{111\\}_{\mathrm{Cu}}||\\{110\\}_{\mathrm{W}}$ (two orthogonal
orientations), and $\\{111\\}_{\mathrm{Cu}}||\\{111\\}_{\mathrm{W}}$
interfaces. We examine the equilibrium configurations and structures having
relative slip across the Cu-W interfaces. We also generate structures for the
experimentally observed $\\{111\\}_{\mathrm{Cu}}||\\{110\\}_{\mathrm{W}}$
orientations by swapping Cu and W atoms in the 1st, 2nd, and 3rd layers away
from the interfaces. We include structures relevant to thermal vibrations by
studying fcc Cu and bcc W with random atom displacements away from the
equilibrium positions. Random displacements have a normal distribution with a
mean of 0 and a standard deviation of 0.15 Å.

The reference dataset contains 4583 structures with 136,928 atomic
environments (78,436 around Cu atoms and 58,492 around W atoms) and 410,784
force components along the cartesian directions. The additional dataset
created in the current study is openly available on the Materials Cloud [63].

###  2.2 DFT methodology

First-principles calculations are performed with the DFT framework of Quantum
Espresso [64]. We adopt the DFT parameters in Marchand  _et al._ [49] to be
consistent with the available Cu reference dataset. The standard generalized
gradient approximation of Perdew  _et al._ [65] (GGA-PBE) described the
electron-electron exchange-correlation effects. Basis sets for Cu and W are
$3p$ $3d$ $4s$ and $5s$ $5p$ $5d$ $6s$ states, respectively. Plane-wave cutoff
energy of 544 eV (40 Ry) and a $\Gamma$-centered Monkhorst-Pack k-point mesh
[66] with a spacing of 1/80 rA-1 are used with a Gaussian integration scheme
with a broadening of 0.6 eV. The Broyden–Fletcher–Goldfarb–Shanno algorithm
optimization was used to ionic and simulation cell relaxations [67, 68, 69,
70].

###  2.3 Neural-network framework and symmetry functions

Various neural network frameworks for the creation of reference structures
(such as active learning), descriptors, and training algorithms are
extensively discussed in the literature [71]. Here, we use the Behler-
Parinello descriptors that represent the local atomic environments through
atomic pairs and triplets with radial and angular symmetry functions. The
open-source code n2p2 [72] is used to train the NNP and is then also
implemented in LAMMPS [73] in further studies and applications. We highlight
the main aspects here.

NNPs consider the energy of an atomic ensemble to be the aggregate of
individual atomic energies. Atomic energies depend primarily on their local
atomic environments, described by the atom-centered symmetry functions
selected based on the atomic coordinates of the reference dataset. Three types
of Behler-Parrinello SFs (radial ($G^{2}_{i}$) SFs for pairs; narrow angular
($G^{\textrm{3}}_{i}$); wide angular ($G^{\textrm{9}}_{i}$) SFs for triplets)
are utilized when developing the NNP. For a central atom $i$, these SFs have
the forms

$G_{i}^{2}=\sum_{j\neq i}e^{-\eta(r_{ij}-r_{s})^{2}}f_{c}\left(r_{ij}\right),$
(1) $\displaystyle G_{i}^{3}=$ $\displaystyle 2^{1-\zeta}\sum_{j,k\neq
i}(1+\lambda\cos{\theta_{ijk}})^{\zeta}e^{-\eta\left(r_{ij}^{2}+r_{ik}^{2}+r_{jk}^{2}\right)}$
(2) $\displaystyle\times
f_{c}\left(r_{ij}\right)f_{c}\left(r_{ik}\right)f_{c}\left(r_{jk}\right),$
$\displaystyle G_{i}^{9}=$ $\displaystyle 2^{1-\zeta}\sum_{j,k\neq
i}(1+\lambda\cos{\theta_{ijk}})^{\zeta}e^{-\eta\left(r_{ij}^{2}+r_{ik}^{2}\right)}$
(3) $\displaystyle\times f_{c}\left(r_{ij}\right)f_{c}\left(r_{ik}\right),$

where $r_{ij}=|\mathbf{r_{j}}-\mathbf{r_{i}}|$ is the radial distance between
atom $i$ and atom $j$ in the local environment and $\theta_{ijk}$ is the angle
between the vectors from atom $i$ to atoms $j$ and $k$, respectively,
capturing the geometries of the neighboring atoms in the local environments.
The hyperparameters $\eta$, $r_{s}$, $\zeta$, and $\lambda$ are selected
through sampling methods proposed in Gastegger  _et al._ [74] and Imbalzano
_et al._ [75]. $f_{c}$ is a cutoff function that reduces the radial components
to zero smoothly at a radial cutoff $r_{\textrm{c}}$, and here we use
$f_{c}(r)=\tanh^{3}\left(1-r/r_{\textrm{c}}\right)$. The most useful SFs
describing the structures in the reference dataset were then obtained with the
PCov-CUR (principle covariant regression CUR) algorithms [75, 76, 77].

With the descriptors (symmetry functions) identified, the energy $E_{i}$ of
atom $i$ in its local environment is calculated as

$\displaystyle E_{i}$
$\displaystyle=f_{1}^{3}\left\\{b_{1}^{3}+\sum_{k=1}^{M_{\mathrm{layer},2}}w_{n1}^{23}f_{n}^{2}\right.$
(4)
$\displaystyle\times\left.\left[b_{n}^{2}+\sum_{m=1}^{M_{\mathrm{layer},1}}w_{mn}^{12}f_{m}^{1}\left(b_{m}^{1}+\sum_{l=1}^{M_{\mathrm{sym}}}w_{lm}^{01}G_{i,l}\right)\right]\right\\},$

where $G_{i,l}$ is the value of the symmetry function around atom $i$ for the
$l^{th}$ set of hyperparameters, $f(\cdot)$ are the activation functions for
each node (for the current study $f_{m}$ and $f_{n}$ are softplus functions
and $f_{1}$ is the identity function), $M_{\mathrm{layer},p}$ is the number of
nodes in the $p^{th}$ hidden layer, $M_{\mathrm{sym}}$ is the number of SFs.
The weights $w^{uv}_{pq}$ connect the $p^{th}$ node in the $u^{th}$ layer to
the $q^{th}$ node in the $v^{th}$ layer and biases $b^{v}_{q}$ are added to
the $q^{th}$ node in the $v^{th}$ layer of the neural network. Training is
carried out to optimize the weights and biases of the neural network through
an iterative process to obtain the best fit to obtain the energy and forces
for the structures in the training dataset. The optimization minimizes the
loss function

$\Gamma=\sum^{N_{\mathrm{struct}}}_{n=1}(E_{\mathrm{NN}}^{n}-E_{\mathrm{ref}}^{n})^{2}+\beta^{2}\sum^{N_{\mathrm{struct}}}_{n=1}\sum_{m=1}^{3N_{\mathrm{atom}}^{n}}(F_{n,\mathrm{NN}}^{m}-F_{n,\mathrm{ref}}^{m})^{2},$
(5)

where $E$ and $F$ are the energies of the structures and the forces of the
atoms in each structure, respectively, with the subscripts “ref” and ‘NN” for
DFT reference values and the NNP predictions. The parameter $\beta$ (with Å
units) is used to scale the weight of the forces (which are in units of eV/Å)
relative to the energies (in eV).

Infinite combinations of symmetry functions are available to describe atoms’
local atomic neighborhoods. However, the selection will decide the accuracy,
efficiency, and possibility of overfitting. Hence, the symmetry functions need
to be chosen carefully. Symmetry functions in Marchand  _et al._ [49] for Cu
did not provide very good predictions regarding the interfaces. Hence, we used
$r_{c}=7.0$ Å, $\lambda=\left[1,-1\right]$, and $\zeta=\left[1,3,12,64\right]$
with a sampling of $\eta$ and $r_{s}$ as defined by Gastegger  _et al._ [74]
and Imbalzano  _et al._ [75]. This led to 326 symmetry functions each for Cu
and W, respectively. Using the PCov-CUR decomposition algorithms and
geometries of the input structures, we selected the 48 ”most useful” SFs
(independent SFs that least affect feature identification) for Cu and for W to
describe the local atomic environments in the training set.

The training uses a Kalman filter optimization carried out for a fixed number
of 400 epochs. The training dataset is selected as 90% of the reference
dataset. The remainder is kept to test the developed NNP for overfitting and
any atomic environment of the reference dataset unrepresented in the training
dataset. 2% of the force components from the training dataset were used for
training to prevent the force components (3 per atom) from overwhelming the
energies (one energy per structure), with 100% of the energies used to update
the NNP per each epoch. For the current NNP, $\beta=1$ Å is used for the force
weighting in eq. 5 such that a 1 meV/Å force error is given the same weight as
a 1 meV energy error.

##  3 Results and discussion

An NNP must be examined for indications of overfitting, outliers in the
testing dataset, and interplay of different atomic environments in the
configurational space of the reference dataset. Further tests are then
required to determine the suitability and accuracy of the NNP for any
metallurgical applications. 20 NNPs were trained using different initial
weights and biases, which didn’t show any outliers, deficiency in the
selection of the training dataset, or overfitting. For each NNP, a root mean
percentage square error (RMPSE) was calculated for the properties of fcc Cu,
bcc W, and Cu-W interface structures. The NNP designated as ”NNP01” in the
supplementary material has the lowest RMPSE and shows the best overall
behavior across the test configurations, and so it is the focus of our results
for the remainder of this paper. The properties calculated for all the 20 NNPs
are shown in the supplementary material.

