# Influence of Crystalline Nanoprecipitates on Shear-Band Propagation in Cu-
Zr-Based Metallic Glasses

Tobias Brink<EMAIL_ADDRESS>Fachgebiet Materialmodellierung,
Institut für Materialwissenschaft, Technische Universität Darmstadt, Jovanka-
Bontschits-Straße 2, D-64287 Darmstadt, Germany  Martin Peterlechner  Institut
für Materialphysik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-
Straße 10, D-48149 Münster, Germany  Harald Rösner  Institut für
Materialphysik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-
Straße 10, D-48149 Münster, Germany  Karsten Albe  Fachgebiet
Materialmodellierung, Institut für Materialwissenschaft, Technische
Universität Darmstadt, Jovanka-Bontschits-Straße 2, D-64287 Darmstadt, Germany
Gerhard Wilde  Institut für Materialphysik, Westfälische Wilhelms-Universität
Münster, Wilhelm-Klemm-Straße 10, D-48149 Münster, Germany

(May 6, 2016)

###### Abstract

The interaction of shear bands with crystalline nanoprecipitates in Cu-Zr-
based metallic glasses is investigated by a combination of high-resolution TEM
imaging and molecular-dynamics computer simulations. Our results reveal
different interaction mechanisms: Shear bands can dissolve precipitates, can
wrap around crystalline obstacles, or can be blocked depending on the size and
density of the precipitates. If the crystalline phase has a low yield
strength, we also observe slip transfer through the precipitate. Based on the
computational results and experimental findings, a qualitative mechanism map
is proposed that categorizes the various processes as a function of the
critical stress for dislocation nucleation, precipitate size, and distance.

Published in:
T. Brink et al., Phys. Rev. Applied 5, 054005 (2016) DOI:
10.1103/PhysRevApplied.5.054005
This article is available under the terms of the Creative Commons Attribution
3.0 License. Further distribution of this work must maintain attribution to
the authors and the published article’s title, journal citation, and DOI.

Materials Science, Mechanics

##  I Introduction

Metallic glasses (MGs) have advantageous mechanical properties, such as a high
yield strength and a large elastic limit, but suffer from brittle failure,
especially under tension, at temperatures significantly below the glass
transition.Inoue (2000); Ashby and Greer (2006) Under compression, improved
ductility is found for composites of MGs and crystalline secondary phases,
namely, for Cu-Zr-based,Fan and Inoue (1997); Lee  _et al._ (2006); Hajlaoui
_et al._ (2007); Fornell  _et al._ (2010); Li  _et al._ (2011) Cu-Ti-
based,Calin  _et al._ (2003) and Zr-Ti-based MGs.Hays  _et al._ (2000, 2001);
Hofmann  _et al._ (2008) Under tension, a small ductility with 1% to 2% strain
is observed for Cu-Zr composites containing nanocrystals.Pauly  _et al._
(2010a); Barekar  _et al._ (2010); Pauly  _et al._ (2010b) With a higher
volume fraction of the crystalline phase, not only compressive but also
significant tensile ductility is reported for Zr-Ti-based,Hofmann  _et al._
(2008); Qiao  _et al._ (2013) Ti-based,Kim  _et al._ (2011) and Cu-Zr-based
MGs.Liu  _et al._ (2012); Wu  _et al._ (2014)

For dendritic precipitates, there is a correlation between the location of
dendrites and the occurrence of shear-band patterns.Hays  _et al._ (2000,
2001) The improved ductility is generally ascribed to the increased number of
shear bands and their limited length given by the constraints of the
crystalline phase.Hofmann  _et al._ (2008) Thus, a high volume fraction of
ductile crystalline phase improves the ductility in compression and
microindentation tests, while a brittle secondary phase does not.Fu  _et al._
(2007) This is confirmed by Song et al., who suggest a crystalline volume
fraction between $40\%$ and $80\%$ in Cu-Zr-based MGs for obtaining good
mechanical properties,Song  _et al._ (2013) which is consistent with the fact
that tensile ductility is observed only in glasses with high volume fractions
of ductile crystalline phases.Hofmann  _et al._ (2008); Kim  _et al._ (2011);
Liu  _et al._ (2012); Wu  _et al._ (2014)