###  3.1 Structures in the reference dataset

The distribution of errors of energies and forces predicted by the NNP for the
structures in the training test and testing set is shown in fig. 1. The
training and testing set errors are similar, with almost identical root mean
square errors (RMSE). The RMSE for energy is higher than in other studies [49,
50, 51, 52, 53], but still within a few meV/atom. Most structures with energy
errors $>5$ meV/atom comprise metastable Cu-W and W with large strains
($V/V_{0}>20\%$). The RMSE would be comparable to other studies if these
structures were neglected. Similar observations can be made for the force
errors, with all force errors $>$400 meV/Å corresponding to structures far
from their equilibrium states, specifically unrelaxed structures with
interstitials and structures with atoms displacements far from their relaxed
configurations. Similar observations are reported in other studies, with
minimal effect on the linear and non-linear mechanical behavior [53].

(a) Errors in energy

(b) Error in forces

Figure 1: Histogram of errors for energies and forces of the structures in the
training and testing datasets as predicted by the NNP. Values corresponding to
the training and testing datasets are shown in blue and red, respectively.

Further tests to examine the NNP performance versus DFT and analytical results
are discussed in the following section.

###  3.2 Elemental Cu and W

Table 1: Comparison of basic mechanical properties of Cu and W predicted by the NNP and by DFT. Property | Cu | W
---|---|---
DFT | NNP | DFT | NNP
Lattice properties
a (Å) | 3.625 | 3.626 | 3.185 | 3.186
V/atom (Å3) | 11.904 | 11.914 | 16.155 | 16.164
Ecoh (eV/atom) | -3.781 | -3.779 | -10.627 | -10.628
Elastic Properties (GPa)
C11 | 180 | 172 | 535 | 528
C12 | 125 | 127 | 208 | 199
C44 | 80 | 72 | 150 | 151
Surface energy (J/m2)
{100} | 1.486 | 1.481 | 4.002 | 3.999
{110} | 1.571 | 1.570 | 3.244 | 3.318
{111} | 1.329 | 1.323 | 3.563 | 3.641
Other properties
Evac (eV) | 1.088 | 1.020 | 3.308 | 3.512
ESIA \- octahedral (eV) | 3.332 | 3.235 | 12.153 | 11.992
ESIA \- tetrahedral (eV) | 3.716 | 3.525 | 11.565 | 11.053

(a) fcc Cu

(b) bcc W

Figure 2: Equation of states obtained for fcc Cu and bcc W with DFT and the
NNP

The NNP shows excellent agreement with DFT for the properties shown in table 1
with most errors $<10$%. The lattice parameters, most of the elastic
properties, and surface energies are in excellent agreement between DFT and
The NNP. The shear modulus C44 for Cu shows the most significant error of 11%.
However, this is a difficult property to predict accurately with NNPs, with
other studies reporting more significant errors [49]. The vacancy formation
energies (Evac) and self-interstitial energies (ESIA) show good agreement with
the DFT values with maximum error $\sim$5%. This indicates that the NNP models
vacancies and interstitials with reasonable accuracy at the equilibrium
states, even though the predicted forces of the non-equilibrium structures in
the reference dataset have a noteworthy error. Figure 2 shows the equation of
states for the fcc and bcc phases of Cu and W, respectively. The agreement is
excellent for an extensive range of strains ($a/a_{0}=\pm$20%).

(a) fcc Cu

(b) bcc W

Figure 3: Decohesion curves for $\\{100\\}$,$\\{110\\}$, and $\\{111\\}$
families of planes for fcc Cu and bcc W comparing DFT and the NNP. Results
from DFT are shown in dashed lines with $\times$, and results from the NNP are
shown in solid lines. All DFT structures here are included in the reference
dataset, with the non-DFT results corresponding to relaxed structures using
the NNP

(a) (111) plane in fcc Cu with Burgers vector $\mathbf{b}=a/2\langle
11\bar{2}\rangle$

(b) (110) plane in bcc W with Burgers vector $\mathbf{b}=a/2\langle
11\bar{1}\rangle$

Figure 4: GSFE curves for the (111) plane in fcc Cu and the (110) plane in bcc
W; comparing DFT and the NNP developed in the current study. * shows the
location and energy of the stable stacking fault in fcc Cu. All DFT structures
here are contained in the training dataset, with the non-DFT results
corresponding to relaxed structures using the NNP

Decohesion and stacking fault energies are important in metallurgical
applications to study plasticity and fracture. Figure 3 shows the decohesion
curves for fcc Cu and bcc W for the $\\{100\\}$,$\\{110\\}$, and $\\{111\\}$
families of planes. The agreement between DFT and the NNP is excellent,
especially for small displacements ($<$2 Å). There are variances of $\sim 2\%$
for large displacements, which have no significant effects in any application.
The Generalized Stacking Fault Energies (GSFE) for $\\{100\\}$, $\\{110\\}$,
and $\\{111\\}$ planes for fcc Cu and bcc W are included in the training data,
and overall agree well with DFT. The most relevant cases for dislocations are
the $\\{111\\}$ planes in Cu shown in fig. 4a and the $\\{110\\}$ planes in W
shown in fig. 4b. For Cu, the NNP unstable stacking fault energy is $\sim$1.5%
lower than the DFT value, which is sufficient accuracy for modeling
dislocation emission from crack tips. However, the NNP stable stacking fault
energy (SSFE) is $\sim 40\%$ lower than the DFT value, which leads to a larger
partial dislocation dissociation distance for the lattice dislocations (see
below) and can strongly inhibit cross-slip. Previous results for the Cu SSFE
for NNPs developed using only the Cu dataset are 15-35% lower than the DFT
value [52], so training here with the addition of W leads to slightly worse
agreement. For W, the unstable stacking fault energy estimated by NNP is just
5% higher than the DFT value, which is sufficiently accurate to capture the
emission behavior of crack tips.

The accurate modeling of dislocation core structures is essential in
understanding the plastic behavior of metals. Dislocation core structures
combine atomic environments related to elastic and stacking fault
deformations. The main operative slip planes are the most densely packed
planes ($\\{111\\}$ for fcc Cu; $\\{110\\}$ for bcc W) and so we examine the
edge and screw dislocations for these slip systems. Initial dislocation
structures are created using the anisotropic Volterra elastic displacement
field [78] in cuboidal simulation cells with $30\times 30$ Å2 in-plane
dimensions and one periodic unit along the dislocation line direction. For
dislocations in fcc Cu, several initial partial separation distances are used
(up to 60 Å in steps of 5 Å). Atoms in a 10 Å layer along the outer boundary
are fixed and the remaining atoms relaxed to the minimum-energy configuration.
The relaxed dislocation core structures were analyzed with the Nye tensor [79,
80], differential displacement maps [81], and disregistry along the slip plane
using the atomman python package [82]. Gradients of the disregistries enable
understanding Burgers vector distribution along slip planes and provide an
estimate of the partial separation distances [83]. A theoretical estimate for
the partial separation distance can be obtained using the elastic properties
and stable stacking fault energy along the slip direction as

$\mathrm{d}_{\mathrm{theory}}=\frac{\mathbf{b}^{(1)}_{i}\mathbf{K}_{ij}\mathbf{b}^{(2)}_{j}}{2\pi\gamma_{\mathrm{ssf}}},$
(6)

where $\mathbf{b}^{(1)}$ and $\mathbf{b}^{(1)}$ are the Burgers vectors of the
partial dislocations, $\mathbf{K}$ is the Stroh matrix [84, 85], and
$\gamma_{\mathrm{ssf}}$ is the stable stacking fault energy of the respective
plane.

The stable dislocation cores are shown in figs. 5 and 6 and exhibit
physically-reasonable structures similar to those obtained previously using
other potentials and/or DFT. The dissociation distance in Cu agrees well with
the theory, as indicated, computed using the NNP properties. Using the DFT
properties, dtheory is $/sim$ 40 and 12 Å for edge and screw dislocations,
respectively, which is much smaller than the NNP-derived values due to the NNP
error in the SSFE. For W, the screw dislocation shows the expected compact
symmetric structure and the edge dislocation shows a small spreading in the
glide plane. The NNPs are thus broadly suitable for modeling dislocation
plasticity in Cu and W.

(a) Edge dislocation

(b) Screw dislocation

Figure 5: Atomistic core structures for $\mathbf{b}=0.5\langle
1\bar{1}0\rangle$ dislocations of $\\{111\\}$ planes in fcc Cu for the NNP,
with the Nye tensors (for the dominant direction) and differential
displacement maps overlaid. Green crosses indicate the centers of the partial
dislocations identified by the disregistry plots. Theoretical and observed
dislocation separation distances are indicated as $d_{\mathrm{theory}}$ and
$d_{\mathrm{observed}}$ respectively.

(a) Edge dislocation

(b) Screw dislocation

Figure 6: Atomistic core structures for
$\mathbf{b}=0.5\langle\bar{1}11\rangle$ dislocations of $\\{110\\}$ planes in
bcc W for the NNP, with the Nye tensor (for the dominant direction) and
differential displacement maps overlaid. Green crosses indicate the centers of
dislocations.

Due to the high temperatures involved in the manufacturing of Cu-W MMCs,
accurate thermodynamic properties of Cu and W are essential for many
applications. Various properties are estimated using periodic arrangements of
$8\times 8\times 8$ simulation cells (fcc and bcc unit cells for Cu and W,
respectively) and executing molecular dynamics simulations in the
isothermal–isobaric ensemble with a time step of 0.5 fs. Before evaluating any
properties, a system is equilibrated for 50,000 time steps (25 ps). Melting
points are estimated using the two-phase method, where the migration of the
solid/liquid interface is monitored, with no net motion at the melting
temperature. The heat of fusion EHF is the energy absorbed to change the metal
from solid to liquid at the melting temperature. The thermal expansion
coefficient of the crystalline phase at T=300K was determined using the
lattice constants measured in simulations at 250 K, 300 K, and 350 K. The
above simulated properties are compared with experimental observations in
table 2, showing good agreement. The melting points of Cu and W are estimated
as $1245\pm 5$ K and $3537\pm 13$ K, respectively, which are 100-150 K lower
than experimental values. The NNP estimates of EHF show errors of $\sim 20\%$
and the thermal expansion values differ by roughly similar factors. Overall,
these estimates are satisfactory since no liquid or high-temperature
configurations are included in the reference training dataset.

Table 2: Comparison of the thermodynamic properties obtained for the trained NNP potential for Cu and W with the corresponding experimental properties. Property | Cu | W
---|---|---
Exp. | NNP | Exp. | NNP
Tm (K) | 1358 | 1245 | 3687 | 3538
EHF (kJ mol-1) | 13.26 111Dean [86] | 11.51 | 52.31 111Dean [86]222Lide [87] | 63.93
$\alpha_{300K}$ ($\times 10^{-6}$) | 16.76 333Wang and Reeber [88] | 20.4 | 4.6 444Hidnert and Sweeney [89] | 4.3

###  3.3 Cu-W systems

Cu and W mainly form metal composites due to the high positive mixing enthalpy
under equilibrium conditions, possibly with complex interfaces [40, 42, 32,
41, 11]. However, Cu-W can form metastable intermetallic structures under non-
equilibrium conditions [1, 8, 9, 10, 4, 11]. Some manufacturing processes
involve liquid Cu and crystalline W. To assess the use of the NNP for
applications across relevant situations arising in the processing and
applicatin of Cu-W systems, we examine NNP predictions for (i) substitution of
Cu in W and W in Cu, (ii) the properties of many metastable alloys, (iii)
solid-state interfaces between fcc Cu and bcc W, and (iv) interactions between
solid W and liquid Cu.

####  3.3.1 Solute atoms

Although Cu and W are largely immiscible, some mixing has been observed in
Cu-W NMLs at high temperatures. The solute solution energy energy
E$\mathrm{{}^{1sol}_{X}}$ (X=Cu or W) is defined as

$\mathrm{E^{1sol}_{X}}=\mathrm{E^{1sol}_{(N-1)Y,X}}-\mathrm{E_{X}}-\mathrm{(N-1)E_{Y}}$
(7)

where E${}_{\mathrm{{}^{1sol}_{(N-1)Y,X}}}$ is the total energy of the
equilibrium simulation cell containing a single solute and N-1 atoms of the
solvent Y=W or Cu. EX and EY are the energies of solute and solvent atoms in
their equilibrium phases (fcc Cu or bcc W). Solute-solute interaction, or
binding, energies E${}^{\mathrm{bind}}_{\mathrm{X}}$ are also examined for
solute pairs (of X=Cu or W) at first, second, and third nearest-neighbor
locations (1 NN, 2 NN, and 3 NN, respectively), defined as

$\mathrm{E^{bind}_{X}}=\mathrm{E^{2sol}_{(N-2)Y,2X}}-2\mathrm{E^{1sol}_{(N-1)Y,X}}+\mathrm{N\
E_{Y}}$ (8)

with the same definitions as eq. 7. The necessary energies are computed using
fully-periodic $5\times 5\times 5$ simulation cells (fcc and bcc unit cells
for Cu and W, respectively). A negative E${}^{\mathrm{bind}}_{\mathrm{X}}$
indicates attractions. Results for E$\mathrm{{}^{1sol}_{X}}$ and
E${}^{\mathrm{bind}}_{\mathrm{X}}$ obtained with the NNP and DFT are shown in
table 3. The agreement between DFT and the NNP is good, with the high positive
values of E$\mathrm{{}^{1sol}_{X}}$ consistent with the high immiscibility.
Correspondingly, the solute-solute interactions are strongly attractive at
first neighbors, consistent with the high immiscibility that drives phase
separation into Cu and W. The NNP binding energies at 2 NN and 3 NN show some
differences with DFT (including cases with a change in sign), but overall, the
values are very small compared to the 1 NN attraction and so have no
significant consequences on physical behavior.

Table 3: E$\mathrm{{}^{1sol}_{X}}$ and E${}^{\mathrm{bind}}_{\mathrm{X}}$ obtained with the NNP for substitution of Cu in bcc W and W in fcc Cu in eV Energy | Cu in bcc W | W in fcc Cu
---|---|---
DFT | NNP | DFT | NNP
E$\mathrm{{}^{1sol}_{X}}$ | 1.464 | 1.568 | 1.876 | 1.816
E${}^{\mathrm{bind}}_{\mathrm{X}}$ |  |  |  |
1 NN | -0.336 | -0.479 | -0.823 | -0.728
2 NN | 0.070 | 0.058 | 0.016 | -0.041
3 NN | 0.070 | -0.081 | -0.082 | -0.017

####  3.3.2 Metastable intermetallic compounds and solid solutions

(a) A15 Cu3W

(b) B2 CuW

(c) D019 Cu3W

(d) D022 Cu3W

(e) L10 CuW

(f) L12 Cu3W

Figure 7: Atomic arrangements for possible metastable Cu-W alloys. Cu and W
are shown in gold and cyan color spheres, respectively. For (a), (c), (d), and
(f), structures with CuW3 stoichiometry are obtained by inverting the Cu and W
atoms.

Intermetallics of Cu and W are meticulously achieved through non-equilibrium
techniques such as ion beam mixing [9]. We have investigated the structures
depicted in fig. 7, which are candidates for ordered metastable alloys in Cu-W
[9, 10, 90]. These structures are included in the reference training dataset.
We have computed the heat of formation ($\mathrm{\Delta H}_{f}$) versus alloy
volume, where $\mathrm{\Delta H}_{f}$ is calculated as

$\mathrm{\Delta
H}_{f}=\frac{\mathrm{E}_{\mathrm{Cu}_{x}\mathrm{W}_{y}}-x\mathrm{E}_{\mathrm{Cu}}-y\mathrm{E}_{\mathrm{W}}}{x+y}$
(9)

where E${}_{\mathrm{Cu}_{x}\mathrm{W}_{y}}$ is the energy of a metastable Cu-W
solid solution, and ECu and EW are the energies of atomic Cu and W in their
equilibrium bulk configuration, fcc and bcc for Cu and W, respectively. The
NNP aligns remarkably well with DFT, as shown in fig. 8. For Cu3W, D019 is the
most stable structure closely followed by D022. For CuW, fct L12 shows
significant stability relative to bcc-like B2. For CuW$3$, $\mathrm{\Delta
H}_{f}$ is lower than for other compositions and all structures are much
closer in formation energy with A15 being lowest. This provides confidence in
applications of the NNP to experimentally-observed interface diffusion and
intermixing at the interfaces [40, 32, 41, 11].

(a) Cu3W

(b) CuW

(c) CuW3

Figure 8: The heats of formation ($\mathrm{\Delta H}_{f}$)-volume curves for
the metastable phases. DFT values are shown with the thick continuous lines,
and the NNP values are shown in the dashed lines. The volumes correspond to
the volumes of structures in fig. 7.

Metastable solid solutions of Cu-W are mainly observed in Cu-rich fcc
structures and W-rich bcc structures. Experimentally, the bcc$\rightarrow$fcc
transformation is observed for $>70$% Cu. We thus examined random CuxW100-x
solid solutions in both fcc and bcc structures. Simulation cell sizes of
$10\times 10\times 10$ unit cells were relaxed isotropically to preserve the
initial crystal structure. For $x<20$% Cu, the system was unstable in the fcc
phase and so the simulation size was reduced to $3\times 3\times 3$ unit cells
to retain the fcc phase. The computed $\mathrm{\Delta H}_{f}$ are shown in
fig. 9. The fcc phase becomes energetically stable for $>80$% Cu, in
reasonable agreement with experiments. However, the NNP shows some differences
versus the DFT-computed $\mathrm{\Delta H}_{f}$ by Liang  _et al._ [43] using
single realizations of 16-atom special quasirandom structure (SQS) (see fig.
9). DFT shows the bcc/fcc transition $60$% Cu, also differing slightly from
experiments. More broadly, the bcc-fcc energy difference versus composition
differs between NNP and DFT, especially in the range $20-70$% Cu, but the NNP
shows good agreement for the formation energies of bcc Cu2W14 (12.5% Cu atm)
and Cu14W2 (87.5% Cu atm) Liang  _et al._ [43]. Some differences between NNP
and DFT can be due to the small size of the DFT simulation cells.

Overall, we conclude that the NNP should be very useful in studying many
aspects of metastable Cu-W alloy behavior and phenomena.

Figure 9: $\mathrm{\Delta H}_{f}$ for disordered CuxW100-x solid solutions
obtained with the NNP. Data points for $10\times 10\times 10$ and $3\times
3\times 3$ simulation sizes are shown in * and $\times$ respectively. The
best-fit lines for these data points are also included. $\mathrm{\Delta
H}_{f}$ for DFT calculations from Liang  _et al._ [43] are also shown here
with values for structures obtained with the special quasirandom structure
(SQS) method are shown * and cluster expansion values are shown in dashed
lines.