While the enhancement of macroscopic ductility of MGs with high volume
fractions of a ductile crystalline phase can be explained by simple composite
models, the influence of nanoprecipitates on the mechanical properties of MGs
containing a much lower volume fraction of crystalline matter is still not
clear. Similar to the case of dendritic precipitates, shear-band patterns were
also observed in glasses containing small spherical crystallites with sizes
around $2\,\mathrm{nm}$.Hajlaoui  _et al._ (2007) These nanoprecipitates can
grow during deformation in certain metallic glasses.Lee  _et al._ (2006); Cao
_et al._ (2007); Pauly  _et al._ (2010a); Barekar  _et al._ (2010); Pauly  _et
al._ (2010b); Fornell  _et al._ (2010) An increased growth rate of
nanocrystallites in shear bands is observed,Chen  _et al._ (2006) which has
been related to enhanced atomic mobility inside shear bands.Wilde and Rösner
(2011); Bokeloh  _et al._ (2011) Deformation-grown nanocrystallites are
observed to contain twins,Cao  _et al._ (2007); Pauly  _et al._ (2010a, b)
which occur only in larger crystallites, e.g., with a size greater than
$20\,\mathrm{nm}$ in a Cu-Zr-Al MG.Pauly  _et al._ (2010b) These deformation-
grown precipitates are the possible reason for strain hardening during
nanoindentation, Fornell  _et al._ (2010) as well as increased plastic strain
during compression.Lee  _et al._ (2006) It is proposed that the participation
of the crystallites in the plastic deformation is the reason for the enhanced
ductility: Wu et al. demonstrate that reducing the stacking-fault energy of B2
CuZr by alloying leads to increased twinning and higher ductility under
tension.Wu  _et al._ (2012) Pauly et al. propose that a martensitic
transformation from the B2 phase to the B$19^{\prime}$ phase with a subsequent
volume change is responsible for toughening in Cu-Zr-based metallic
glasses.Pauly  _et al._ (2010a) This interpretation, however, is not generally
accepted. Corteen et al., in contrast, note that the volume change of the
martensitic transformation is very small and cannot contribute significantly
to toughening.Corteen  _et al._ (2011) They instead suggest that precipitates
increase plasticity by favoring the nucleation of new shear bands over the
growth of critical shear bands. Indeed, recent simulation and experimental
results provide evidence for the fact that crystal–glass interfaces serve as
nucleation sites for shear bands and are therefore responsible for the
simultaneous nucleation of multiple shear bands.Albe  _et al._ (2013); Zaheri
_et al._ (2014); Wang  _et al._ (2014a) In tensile tests and corresponding
molecular dynamics (MD) simulations of nanolaminates of copper nanocrystals
separated by thin Cu-Zr glass layers, the crystal–glass interface acts as a
source or sink for dislocations. Shear transformation zones (STZs) are
activated by interactions with dislocations.Wang  _et al._ (2007); Arman  _et
al._ (2011); Brandl  _et al._ (2013)

Figure 1: An undeformed Zr53.8Cu31.6Ag7.0Al7.6 sample after annealing in a TEM
bright-field image. An overview of the sample is shown in (a), with clearly
visible crystalline precipitates. The inset (b) shows a magnified view,
indicating the transition from globular to dendritic morphologies during
precipitate growth. The different brightness of parts of one precipitate in
(b) is most likely due to the different geometrical orientations of different
parts of the same precipitate with respect to the incident electron beam. The
strong black-white contrast change observed for some nanocrystals in (a) is
most likely due to twinning.