####  3.3.3 Cu-W interface structures

Cu-W interfaces are the most critical features in binary Cu and W since they
exist primarily as immiscible composites under equilibrium conditions. The
higher density of interfaces in NML systems can significantly affect the
properties of the composites; for example, Cu-W NML composites have been
observed to degrade when annealed to temperatures $>400\ ^{\circ}$C. Further,
the experimentally observed interface stresses at Cu-W interfaces show a
variation that exceeds expectations of sharp interfaces [39]. Understanding
the underlying atomistic mechanisms is critical in realizing possible
applications of these NMLs. Highly incoherent interfaces and nanometric layer
thicknesses beyond 2 nm make first-principles simulations impractical [39],
and hence Ab initio calculations in the literature were carried out using very
modest-size systems with low complexity.

The interface structures in the reference dataset were fully relaxed (cell
shape; ionic coordinates) with DFT and the NNP. Six different interfaces were
considered, with the notation “Interface $x$” with $x=1,2,3,4,5,6$, for the
interface planes and in-plane dimensions of the simulation cells shown in
table 4. Interfaces 4 and 5 are more critical due to the
$(111)_{\mathrm{Cu}}//(110)_{\mathrm{W}}$ texture observed in the experiments
[32, 33, 34, 20, 21, 23, 35, 36] with Interface 4 in the NW orientation and
Interface 5 rotated 90∘ from the NW orientation. The KS orientation is between
interfaces 4 and 5, with an inclination of 5.26∘ from interface 4 (NW
orientation). All interface studies use 8 atomic planes each for layers with
$\\{100\\}$ and $\\{110\\}$ surfaces and 12 atomic planes for layers with
$\\{111\\}$ surfaces. The misfit strains of Cu and W
($\varepsilon_{\mathrm{Cu}}$ and $\varepsilon_{\mathrm{W}}$ for the in-plane
x,y directions in table 4, with tensile strains taken as positive) and the
interface energy (EIE) were compared between DFT and the NNP (see table 5).
EIE was determined as

$\mathrm{E_{IE}}=\frac{\mathrm{E_{Cu-W}}-\mathrm{E^{\prime}_{Cu}}-\mathrm{E^{\prime}_{W}}}{\mathrm{2A}}$
(10)

where $\mathrm{E_{Cu-W}}$ is the energy of the simulation cell structure
containing the interface and $\mathrm{E^{\prime}_{Cu}}$ and
$\mathrm{E^{\prime}_{W}}$ are the energies of bulk Cu and W strained to the
dimensions of the relaxed cell containing two interfaces each with area A.

Table 4: Interface planes and the in-plane dimensions of the simulation cells for the interface structures used in training the NNP. The planes and the dimensions use the Miller index notations of the constituent fcc Cu and bcc W layers. Interface | Interface plane - z | In-plane dimensions
---|---|---
x | y
Interface 1 | $(100)_{\mathrm{Cu}}//(100)_{\mathrm{W}}$ | $\langle 010\rangle_{\mathrm{Cu}}//\langle 010\rangle_{\mathrm{W}}$ | $\langle 001\rangle_{\mathrm{Cu}}//\langle 001\rangle_{\mathrm{W}}$
Interface 2 | $(100)_{\mathrm{Cu}}//(100)_{\mathrm{W}}$ | $\langle 010\rangle_{\mathrm{Cu}}//\langle 011\rangle_{\mathrm{W}}$ | $\langle 001\rangle_{\mathrm{Cu}}//\langle 0\bar{1}1\rangle_{\mathrm{W}}$
Interface 3 | $(110)_{\mathrm{Cu}}//(110)_{\mathrm{W}}$ | $\langle 001\rangle_{\mathrm{Cu}}//\langle 001\rangle_{\mathrm{W}}$ | $\langle 1\bar{1}0\rangle_{\mathrm{Cu}}//\langle 1\bar{1}0\rangle_{\mathrm{W}}$
Interface 4 | $(111)_{\mathrm{Cu}}//(110)_{\mathrm{W}}$ | 0.5$\langle\bar{1}10\rangle_{\mathrm{Cu}}//\langle 001\rangle_{\mathrm{W}}$ | 0.5$\langle 11\bar{2}\rangle_{\mathrm{Cu}}//\langle 1\bar{1}0\rangle_{\mathrm{W}}$
Interface 5 | $(111)_{\mathrm{Cu}}//(110)_{\mathrm{W}}$ | $\langle 11\bar{2}\rangle_{\mathrm{Cu}}//\langle 001\rangle_{\mathrm{W}}$ | $\langle\bar{1}10\rangle_{\mathrm{Cu}}//\langle 1\bar{1}0\rangle_{\mathrm{W}}$
Interface 6 | $(111)_{\mathrm{Cu}}//(111)_{\mathrm{W}}$ | 0.5$\langle 11\bar{2}\rangle_{\mathrm{Cu}}//\langle 1\bar{1}0\rangle_{\mathrm{W}}$ | $1.5\langle 1\bar{1}0\rangle_{\mathrm{Cu}}//\langle 11\bar{2}\rangle_{\mathrm{W}}$
Table 5: Comparison of the interface energies and the misfit strains obtained for the NNP with the corresponding properties obtained from DFT calculations Interface | EIE (mj/m-2) |  $\varepsilon_{\mathrm{Cu,x}}$ (%) |  $\varepsilon_{\mathrm{Cu,y}}$ (%) |  $\varepsilon_{\mathrm{W,x}}$ (%) |  $\varepsilon_{\mathrm{W,y}}$ (%)
---|---|---|---|---|---
DFT | NNP | DFT | NNP | DFT | NNP | DFT | NNP | DFT | NNP
Interface 1 | 1979 | 1973 | -6.8 | -6.2 | -6.8 | -6.2 | 6.1 | 6.7 | 6.1 | 6.7
Interface 2 | 1189 | 1258 | 22.6 | 22.2 | 22.6 | 22.2 | -1.3 | -1.7 | -1.3 | -1.7
Interface 3 | 1637 | 1474 | -14.6 | -15.4 | -7.6 | -6.9 | -2.8 | -3.7 | 5.1 | 5.9
Interface 4 | 844 | 830 | 19.4 | 19.8 | -0.6 | -0.5 | -3.9 | -3.6 | -2.0 | -1.9
Interface 5 | 2255 | 2189 | -5.0 | -4.8 | -1.2 | 0.3 | 8.1 | 8.3 | 37.7 | 39.7
Interface 6 | 2325 | 2296 | 0.3 | 0.6 | 0.3 | 0.6 | -1.2 | -0.8 | -1.2 | -0.8

(a) Interface 4 - DFT

(b) Interface 4 - NNP

(c) Interface 5 - DFT

(d) Interface 5 - NNP

Figure 10: Slip energy surfaces for interfaces 4 and 5 obtained with DFT and
the NNP. The energies were obtained for rigid slips at the interface.

The largest percentage error is for the interface energies for interfaces 2
and 3, showing 5% -10%). However, the agreement for the more critical
Interfaces 4 and 5 is excellent. There is also very good agreement between DFT
and NNP for the misfit strains in all the interfaces. Due to the modest sizes
used with DFT, the misfit strains are extremely large. The only reasonable
misfit strains are for Interface 6, which has not been observed in Cu-W
systems. Due to the high misfit strains in these interfaces, real multilayer
structures will be incoherent. Information on local atomic environments in
these environments was captured in the reference dataset by providing rigid
relative displacements at the interfaces. We compute the slip energy surfaces
for interfaces 4 and 5, which captures the behavior of the common
$\\{111\\}_{\mathrm{Cu}}//\\{110\\}_{\mathrm{W}}$ interfaces, with the NNP and
DFT and find excellent agreement, as shown in fig. 10. Furthermore, the NNP
energy surfaces are smooth and continuous.

The interface structures in the reference dataset have high misfit strains
(the largest being 37%-40% strain in the W layer for interface 5) due to the
modest size of the simulation cells used to compensate for the computational
costs of DFT. However, realistic interfaces will have much smaller misfit
strains to reduce the strain energy of the Cu and W layers, which can have
thicknesses in the nm range. To further validate the ability of the NNP to
model $\\{111\\}_{\mathrm{Cu}}//\\{110\\}_{\mathrm{W}}$ interfaces, we compare
with results from Bodlos  _et al._ [38], who simulated
$\\{111\\}_{\mathrm{Cu}}//\\{110\\}_{\mathrm{W}}$ interfaces with DFT for
larger in-plane dimensions for the NW and KS orientations, albeit with a
single Cu layer and a single W layer connected through a single interface.
Each layer was six atomic planes thick, while the surfaces opposite the
interface faced vacuums. Using the NNP, the misfit strains, interface energies
(EIE), and work of separation (EWoS), and the results are shown in table 6
along with the DFT results of Bodlos  _et al._ [38] (not used in training).
The EIE and EWoS are in good agreement, and some errors could be due to
differences in the DFT parameters Bodlos  _et al._ [38] versus those used here
for training the NNP. The atomic arrangements adjacent to the interfaces for
these structures as computed by the NNP are shown in fig. 11a and fig. 11b for
the KS and NW orientations, respectively. These arrangements match the DFT
results in Bodlos  _et al._ [38]. These results further confirm the ability of
the NNP to model the interfaces with close to DFT accuracy.

(a) KS orientation for the cells size in Bodlos  _et al._ [38]

(b) NW orientation

(c) KS orientation for the cells size with lower misfit strain

Figure 11: Atomic arrangements of the interfaces obtained with the NNP. Only the two atomic layers adjacent to the interfaces are shown for clarity, with the repeating unit cell identified by with the red square. Table 6: EIE, EWoS, $\varepsilon_{\mathrm{Cu}}$, and $\varepsilon_{\mathrm{W}}$, obtained for KS and NW interfaces with NNP. DFT values from Bodlos  _et al._ [38] are shown when available. Property | KS interface | NW interface
---|---|---
DFT [38] | NNP111For the simulation cell size in Bodlos  _et al._ [38] | NNP222For the simulation cell size with lower misfit strain | DFT [38] | NNP111For the simulation cell size in Bodlos  _et al._ [38] | NNP222For the simulation cell size with lower misfit strain
EIE (mJ/m2) | 1320 | 1278 | 1392 | 1170 | 1254 | 1289
EWoS (mJ/m2) | 3220 | 3214 | - | 3400 | 3275 | -
$\varepsilon_{\mathrm{Cu}}(x,y)$ (%) | - | 5.0,2.9 | -0.2,0.2 | - | 0.6,-2.1 | -0.8,1.2
$\varepsilon_{\mathrm{W}}(x,y)$ (%) | - | -2.4,-2.4 | -0.1,-0.2 | - | -0.8,-1.5 | -0.1,-0.3
|  |  |  |  |  |

The misfit strains of these structures (see table 6) could still be quite
large compared to realistic systems. This can cause considerable strain
energies, especially since the layer thicknesses of realistic NMLs are large
(in the nm range). Therefore, we simulated more realistic NMLs for NW and KS
interfaces with alternating Cu and W layers of $\sim 5$ nm thicknesses. The
misfit strains for these structures are reduced compared to the structures
from Bodlos  _et al._ [38]. For the NW interface, where only the layer
thicknesses and periodicity normal to the interfaces are changed, there is a
marginal increase in the interface energy (2.8%) and no significant change in
structure compared to the difference from fig. 11b. However, the EIE for the
KS orientation is noticeably increased (9%) and the atomic arrangement
adjacent to the KS interface in the larger simulation cell is more complex
(see fig. 11c) as compared to fig. 11a. These changes in interface structure
and misfit strains in the layers due to the use of realistic layer thickness
demonstrate that understanding complex interface behavior in real systems,
including microstructure evolution (from NML to nanocomposite), intermixing,
and delamination, requires such large sizes, which in turn requires the use of
the NNP.

With stability, near-DFT accuracy, and reasonable computational costs, this
Cu/W NNP is poised to enable the identification of the underlying atomistic
origins of interface structures, energies, and stresses. Progress in this
direction is discussed in forthcoming work [56].

####  3.3.4 Interactions between solid W and liquid Cu

Due to a positive enthalpy of mixing, the formation of Cu and W solutions is
discouraged. Differences in the melting points of Cu and W facilitate the
infiltration of liquid Cu into porous W matrices when manufacturing Cu-W MMCs.
Hence, we investigate the ability of the NNP to simulate solid W/liquid Cu
interfaces. We study four configurations: (i) a liquid Cu layer ($\sim 15$ Å
thick) sandwiched between bcc W layers with $\\{110\\}$ surfaces at 1600 K;
(ii) a spherical W precipitate (radius of 20 Å) suspended in liquid Cu at 1600
K; (iii) separation of bcc W in an initial liquid Cu-W mixture (95% Cu and 5%
W) at 1600 K; and (iv) wetting angles for Cu in a W (110) surface at 1350 K.
Simulations for configurations (i) and (ii) use the NPT ensemble with 0.0005
ps timestep for 150 ps with Cu and W in liquid and solid states, respectively.
Figure 12 shows snapshots of typical arrangements after the systems reach
equilibrium. We see no changes in the Cu or W phases during the simulations.
The NNP simulates these structures well, with small deviations around the mean
and no irregular energy variations.

(a) Liquid Cu sandwiched between $\\{110\\}$ W surfaces

(b) W precipitate suspended in liquid Cu

Figure 12: Typical atomic arrangements observed for liquid Cu and solid W
systems under equilibrium. Gold and cyan circles represent Cu and W,
respectively.

The separation of bcc W from the Cu-W liquid mixture is initiated by
introducing a small spherical bcc W precipitate (radius of 10 Å) to a liquid
Cu-W mixture (5% W). The simulation is executed in the NPT ensemble with 0.002
ps timestep at 1600 K for a period of 8000 ps. Figure 13 shows snapshots of
the system at different times. Initially, the W atoms aggregate to form
smaller precipitates. As the system evolves, the smaller precipitates
aggregate to form larger aggregates while the initial bcc W precipitate grows.
The crystal structures of the small precipitates cannot be classified, which
may be due to artifacts in the potential or simply the large number of
interface atoms. However, the bcc precipitate grows steadily while the system
energy is decreasing. The surfaces of the final bcc W precipitate are
dominated by $\\{110\\}$ planes, as expected, since this is the lowest-energy
surface of bcc W.

(a) t = 0

(b) t = 400 ps

(c) t = 4000 ps

(d) t = 8000 ps

Figure 13: Precipitation of W in a liquid mixture of Cu and W. Only the W
atoms are shown here. Blue, red, and yellow circles represent atoms in bcc,
hcp, and icosahedral crystal structures, respectively, with white circles
representing atoms that cannot be classified to any structural system as
identified by common neighbor analysis [91, 92]

Finally, we simulate the wetting angle of a Cu droplet on a (110) W surface. A
spherical Cu droplet with 8000 atoms is placed on a $\sim 15$ Å thick W layer
with $\sim 190\times 190$ Å2 surface area. The simulation is executed in the
NPT ensemble with 0.002 ps timestep at 1350 K for 350 ps. The system reaches a
quasi-equilibrium state in 40 ps. Figure 14 shows the probability distribution
of Cu atoms for 5 Å thick layers in the orthogonal $\langle 001\rangle$ and
$\langle\bar{1}10\rangle$ directions in the (110) W surface. The wetting
angles for the two directions are very similar (103.5∘; 101.5∘) but higher
than the observed stabilized wetting angle [93]. The system also forms a one-
atom-thick “precursor” Cu film spread over the entire W (110) surface, for
which there is no evidence in Cu-W.

(a) Normal to $\langle 001\rangle$ in the W surface

(b) Normal to $\langle\bar{1}10\rangle$ in the W surface

Figure 14: The probability distribution of Cu atoms for a 5 Å thick layer at
the center of mass of the Cu droplet. The space was meshed with $0.5\times
0.5$ Å2 units. A circle was fitted to the surface of the Cu droplet with the
blue color dashed curve.

The Cu monolayer on (110) W may be driven by a Cu adsorption energy that is
stronger in the NNP then in DFT. The adsorption energy of a single Cu atom on
the (110) surface (Eads) is calculated at 0 K as

$\mathrm{E}_{\mathrm{ads}}=\mathrm{E}_{\mathrm{sys}}-\mathrm{E}_{\mathrm{slab}}-\mathrm{E}_{\mathrm{Cu}}$
(11)

where Esys, Eslab, and ECu are the energies of the system with an adsorbed Cu
atom, the system with clean surfaces, and a Cu atom in fcc bulk system. Eads
computed from DFT and NNP are -0.033 eV and -0.378 eV, respectively. The much
lower NNP value drives the formation of the precursor Cu monolyaer film and
then likely influences the observed wetting angles. The energetics of adsorbed
Cu on W surfaces are not represented in the reference dataset for the current
NNP. If Cu wetting becomes an important application issue, then a new NNP can
be created that incorporates the relevant DFT information in the training
data.

##  4 Conclusions

Composites of Cu and W have recently gained significant attention and have
potential applications in brazing fillers and shielding materials for plasma
and radiation. The immiscibility of Cu-W composites leads to their fabrication
as nano-multilayers, nano-composites, and metal matrix composites, sparking
interest in understanding the behavior of interfaces in the solid state and
between liquid W and solid Cu. We have developed a neural network potential
using the Behler-Parrinello framework to enable the detailed study of
atomistic mechanisms associated with these Cu-W materials. This potential,
created with a meticulously curated reference dataset and an optimized set of
descriptors, provides very good accuracy versus first-principles methods while
overcoming their computational limitations.

The NNP was rigorously validated for the mechanical and thermodynamic behavior
of bulk Cu and W. We also assessed its performance for binary Cu-W systems,
including substitutions, metastable alloys, and standard Cu-W interfaces. The
thermodynamics of binary systems, particularly the interactions between solid
W and liquid Cu, were also examined. The NNP demonstrates excellent quality in
these studies, showing no instabilities and maintaining good agreement with
first-principles values and experimental observations where applicable. This
successful validation instills confidence in the reliability and accuracy of
the NNP.

The ability of the NNP to predict a comprehensive range of behavior opens up
many new possibilities for modeling interfaces. In particular, the NNP can be
used to investigate intermixing, phase stability, and the large observed
variation in interface stresses versus processing conditions, enhancing our
understanding of these phenomena; we will report on progress in this direction
in forthcoming work [56]. The Cu-W NNP can also elucidate the effect of layer
thickness, in-plane stress, macroscopic porosity, temperature, grain
boundaries, and non-equilibrium vacancies in physical vapor-deposited Cu/W
nano-multilayers on Young’s modulus, explaining contradictory experimental
measurements in relation to different deposition conditions and resulting
microstructures; we will report on progress in this direction in another
forthcoming work [57]. More broadly, the versatility of this NNP, and others
that have been developed recently, demonstrates that machine-learned
potentials provide a valuable tool for studying many atomic-scale problems in
materials science and engineering.

###### Acknowledgements.

This research was supported by the NCCR MARVEL, a National Centre of
Competence in Research, funded by the Swiss National Science Foundation (grant
number 205602).

##  Appendix A Symmetry function hyper-parameters used in developing Cu-W NNP
potential

Hyper-parameters of the symmetry functions used when developing the Cu-W
potential are shown in LABEL:tab:rad_syms and LABEL:tab:ang_syms. Symmetry
function types 2, 3, and 9 correspond to $G^{2}$, $G^{3}$, and $G^{9}$
symmetry functions as shown in eqs. 1, 2 and 3 respectively.