Computational studies on the interaction of crystalline precipitates with
shear bands provide further insights into nanoscale mechanisms. Lund and Schuh
conduct quasi-2D molecular-statics simulations of a binary Lennard-Jones
system with a nanocrystal inclusion.Lund and Schuh (2007) They identify three
mechanisms of deformation, depending on the ratio of shear-band thickness to
crystal size. For small crystals, the deformation is accommodated either in
the interface (for example, by rotation) or by dissolution of the crystal. For
wide shear bands and intermediate crystal sizes, dislocations in the crystal
nucleate at the interface. Finally, for crystals larger than the shear band,
they observe homogeneous dislocation nucleation due to stress building in the
nanocrystal center. However, it is somewhat unclear how the observed
homogeneous dislocation nucleation depends on the artificially induced shear
band and the resulting stress state in the system. Shi and Falk conduct
molecular-dynamics simulations on a monoatomic amorphous model system with a
high fraction of bcc nanocrystallites.Shi and Falk (2008) They find that
deformation is induced at the interfaces and that shear bands bend around
crystallites away from a direction of maximum resolved shear stress. They also
observe blocking of shear bands by crystallites. Because of the high fraction
of crystalline phase, the system more closely resembles a nanocrystalline
structure. The observation of the initiation of plastic deformation at
interfaces still matches the simulations by Albe et al. Albe  _et al._ (2013)
and underlines the importance of the crystal–glass interface in these
composite systems.

While it has been shown that interfaces promote shear-band nucleation and that
precipitates can act as obstacles or can deform together with the matrix,
there is no comprehensive study that investigates the influence of the size
and number density of the precipitates. Furthermore, some mechanisms governing
the interaction between a propagating shear band and a preexisting precipitate
have been observed but not investigated and discussed in detail. Therefore the
goal of this study is to investigate the interaction of a shear band with
preexisting precipitates in Cu-Zr-based MGs. In the experimental part of the
study, we anneal Zr-Cu-Ag-Al melt-spun ribbons to induce the formation of
nanocrystalline precipitates. We present transmission electron microscopy
(TEM) images of the samples before and after deformation by cold rolling and
identify the effects of crystalline precipitates on the shear-band
propagation. Using molecular-dynamics computer simulations, we model composite
systems with a metallic-glass matrix and crystalline precipitates. We control
the initiation of a shear band using a stress concentrator and put
precipitates in its propagation path. The focus of the simulations is on the
size effects of “hard” precipitates that do not partake in the plastic
deformation. Additionally, we study the shear-band interaction with “soft,”
plastically deformable precipitates. Finally, we derive a deformation map from
the combined observations of simulations and experiments that classifies the
observed mechanisms.

##  II Experiment

###  II.1 Experimental setup

We prepare metallic-glass samples of nominal composition
Zr53.8Cu31.6Ag7.0Al7.6 from pure components (Cu: 99.999%, Zr: 99.998%, Ag:
99.999%, Al: 99.999%; all in at.%) by prealloying using arc melting. After
repeated arc melting with intermittent turning of the specimen to enhance
homogenization, the entire ingots are inserted into quartz-glass crucibles for
melt spinning. The weight loss during alloying is minimal and subsequent
composition analyses by energy-dispersive x-ray diffraction (EDX) confirm that
the composition of the material is equal to the nominal composition within the
accuracy of the measurement. For melt spinning, the ingots are inductively
melted under an Ar atmosphere and the melt is ejected onto a rotating Cu wheel
(tangential wheel velocity: $30\,\mathrm{m/s}$), resulting in completely
amorphous thin ribbon samples of approximately $80$-$\mathrm{\upmu{}m}$
thickness. X-ray diffraction on the as-quenched ribbon samples does not
indicate the presence of any crystalline fraction exceeding the sensitivity
threshold of this method. Parts of the ribbon samples are cut to perform
differential scanning calorimetry (DSC) measurements in a Perkin Elmer Diamond
DSC device. Both isochronal and isothermal measurements under a purified Ar
gas flow are conducted and in conjunction with microstructure analyses the
time and temperature dependence of the evolving crystalline fraction is
determined. On the basis of these results, the samples for deformation
processing are annealed in the DSC device at $410\,^{\circ}\mathrm{C}$ for
$3\,\mathrm{h}$. This thermal treatment results in the formation of
nanocrystalline precipitates with an average diameter of about
$70\,\mathrm{nm}$ and a number density on the order of
$10^{20}\,\mathrm{m^{-3}}$. The resulting microstructure is shown in Fig. 1.
Given that the width of shear bands is on the order of
$10\,\mathrm{nm}$,Rösner  _et al._ (2014) there is no straight path for
propagating shear bands to avoid the interaction with nanoprecipitates in
these samples.