Table 7: Hyper-parameters for the radial symmetry functions ($G^{2}$) selected by the PCov-CUR decomposition algorithm [76] as described in eq. 1. $r_{\textrm{c}}=7.0$ for all the symmetry functions Central | Neighbor | $\eta$ (Å-2) | $r_{\textrm{s}}$ (Å)
---|---|---|---
atom type | atom type
Cu | W | 2.0000 | 1.50
Cu | W | 2.0000 | 2.00
Cu | W | 2.0000 | 2.50
Cu | W | 2.0000 | 3.00
Cu | W | 2.0000 | 3.50
Cu | W | 2.0000 | 4.00
Cu | W | 2.0000 | 4.50
Cu | W | 2.0000 | 5.00
Cu | W | 2.0000 | 5.50
Cu | W | 2.0000 | 6.00
Cu | Cu | 2.0000 | 2.00
Cu | Cu | 2.0000 | 2.50
Cu | Cu | 2.0000 | 3.00
Cu | Cu | 2.0000 | 3.50
Cu | Cu | 2.0000 | 4.00
Cu | Cu | 2.0000 | 4.50
Cu | Cu | 2.0000 | 5.00
Cu | Cu | 2.0000 | 5.50
Cu | Cu | 2.0000 | 6.00
Cu | Cu | 2.0000 | 6.50
Cu | Cu | 0.0204 | 0.00
W | W | 2.0000 | 2.00
W | W | 2.0000 | 2.50
W | W | 2.0000 | 3.00
W | W | 2.0000 | 3.50
W | W | 2.0000 | 4.00
W | W | 2.0000 | 4.50
W | W | 2.0000 | 5.00
W | W | 2.0000 | 5.50
W | W | 2.0000 | 6.00
W | W | 2.0000 | 6.50
W | W | 0.0204 | 0.00
W | Cu | 2.0000 | 2.00
W | Cu | 2.0000 | 2.50
W | Cu | 2.0000 | 3.00
W | Cu | 2.0000 | 3.50
W | Cu | 2.0000 | 4.00
W | Cu | 2.0000 | 4.50
W | Cu | 2.0000 | 5.00
W | Cu | 2.0000 | 5.50
W | Cu | 2.0000 | 6.00
W | Cu | 0.0204 | 0.00
Table 7: (Continued.) Table 8: Hyper-parameters for the angular symmetry functions ($G^{3}$ and $G^{9}$) for selected by the PCov-CUR decomposition algorithm [76] as described in eqs. 2 and 3. $r_{\textrm{c}}=7.0$ for all the symmetry functions Central | Type | Neighbor | Neighbor | $\eta$ (Å-2) | $\lambda$ | $\zeta$
---|---|---|---|---|---|---
atom type | atom 1 type | atom 2 type
Cu | 9 | W | W | 0.0204 | 1 | 1
Cu | 9 | W | W | 0.0204 | 1 | 3
Cu | 9 | W | W | 0.0204 | -1 | 1
Cu | 9 | W | W | 0.0204 | -1 | 12
Cu | 9 | W | W | 0.0204 | -1 | 64
Cu | 9 | W | W | 0.0740 | 1 | 1
Cu | 9 | Cu | W | 0.1408 | -1 | 1
Cu | 9 | Cu | W | 0.0204 | 1 | 1
Cu | 9 | Cu | W | 0.0204 | 1 | 3
Cu | 9 | Cu | W | 0.0204 | 1 | 12
Cu | 9 | Cu | W | 0.0204 | 1 | 64
Cu | 9 | Cu | W | 0.0204 | -1 | 1
Cu | 9 | Cu | W | 0.0204 | -1 | 12
Cu | 9 | Cu | W | 0.0204 | -1 | 64
Cu | 9 | Cu | W | 0.0740 | 1 | 1
Cu | 9 | Cu | W | 0.0740 | -1 | 1
Cu | 9 | Cu | W | 0.0740 | -1 | 12
Cu | 9 | Cu | Cu | 0.0204 | 1 | 1
Cu | 9 | Cu | Cu | 0.0204 | 1 | 3
Cu | 9 | Cu | Cu | 0.0204 | 1 | 12
Cu | 9 | Cu | Cu | 0.0204 | 1 | 64
Cu | 9 | Cu | Cu | 0.0204 | -1 | 1
Cu | 9 | Cu | Cu | 0.0204 | -1 | 12
Cu | 9 | Cu | Cu | 0.0204 | -1 | 64
Cu | 9 | Cu | Cu | 0.0740 | 1 | 1
Cu | 9 | Cu | Cu | 0.0740 | -1 | 1
Cu | 9 | Cu | Cu | 0.0740 | -1 | 64
W | 9 | W | W | 0.0204 | 1 | 1
W | 9 | W | W | 0.0204 | 1 | 12
W | 9 | W | W | 0.0204 | 1 | 64
W | 9 | W | W | 0.0204 | -1 | 1
W | 9 | W | W | 0.0204 | -1 | 3
W | 9 | W | W | 0.0204 | -1 | 12
W | 9 | W | W | 0.0204 | -1 | 64
W | 9 | W | W | 0.0740 | 1 | 1
W | 9 | W | W | 0.0740 | -1 | 1
W | 9 | W | W | 0.0740 | -1 | 64
W | 9 | Cu | W | 0.1408 | -1 | 1
W | 9 | Cu | W | 0.0204 | 1 | 1
W | 9 | Cu | W | 0.0204 | 1 | 3
W | 9 | Cu | W | 0.0204 | 1 | 12
W | 9 | Cu | W | 0.0204 | 1 | 64
W | 9 | Cu | W | 0.0204 | -1 | 1
W | 9 | Cu | W | 0.0204 | -1 | 12
W | 9 | Cu | W | 0.0204 | -1 | 64
W | 9 | Cu | W | 0.0740 | 1 | 1
W | 9 | Cu | W | 0.0740 | -1 | 1
W | 9 | Cu | Cu | 0.0204 | 1 | 1
W | 9 | Cu | Cu | 0.0204 | 1 | 3
W | 9 | Cu | Cu | 0.0204 | 1 | 12
W | 9 | Cu | Cu | 0.0204 | -1 | 1
W | 9 | Cu | Cu | 0.0204 | -1 | 12
W | 9 | Cu | Cu | 0.0204 | -1 | 64
W | 9 | Cu | Cu | 0.0740 | -1 | 1
Table 8: (Continued.)