Figure 2: TEM image of a shear band in a Zr-Cu-Ag-Al ribbon after deformation.
The green arrows mark the positions of crystalline precipitates.

After the DSC heat treatment, the partially crystalline material is deformed
by cold rolling at room temperature in one step to true strain values of about
$\varepsilon=5\%$ at room temperature, applying a strain rate of the order of
$\dot{\varepsilon}=1\,\mathrm{s^{-1}}$. Subsequently, specimens for TEM
investigations are prepared by grinding, dimpling, and finally precision ion
polishing (PIPS, Gatan), using a low acceleration voltage of
$2.5\,\mathrm{kV}$ and low incidence angles ($<4^{\circ}$) to minimize damage
by the preparation process. The electron transparent samples are then analyzed
in dark-field, bright-field, and high-resolution transmission electron
microscopy modes in a Zeiss Libra 200FE TEM operated at an acceleration
voltage of $200\,\mathrm{kV}$. Since intrinsic shear bands formed upon cold
rolling could not be found during the TEM inspections of the thin foil
regions, the shear bands generated as a result of sample preparation and/or
subsequent sample manipulation are studied instead. Thus, the final sample
state resembles the state in an in situ TEM experiment comparable to the work
in Ref. Wilde and Rösner, 2011, where the shear bands are generated at crack
tips in the thin TEM foil.

###  II.2 Experimental results

Figure 3: TEM and HRTEM images of a shear band. In the upper left corner, the
electron transparent hole is visible, which stems from the sample preparation
for TEM. (a–b) Bright-field images of the same area, using a different tilt
angle. The shear band appears as a white stripe, with part of it already
cracked (bright white). (c) Dark-field image of the same area: The
nanoprecipitates are visible and marked by circles. The crack stops at
precipitate I; the shear band continues from there to precipitate II. (d) The
power spectrum of precipitate II along the $\langle\overline{1}10\rangle$ zone
axis revealing superlattice reflections of the martensitic B$19^{\prime}$
structure. (e) HRTEM image of precipitate II, showing the shear band stopping
at the precipitate. The contrast in (c) and (e) is enhanced to improve the
visibility of the precipitates and the shear band.

Figure 2 shows a shear band in a sample after TEM preparation. It is
noticeable that the shear band switches propagation directions in the vicinity
of precipitates. Between precipitates 2 and 3, the shear band has an
additional bend. The propagation path, in general, suggests that the shear
band is “attracted” to the precipitates, possibly due to a stress field
resulting from the density change on crystallization, which explains the
change of direction between precipitates 2 and 3. Because of the processing of
the samples, it can be excluded that the repeated bending of the shear band
results from a change of the external stress state: The deformation by cold
rolling is performed in a single step, and the observed shear bands were
created during TEM preparation, resembling an in situ experiment. As the path
change in the presence of precipitates is rather large (in Fig. 2 around
$45^{\circ}$) and correlated to the position of the precipitates, it is most
likely induced by the presence of the precipitates. The literature supports
this, as shear bands in homogeneous metallic glasses (Cu-Zr-based or
otherwise) are straight on the length scale presented here.Kumar  _et al._
(2007); Wang  _et al._ (2008); Rösner  _et al._ (2014); Schmidt  _et al._
(2015) It is therefore clear that the precipitates play a major role in
influencing shear-band propagation and thereby the macroscopic plastic
deformation of the material. TEM images of a second shear band, shown in Fig.
3, shed more light on this interaction. The shear band interacts with two
precipitates: A crack follows the path of the shear band up to the first
precipitate and the shear band continues from the first to the second
precipitate. Because of the visible crack opening near the first precipitate,
it can be ruled out that the shear band originates from the second
precipitate. While the first precipitate is passed, the second precipitate,
which is encountered centrally, stops the shear band. This can also be
observed at the end of the shear band in Fig. 2. A detailed analysis of the
gray-scale intensity distribution of the high-resolution TEM (HRTEM) image in
Fig. 3(e) indicates that the shear band changes its path slightly near the
crystalline precipitate but does not proceed further or shows slip transfer
into the precipitate. Additionally, a smaller, shear-band-like region emerges
almost perpendicular to the previous propagation direction (yellow arrow).
This indicates either a shear-band deflection, or a nucleation of a new,
perpendicular shear band. Based on the highly local nature of the intensity
distribution and based on the comparison of the contrast of other
precipitates, preparation artifacts can be excluded. Thus, this type of
interaction observed here is part of the intrinsic interaction mechanism
between precipitates and advancing shear bands. The power spectrum in Fig.
3(d) indicates a precipitate with B$19^{\prime}$ crystal structure,Schryvers
_et al._ (1997) which is consistent with prior observations in Cu-Zr-based
MGs.Pauly  _et al._ (2010a)

It is clear that the interaction of a propagating shear band with a
distribution of crystalline precipitates depends on the actual stress state
near the shear-band tip, the already-accommodated stress by the shear-band
propagation and the details of the local distribution of crystalline
precipitates, their sizes, and the residual stresses in the glass matrix due
to the formation of the precipitates. Depending on these factors, there are
various explanations for the observed paths: The path changes are caused
either by a deflection of the shear band, by the blocking and subsequent
renucleation of a new shear band, or by several nascent shear bands growing
together. The unhindered “passing” of a precipitate can be explained either by
a temporary path change of the shear band or by the participation of the
precipitate in the plastic deformation. Because of the necessarily limited
amount of data that can be obtained in the experiment, we undertake MD
simulations to test the aforementioned hypotheses and investigate their
relation to the sample geometry.

##  III Simulation

###  III.1 Simulation setup and analysis

Figure 4: Schematic simulation setup. The picture shows a cut through the
three-dimensional simulation box in the $xz$ plane at $l/2$. A notch is
inserted to control the origin of the shear band (yellow). The spherical
precipitate is shown in blue. The box has open boundaries in the $x$ direction
and is otherwise periodic. A constant strain rate is applied in the $z$
direction. Figure 5: Snapshots of a simulation with a $30$-$\mathrm{nm}$ CuZr
precipitate. The shear band wraps around the precipitate and continues
unhindered. The glass matrix is colored according to the atomic strain. The
precipitate atoms are shown in blue if they appear in the B2 structure; no
defects are visible. The left column shows a cut through the middle of the
precipitate. On the right, all atoms with $\eta_{i}<0.3$ are deleted. A video
version is provided in Video 1.

List of myvideos 1 Simulation of a shear band wrapping around a
$30$-$\mathrm{nm}$ CuZr precipitate as shown in Fig. 5.

In order to gain more insights into the nanoscale mechanisms of shear-band
interaction with precipitates, we perform a number of MD computer simulations,
which allow for an “in situ” observation of shear-band propagation. We use the
software lammps Plimpton (1995) to quench metallic-glass samples, insert
precipitates, and perform mechanical testing. The simulated metallic glass is
a $\mathrm{Cu_{64}Zr_{36}}$ alloy modeled with a Finnis–Sinclair-type
potential by Mendelev et al.Mendelev  _et al._ (2009) Metallic-glass samples
with dimensions $10\times 10\times 10\,\mathrm{nm}^{3}$ and $20\times 20\times
20\,\mathrm{nm}^{3}$ are prepared by melting the material at
$2000\,\mathrm{K}$ and subsequent quenching to $50\,\mathrm{K}$ with a cooling
rate of $0.01\,\mathrm{K/ps}$.