## References

  * Kong  _et al._ [2002] L. T. Kong, X. Y. Li, W. S. Lai, J. B. Liu, and B. X. Liu, Interfacial reaction of w/cu examined by an n-body potential through molecular dynamics simulations, Jpn. J. Appl. Phys 41, 4503 (2002).
  * Tsakiris  _et al._ [2014] V. Tsakiris, M. Lungu, E. Enescu, D. Pavelescu, G. H. Dumitrescu, A. Radulian, and N. Mocioi, Nanostructured w-cu electrical contact materials processed by hot isostatic pressing, Acta Phys. Pol., A 125, 348 (2014).
  * Thomas  _et al._ [2017] K. Thomas, A. A. Taylor, R. Raghavan, V. Chawla, R. Spolenak, and J. Michler, Microstructure and mechanical properties of metastable solid solution copper‑tungsten films, Thin Solid Films 642, 82 (2017).
  * Wei  _et al._ [2019] W. Wei, L. Chen, H. R. Gong, and J. L. Fan, Strain-stress relationship and dislocation evolution of w-cu bilayers from a constructed n-body w-cu potential, J. Condens. Matter Phys. 31, 305002 (2019).
  * Li  _et al._ [2023] R. Li, W. Chen, K. Zhou, Y. Sun, Y. Wen, Z. Wang, A. Elmarakbi, and Y.-Q. Fu, Freeze-casted tungsten skeleton reinforced copper matrix composites, J. Alloys Compd. 960, 170859 (2023).
  * Petrovskiy  _et al._ [2020] P. Petrovskiy, M. Doubenskaia, A. Sova, and A. Travyanov, Analysis of copper-tungsten cold spray coating: Kinetics of coating formation and its thermal properties, Surf. Coat. Technol. 385, 125376 (2020).
  * Lorenzin  _et al._ [2022] G. Lorenzin, M. S. B. Hoque, D. Ariosa, L. P. H. Jeurgens, E. R. Hoglund, J. A. Tomko, P. E. Hopkins, and C. Cancellieri, Tensile and compressive stresses in cu/w multilayers: Correlation with microstructure, thermal stability, and thermal conductivity, Acta Mater. 240, 118315 (2022).
  * Gong  _et al._ [2003] H. R. Gong, L. T. Kong, W. S. Lai, and B. X. Liu, Glass-forming ability determined by an n-body potential in a highly immiscible cu-w system through molecular dynamics simulations, Phys. Rev. B 68, 144201 (2003).
  * Zhang  _et al._ [2003] R.-F. Zhang, Z.-C. Li, and B.-X. Liu, Metastable phase formed in immiscible cu-w multilayers by ion mixing, Jpn. J. Appl. Phys 42, 7009 (2003).
  * Zhang  _et al._ [2004] R. F. Zhang, L. T. Kong, H. R. Gong, and B. X. Liu, Comparative study of metastable phase formation in the immiscible cu-w system by ab initio calculation and n-body potential, J. Condens. Matter Phys. 16, 5251 (2004).
  * Zeng  _et al._ [2021] T. Zeng, F. Li, and Y. Huang, Construction of an n-body potential for revealing the atomic mechanism for direct alloying of immiscible tungsten and copper, Mater. 14, 5988 (2021).
  * Ubale and Deshmukh [2018] S. B. Ubale and S. D. Deshmukh, Experimental Investigation and Modelling of Wire Electrical Discharge Machining Process on W-Cu Metal Matrix Composite, Mater. Today Proc. 5, 84 (2018).
  * Tejado  _et al._ [2018] E. Tejado, A. V. Müller, J. H. You, and J. Y. Pastor, The thermo-mechanical behaviour of W-Cu metal matrix composites for fusion heat sink applications: The influence of the Cu content, J. Nucl. Mater. 498, 468 (2018).
  * Tejado [2020] E. Tejado, W–Cu metal matrix composites for next generation fusion devices, Mater. Today 38, 136 (2020).
  * Ho  _et al._ [2008] P. W. Ho, Q. F. Li, and J. Y. Fuh, Evaluation of W-Cu metal matrix composites produced by powder injection molding and liquid infiltration, Mater. Sci. Eng. A 485, 657 (2008).
  * Misra and Krug [2001] A. Misra and H. Krug, Deformation behavior of nanostructured metallic multilayers, Adv. Eng. Mater. 3, 217 (2001).
  * Misra  _et al._ [2005] A. Misra, J. P. Hirth, and R. G. Hoagland, Length-scale-dependent deformation mechanisms in incoherent metallic multilayered composites, Acta Mater. 53, 4817–4824 (2005).
  * Misra  _et al._ [2007] A. Misra, M. J. Demkowicz, X. Zhang, and R. G. Hoagland, The radiation damage tolerance of ultra-high strength nanolayered composites, Small-Scale Mech. Behav. 59, 62–65 (2007).
  * Girault  _et al._ [2006] B. Girault, P. Villain, E. Le Bourhis, P. Goudeau, and P.-O. Renault, X-ray diffraction analysis of the structure and residual stresses of w/cu multilayers, Surf. Coat. Technol. 201, 4372 (2006).
  * Cancellieri  _et al._ [2016] C. Cancellieri, F. Moszner, M. Chiodi, S. Yoon, J. Janczak-Rusch, and L. P. H. Jeurgens, The effect of thermal treatment on the stress state and evolving microstructure of cu/w nano-multilayers, J. Appl. Phys. 120, 195107 (2016).
  * Moszner  _et al._ [2016a] F. Moszner, C. Cancellieri, M. Chiodi, S. Yoon, D. Ariosa, J. Janczak-Rusch, and L. P. H. Jeurgens, Thermal stability of cu/w nano-multilayers, Acta Mater. 107, 345 (2016a).
  * Moszner  _et al._ [2016b] F. Moszner, C. Cancellieri, C. Becker, M. Chiodi, J. Janczak-Rusch, and L. P. H. Jeurgens, Nano-structured cu/w brazing fillers for advanced joining applications, J. Mater. Sci. Eng. B. 6, 226 (2016b).
  * Druzhinin  _et al._ [2019] A. V. Druzhinin, D. Ariosa, S. Siol, N. Ott, B. B. Straumal, J. Janczak-Rusch, L. P. H. Jeurgens, and C. Cancellieri, Effect of the individual layer thickness on the transformation of cu/w nano-multilayers into nanocomposites, Mater. 7, 100400 (2019).
  * Druzhinin  _et al._ [2021a] A. V. Druzhinin, G. Lorenzin, D. Ariosa, S. Siol, B. B. Straumal, J. Janczak-Rusch, L. P. H. Jeurgens, and C. Cancellieri, The effect of the graded bilayer design on the strain depth profiles and microstructure of cu/w nano-multilayers, Mater. Des. 209, 110002 (2021a).
  * Druzhinin  _et al._ [2021b] A. V. Druzhinin, C. Cancellieri, L. P. H. Jeurgens, and B. B. Straumal, The effect of interface stress on the grain boundary grooving in nanomaterials: Application to the thermal degradation of cu/w nano-multilayers, Scr. Mater. 199, 113866 (2021b).
  * Li  _et al._ [2022] H. Li, Z. cheng Xing, B. jin Li, X. sheng Liu, B. Lehmert, M. Matthias, Z. xin Li, and W. Tillmann, Microstructural evolution of Cu/W nano-multilayers filler metal during thermal treatment, Vacuum 200, 111007 (2022).
  * Troncoso  _et al._ [2024] J. F. Troncoso, G. Lorenzin, C. Cancellieri, and V. Turlo, Explaining the effect of in-plane strain on thermal degradation kinetics of Cu/W nano-multilayers, Scr. Mater. 242, 115902 (2024).
  * Liang  _et al._ [2013] C. P. Liang, J. L. Fan, H. R. Gong, X. Liao, X. Zhu, and S. Peng, Interface structure and work function of W-Cu interfaces, Appl. Phys. Lett. 103, 10.1063/1.4833249 (2013).
  * Monclús  _et al._ [2014] M. A. Monclús, M. Karlik, M. Callisti, E. Frutos, J. Llorca, T. Polcar, and J. M. Molina-Aldareguía, Microstructure and mechanical properties of physical vapor deposited Cu/W nanoscale multilayers: Influence of layer thickness and temperature, Thin Solid Films 571, 275 (2014).
  * Callisti  _et al._ [2016] M. Callisti, M. Karlik, and T. Polcar, Bubbles formation in helium ion irradiated Cu/W multilayer nanocomposites: Effects on structure and mechanical properties, J. Nucl. Mater. 473, 18 (2016).
  * Ma  _et al._ [2018] G. C. Ma, J. L. Fan, and H. R. Gong, Mechanical behavior of Cu-W interface systems upon tensile loading from molecular dynamics simulations, Comput. Mater. Sci. 152, 165 (2018).
  * Taylor [1966] N. J. Taylor, A LEED study of the epitaxial growth of copper on (110) surface of tungsten, Surf. Sci. 4, 161 (1966).
  * Moss and Blott [1969] A. R. L. Moss and B. H. Blott, The epitaxial growth of copper on the (110) surface of a tungsten single crystal studied by LEED, Auger electron, and work function techniques, Surf. Sci. 17, 240 (1969).
  * Gładyszewski  _et al._ [1994] G. Gładyszewski, J. Pacaud, P. Goudeau, C. Jaouen, A. Naudon, and J. Grilhé, Interface of the CuW multilayers, Vacuum 45, 285 (1994).
  * Yang  _et al._ [2021] M. Yang, T. Xie, L. Fu, J. Zhu, W. Yang, D. Li, and L. Zhou, Electric transmission behavior of self-assembled Cu–W nano multilayers, Prog. Nat. Sci.: Mater. Int. 31, 25 (2021).
  * Yeom  _et al._ [2023] J. Yeom, G. Lorenzin, C. Cancellieri, and J. Janczak-Rusch, Evaluation of residual stresses in cu/mo, cu/nb nanomultilayers with a strong fiber texture, Mater. Lett. 352, 135074 (2023).
  * Reshöft  _et al._ [1999] K. Reshöft, C. Jensen, and U. Köhler, Atomistics of the epitaxial growth of Cu on W(110), Surf. Sci. 421, 320 (1999).
  * Bodlos  _et al._ [2022] R. Bodlos, V. Fotopoulos, J. Spitaler, A. L. Shluger, and L. Romaner, Energies and structures of cu/nb and cu/w interfaces from density functional theory and semi-empirical calculations, Mater. 21, 10.1016/j.mtla.2022.101362 (2022).
  * Lorenzin  _et al._ [2024a] G. Lorenzin, J. F. Troncoso, M. Liyanage, A. V. Druzhinin, L. P. H. Jeurgens, C. Cancellieri, and V. Turlo, Experimental and ab initio derivation of interface stress in nanomultilayered coatings: Application to immiscible Cu / W system with variable in-plane stress, Appl. Surf. Sci. 661, 159994 (2024a).
  * Melmed [1965] A. J. Melmed, Adsorption and surface diffusion of copper on tungsten, J. Chem. Phys. 43, 3057 (1965).
  * Wen  _et al._ [2007] S. P. Wen, R. L. Zong, F. Zeng, Y. Gao, and F. Pan, Evaluating modulus and hardness enhancement in evaporated Cu/W multilayers, Acta Mater. 55, 345 (2007).