We use the sample geometry illustrated in Fig. 4 to investigate the influence
of preexisting precipitates on an approaching shear band. Size effects are
studied by varying the diameter $d$ of the precipitates and by controlling the
distance between periodic precipitate images by varying the box width $l$.
Sample sizes are $120\,\mathrm{nm}\times l\times 60\,\mathrm{nm}$, with
$l=10\,\mathrm{nm}$, $20\,\mathrm{nm}$, $30\,\mathrm{nm}$, and
$40\,\mathrm{nm}$. With a single precipitate per simulation box, this
corresponds to number densities for the precipitates of $1.4\times
10^{22}/\mathrm{m^{3}}$, $6.9\times 10^{21}/\mathrm{m^{3}}$, $4.6\times
10^{21}/\mathrm{m^{3}}$, and $3.5\times 10^{21}/\mathrm{m^{3}}$, respectively.
The Cu-Zr glass samples are replicated to reach the desired box dimensions,
and spherical precipitates with diameters from $3\,\mathrm{nm}$ to
$40\,\mathrm{nm}$ were inserted. For this, a hole is cut into the glass matrix
with a size chosen to accommodate the precipitate without overlapping atoms.
For the CuZr precipitate, we use the experimentally observed B2
structure.Pauly  _et al._ (2007); Sun  _et al._ (2005); Das  _et al._ (2007);
Jiang  _et al._ (2007) The B$19^{\prime}$ structure, which was found in the
precipitates in the experimental part of this paper, is a distortion of the B2
structure.Schryvers  _et al._ (1997) A notch controls the origin of the shear
band and makes sure that it always hits the precipitate. Periodic boundary
conditions are applied in the $y$ and $z$ directions and open boundaries in
the $x$ direction. The resulting structure is equilibrated at $50\,\mathrm{K}$
for $2\,\mathrm{ns}$ with a barostat at ambient pressure in periodic
directions. After equilibration, no long ranging stress field around the
precipitate is left; any mismatches are accommodated by the glass during the
interface creation.

The resulting composite samples are deformed at $50\,\mathrm{K}$ under a
constant tensile strain rate of $\dot{\varepsilon}=4\times 10^{7}/\mathrm{s}$
in the $z$ direction up to a total strain of at least $10\%$. The trajectories
from equilibrated to fully deformed samples are analyzed to observe the shear-
band propagation path. The shear band is identified using the von Mises local
shear invariant $\eta_{i}$ Shimizu  _et al._ (2007) as implemented in the
visualization tool Ovito.Stukowski (2010) Atoms with a local shear greater
than around $0.2$ are assigned to the shear band. To observe plastic
deformation events in the crystalline phase, we perform atomic structure
identification, which can identify crystal structures, stacking faults, and
other defects.Stukowski (2012)

###  III.2 Wrapping and blocking

Figure 6: Comparison of the Orowan mechanism (a) with a shear band wrapping
around precipitates (b). The slip plane is shown in gray, the precipitates in
blue, the dislocation as a black line, and the shear band as a red plane. The
precipitates can stop a dislocation because it must remain in a defined slip
plane, while the shear band can temporarily leave its slip plane and continue
unhindered afterwards.

The first observed interaction mechanism between a propagating shear band and
a preexisting precipitate is shown in Fig. 5 and Video 1. Here, the shear band
wraps around the precipitate like a carpet moving over a small obstacle. The
particle does not deform and simply moves along with one half of the glass
matrix. We call this mechanism the _wrapping mechanism_. This kind of athermal
mechanism is virtually unknown in crystalline materials. The closest analog in
a crystal—dislocation climb—is purely thermally activated. It is therefore
instructive to compare the two material classes, as shown in Fig. 6. In a
crystalline material, a dislocation moves on defined slip planes. A change of
slip plane is possible only for the screw components of the
dislocation,Cottrell (1953) is connected with a high energy barrier, and is
usually observed only in stage III work hardening.Hirth and Lothe (1982)
Therefore, the Orowan mechanism applies: The dislocation is bent around the
obstacle and finally forms dislocation rings [Fig. 6(a)].Gottstein (2004)
These rings may pile up, thereby hardening the material. In a metallic glass,
as in any isotropic material, all slip directions are equivalent. Only an
applied external stress differentiates the directions. Under tensile stress,
the planes oriented in $45^{\circ}$ angles towards the tensile axis experience
the highest resolved stress. An obstacle can be avoided simply by temporarily
and locally changing the slip path, thereby wrapping around the obstacle [Fig.
6(b)]. Depending on the precipitate distance, this wrapping mechanism is
observed for precipitates with diameters smaller than $25\,\mathrm{nm}$ to
$35\,\mathrm{nm}$ in our simulations.

Figure 7: Snapshots of a simulation with a $37.5$-$\mathrm{nm}$ CuZr-
precipitate. The shear band is blocked by the precipitate, while a second
shear band is immediately nucleated in another plane of high resolved shear
stress. The glass matrix is colored according to the atomic strain. The
precipitate atoms are shown in blue if they appear in the B2 structure; no
defects are visible. The left column shows a cut through the middle of the
precipitate. On the right, all atoms with $\eta_{i}<0.3$ are deleted. A video
version is provided in Video 2.