  * Bauer  _et al._ [1974] E. Bauer, H. Poppa, G. Todd, and F. Bonczek, Adsorption and condensation of Cu on W single-crystal surfaces, J. Appl. Phys. 45, 5164 (1974).
  * Liang  _et al._ [2017] C. P. Liang, C. Y. Wu, J. L. Fan, and H. R. Gong, Structural, thermodynamic, and mechanical properties of WCu solid solutions, J. Phys. Chem. Solids 110, 401 (2017).
  * Daw and Baskes [1983] M. S. Daw and M. I. Baskes, Semiempirical, quantum mechanical calculation of hydrogen embrittlement in metals, Phys. Rev. Lett. 50, 1285 (1983).
  * Daw and Baskes [1984] M. S. Daw and M. I. Baskes, Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals, Phys. Rev. B 29, 6443 (1984).
  * Finnis and Sinclair [1984] M. W. Finnis and J. E. Sinclair, A simple empirical n-body potential for transition metals, Phil. Mag. A 50, 45 (1984).
  * Tai  _et al._ [2006] K. P. Tai, X. D. Dai, Y. X. Shen, and B. X. Liu, Formation and structural anomaly of the metastable phases in an immiscible ag-mo system studied by ion beam mixing and molecular dynamics simulation, J. Phys. Chem. B 110, 595–606 (2006).
  * Yang  _et al._ [2022] L. Yang, Y. Shen, S. Mi, J. Fan, and H. Gong, Phase stability and mechanical property of w–cu solid solutions from a newly derived w–cu potential, Physica B Condens. Matter 624, 413436 (2022).
  * Marchand  _et al._ [2020a] D. Marchand, A. Jain, A. Glensk, and W. A. Curtin, Machine learning for metallurgy I. A neural-network potential for Al-Cu, Phys. Rev. Mater. 4, 103601 (2020a).
  * Stricker  _et al._ [2020] M. Stricker, B. Yin, E. Mak, and W. A. Curtin, Machine learning for metallurgy II. A neural-network potential for magnesium, Phys. Rev. Mater. 4, 103602 (2020).
  * Jain  _et al._ [2021] A. C. P. Jain, D. Marchand, A. Glensk, M. Ceriotti, and W. A. Curtin, Machine learning for metallurgy III: A neural network potential for Al-Mg-Si, Phys. Rev. Mater. 5, 053805 (2021).
  * Marchand and Curtin [2022] D. Marchand and W. A. Curtin, Machine learning for metallurgy IV. A neural-network potential for al-cu-mg and al-cu-mg-zn, Phys. Rev. Mater. 6, 053803 (2022).
  * Liyanage  _et al._ [2022] M. Liyanage, D. Reith, V. Eyert, and W. A. Curtin, Machine learning for metallurgy v: A neural-network potential for zirconium, Phys. Rev. Mater. 6, 063804 (2022).
  * Behler and Parrinello [2007] J. Behler and M. Parrinello, Generalized neural-network representation of high-dimensional potential-energy surfaces, Phys. Rev. Lett. 98, 146401 (2007).
  * Behler [2011] J. Behler, Atom-centered symmetry functions for constructing high-dimensional neural network potentials, J. Chem. Phys. 134, 074106 (2011).
  * Hu  _et al._ [2024] Y. Hu, G. Lorenzin, J. Yeom, M. Liyanage, W. Curtin, L. P. Jeurgens, J. Janczak-Rusch, C. Cancellieri, and V. Turlo, Uncovering the origin of interface stress enhancement and compressive-to-tensile transition in immiscible nanomultilayers, unpublished manuscript (2024).
  * Lorenzin  _et al._ [2024b] G. Lorenzin, F. F. Klimashin, J. Yeom, Y. Hu, J. Michler, J. Janczak-Rusch, V. Turlo, and C. Cancellieri, Effect of residual stress and microstructure on mechanical properties of sputter-grown Cu/W nanomultilayers, unpublished manuscript (2024b).
  * Zhang  _et al._ [1999] Z. S. Zhang, M. Zhang, and F. Pan, Microstructure and magnetic behaviors of fe/hf multilayers by vapor deposition, J. Magn. Magn. Mater. 192, 53 (1999).
  * Choe and Shin [2000] S.-B. Choe and S.-C. Shin, Magnetic domain reversal dynamics in co/pd nanomultilayers, J. Magn. Magn. Mater. 209, 84 (2000).
  * Srinivasan and Subramanian [2007] D. Srinivasan and P. R. Subramanian, Kirkendall porosity during thermal treatment of mo–cu nanomultilayers, Mater. Sci. Eng. A 459, 145 (2007).
  * Marchand  _et al._ [2020b] D. Marchand, A. Jain, A. Glensk, and W. A. Curtin, Machine learning for metallurgy: a neural network potential for al-cu, Materials Cloud Archive 2020.62, 10.24435/materialscloud:d3-3r (2020b).
  * Kirklin  _et al._ [2015] S. Kirklin, J. E. Saal, B. Meredig, A. Thompson, J. W. Doak, M. Aykol, S. Rühl, and C. Wolverton, The open quantum materials database (oqmd): assessing the accuracy of dft formation energies, Npj Comput. Mater. 1, 15010 (2015).
  * Liyanage  _et al._ [2024] M. Liyanage, V. Turlo, and W. A. Curtin, Machine learning potential for the cu-w system, Materials Cloud Archive 2024.X, 10.24435/materialscloud:1m-0s (2024).
  * Giannozzi  _et al._ [2009] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari, and R. M. Wentzcovitch, QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials, J. Condens. Matter Phys. 21, 395502 (2009).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  * Monkhorst and Pack [1976] H. Monkhorst and J. Pack, Special points for brillouin-zone integrations, Phys. Rev. B 13, 5188 (1976).
  * Broyden [1970] C. G. Broyden, The convergence of a class of double-rank minimization algorithms 1. general considerations, IMA J. Appl. Math. 6, 76–90 (1970).
  * Fletcher [1970] R. Fletcher, A new approach to variable metric algorithms, Comput. J. 13, 317–322 (1970).
  * Goldfarb [1970] D. Goldfarb, A family of variable-metric methods derived by variational means, Math. Comput. 24, 23 (1970).
  * Shanno [1970] D. F. Shanno, Conditioning of quasi-newton methods for function minimization, Math. Comput. 24, 647 (1970).
  * Mishin [2021] Y. Mishin, Machine-learning interatomic potentials for materials science, Acta Mater. 214, 116980 (2021).
  * Singraber  _et al._ [2019] A. Singraber, J. Behler, and C. Dellago, Library-based LAMMPS implementation of high-dimensional neural network potentials, J. Chem. Theory Comput. 15, 1827–1840 (2019).
  * Plimpton [1995] S. Plimpton, Fast parallel algorithms for short–range molecular dynamics, J. Comput. Phys. 117, 1 (1995).
  * Gastegger  _et al._ [2018] M. Gastegger, L. Schwiedrzik, M. Bittermann, F. Berzsenyi, and P. Marquetand, wACSF—weighted atom-centered symmetry functions as descriptors in machine learning potentials, J. Chem. Phys. 148, 241709 (2018).
  * Imbalzano  _et al._ [2018] G. Imbalzano, A. Anelli, D. Giofré, S. Klees, J. Behler, and M. Ceriotti, Automatic selection of atomic fingerprints and reference configurations for machine-learning potentials, J. Chem. Phys. 148, 241730 (2018).
  * Cersonsky  _et al._ [2021] R. K. Cersonsky, B. A. Helfrecht, E. A. Engel, S. Kliavinek, and M. Ceriotti, Improving sample and feature selection with principal covariates regression, Mach. Learn.: Sci. Technol. 2, 035038 (2021).
  * Goscinski  _et al._ [2023] A. Goscinski, V. P. Principe, G. Fraux, S. Kliavinek, B. A. Helfrecht, P. Loche, M. Ceriotti, and R. K. Cersonsky, scikit-matter: A suite of generalisable machine learning methods born out of chemistry and materials science, Open Res. Eur. 3, 81 (2023).
  * Volterra [1907] V. Volterra, Sur l’équilibre des corps élastiques multiplement connexes, Ann. Sci. de l’Ecole Norm. Superieure 3, 401 (1907).
  * Nye [1953] J. F. Nye, Some geometrical relations in dislocated crystals, Acta Metall. 1, 153 (1953).
  * Hartley and Mishin [2005] C. S. Hartley and Y. Mishin, Characterization and visualization of the lattice misfit associated with dislocation cores, Acta Mater. 53, 1313 (2005).
  * Vítek  _et al._ [1970] V. Vítek, R. C. Perrin, and D. K. Bowen, The core structure of $1/2(111)$ screw dislocations in bcc crystals, Philos. Mag. 21, 1049 (1970).
  * [82] National Institute of Standards and Technology,  _atomman: Atomistic Manipulation Toolkit_ , https://www.ctcms.nist.gov/potentials/atomman/ [Accessed: 2020].
  * Clouet [2020] E. Clouet, Ab Initio models of dislocations, in  _Handbook of Materials Modeling: Methods: Theory and Modeling_, edited by W. Andreoni and S. Yip (Springer International Publishing, Cham, 2020) pp. 1503–1524.
  * Stroh [1958] A. N. Stroh, Dislocations and cracks in anisotropic elasticity, Philos. Mag. 3, 625 (1958).
  * Stroh [1962] A. N. Stroh, Steady state problems in anisotropic elasticity, J. Math. Phys. 41, 77 (1962).
  * Dean [1999] J. A. Dean, ed.,  _Lange’s Handbook of Chemistry_ , 15th ed. (McGraw-Hill, 1999).
  * Lide [2003] D. R. Lide, ed.,  _CRC Handbook of Chemistry and Physics_ , 84th ed. (CRC Press., Boca Raton, Florida, 2003).
  * Wang and Reeber [1996] K. Wang and R. Reeber, Thermal expansion of copper, High Temp. Mater. Sci. 35, 181 (1996).
  * Hidnert and Sweeney [1925] P. Hidnert and W. T. Sweeney, Thermal expansion of tungsten, Sci. Pap. Bur. Std. 20, 483 (1925).
  * Kong  _et al._ [2016] Y. Kong, J. Pi, S. Xiao, and Y. Du, Asymmetric mixing behavior and stability of the predicted phases in the w–cu system, Calphad 53, 116 (2016).
  * Honeycutt and Andersen [1987] J. D. Honeycutt and H. C. Andersen, Molecular dynamics study of melting and freezing of small lennard-jones clusters, J. Phys. Chem. 91, 4950–4963 (1987).
  * Faken and Jónsson [1994] D. Faken and H. Jónsson, Systematic analysis of local atomic structure combined with 3D computer graphics, Comput. Mater. Sci. 2, 279 (1994).
  * Izaguirre  _et al._ [2024] I. Izaguirre, J. de Prado, M. Sánchez, and A. Ureña, Wettability and microstructural evolution of copper filler in w and eurofer brazed joints, Int. J. Adv. Manuf. Tech. 131, 5905–5921 (2024).