List of myvideos 2 Simulation of a shear band being blocked by a
$37.5\,\mathrm{nm}$ CuZr precipitate as shown in Fig. 7.

An alternative mechanism appears for increasing diameters and decreasing
distances between precipitates: The shear band is _blocked_ by the
precipitate. This causes the simultaneous nucleation of a second shear band
perpendicular to the first one on the opposite side of the precipitate as
shown in Fig. 7 and in Video 2. Again, there is no slip transfer to the
crystalline phase. The paths of the shear bands in this simulation and in the
HRTEM image of the experimental sample [Fig. 3(e)] are comparable. Both show
what looks like a shear band that starts to wrap around the precipitate but
does not propagate further. In the experiment, though, no fully formed shear
band appears perpendicular to the original one. This may be a result of the
more complex stress state or the fact that a new shear band can nucleate at
another precipitate that is not visible in the images. Still, a region
resembling a nascent shear band appears perpendicular to the original
propagation direction, strengthening the agreement with the simulation
results.

A systematic investigation of the parameters of precipitate distance and size
reveal a clear correlation with the mechanism. For a more quantitative
analysis, we define an empirical parameter $\Lambda$, which is given by the
ratio of the cross-sectional area $A$ of the precipitate divided by the
distance of precipitate centers $l$ as shown in Fig. 8(a). Similar to the
derivation of the Orowan stress, the crystalline volume fraction
$f=V_{\mathrm{precipitates}}/V$ can be estimated from the average crystallite
distance by the relationGottstein (2004)

$l=\frac{d/2}{\sqrt{f}}\quad\Leftrightarrow\quad f=\frac{d^{2}}{4l^{2}}.$ (1)

Using that, we can express $\Lambda$ only in terms of volume fraction and
precipitate geometry:

$\Lambda=\frac{A}{l}=\frac{A\sqrt{f}}{d/2}.$ (2)

For nonoverlapping precipitates ($d<l$), it is $A=\pi d^{2}/4$ and $\Lambda$
reduces to

$\Lambda=\pi\frac{d}{2}\sqrt{f}.$ (3)

When the precipitates overlap ($d>l$), we reduce the area $A$ to remove the
overlapping circle segments. In the limit $l\rightarrow 0$, $\Lambda$
corresponds to $A/l$ of an infinite cylinder parallel to the shear-band front.
Figure 8(b) shows a contour plot of the $\Lambda$ parameter as a function of
the precipitate diameter and distance. The data points in the plot represent
results from our MD simulations, divided into those showing the wrapping and
those showing the blocking mechanism. For the simulation geometry used in this
work, there exists a given $\Lambda$ that clearly separates the two
mechanisms:

$\Lambda_{\mathrm{crit}}\approx 12.65\,\mathrm{nm}$ (4)

This $\Lambda_{\mathrm{crit}}$ is not universal, as, for example, the distance
between precipitate and notch is not varied. A test simulation finds that
increasing the distance from the notch and therefore increasing shear-band
length favors the wrapping mechanism.

Figure 8: Influence of the precipitate size and distance on the mechanism. (a)
Explanation of the parameter $l$, distance of the precipitates, and $A$, the
area of a cut through the precipitate. (b) shows a contour plot of the
empirical geometry factor $\Lambda=A/l$. The dashed line shows the critical
value of $\Lambda$ for a transition from the wrapping to the blocking
mechanism. Data points are MD simulations. The data points at
$l=0\,\mathrm{nm}$ are infinite cylinders along the $y$ axis, representing the
case of overlapping spheres with infinitesimally small distances between their
centers. Figure 9: Stress-strain curves of samples which exhibit the wrapping
or blocking mechanisms. The arrows indicate when the shear band hits the
precipitate. Figure 10: Mechanical dissolution of a $3$-$\mathrm{nm}$ copper
particle in a shear band. The gray-scale color coding shows atomic strain
$\eta_{i}$ from $0.0$ (black) to $1.0$ (white). The precipitate is shown in
color: Light blue atoms are in the fcc structure, green atoms are in a
stacking fault, and dark blue atoms are disordered. The arrows indicate the
shear direction. Subfigure (a) shows a cut through the middle of the
nanoprecipitate, while (b) shows only the atoms that initially belonged to the
nanoprecipitate.

Using these formulas, we can also estimate $\Lambda$ for the experimental
results. Given a number density of precipitates $n=10^{20}\,\mathrm{m^{-3}}$
and particle diameters of around $70\,\mathrm{nm}$, we obtain

$\displaystyle f$
$\displaystyle=\frac{V_{\mathrm{precipitates}}}{V}=n\frac{4}{3}\pi(35\,\mathrm{nm})^{3}\approx
1.8\%$ (5) $\displaystyle\Lambda_{\mathrm{exp}}$ $\displaystyle=\pi\times
35\,\mathrm{nm}\times\sqrt{1.8\%}\approx
14.7\,\mathrm{nm}>\Lambda_{\mathrm{crit}}.$ (6)

This is consistent with the fact that the shear bands that are observed in the
glass samples are blocked by the precipitates. Still, the value is close to
$\Lambda_{\mathrm{crit}}$, which means that slightly smaller precipitates (or
precipitates that are not hit centrally, virtually decreasing the prefactor
$d$) may be susceptible to wrapping.

Figure 11: Stress-strain curves for the composites with fcc copper
precipitates and for a $10$-$\mathrm{nm}$ copper nanowire. (a) Curves from
simulations with the Mendelev potential. The copper precipitates do not deform
plastically and show similar behavior to the CuZr precipitates discussed
earlier. The curve of the copper nanowire explains this: The yield stress, and
therefore the critical shear stress, is higher than the highest resolved shear
stress in the steady state in the composite. To be able to compare the tensile
stress in the composite with the shear stress in the nanowire, the tensile
stress axis is scaled with a factor $0.5$ corresponding to the Schmid factor
for the plane of highest resolved shear stress. (b) Curves from simulations
with the Ward potential. Now the critical stress for heterogeneous dislocation
nucleation is lower than the steady-state stress in the composite and the
precipitate deforms plastically. (c) The simulation setup for the nanowire.
The wire is sheared in the $[110]$ direction of the fcc crystal structure on
the $(111)$ plane. The red atoms are fixed and the atoms on the top are
shifted with a constant velocity to shear the nanowire.

Figure 9 shows examples of the stress-strain curves of samples that exhibit
either the wrapping or the blocking mechanism. Neither a pronounced ductility
nor significant strain hardening can be observed. This lack of strain
hardening is in accordance with experimental data for tensile tests on Cu-Zr-
based metallic glasses with crystalline precipitates.Pauly  _et al._ (2010a);
Barekar  _et al._ (2010); Pauly  _et al._ (2010b) It also fits with a recent
study of compression tests of Cu-Zr-based metallic glass, that finds an effect
of particle size on mechanical parameters but no particle hardening.Wang  _et
al._ (2014b)

###  III.3 Plastic deformation of the crystalline phase

Figure 12: Snapshots of a simulation with a $30$-$\mathrm{nm}$ copper
precipitate. This simulation uses the Ward potential, in which the critical
stress for dislocation nucleation is realistic. As the precipitate is
sufficiently soft, the shear band can cut through it. The glass matrix is
colored according to the atomic strain (the same scale as Figs. 5 and 7).
Defects in the fcc crystal structure are colored according to the legend. The
left column shows a cut through the middle of the precipitate. On the right,
all atoms with $\eta_{i}<0.3$ and all fcc-coordinated atoms are deleted. A
video version is provided in Video 3.

List of myvideos 3 Simulation of a shear band cutting a $30$-$\mathrm{nm}$
copper precipitate as shown in Fig. 12.

The precipitates discussed until now were all “hard,” i.e., not susceptible to
plastic deformation under the simulation conditions: Because of the high
antiphase-boundary energy in the B2 structure, superdislocations or twinning
with respect to a martensitic transformation would be needed for a plastic
deformation of the precipitates. The stress available at the shear-band tip is
not sufficient to nucleate these defects, which explains why no plastic
deformation of the crystalline phase can be observed in our simulations. In
experiments, deformation-grown precipitates show twinning defects which may
cause softening effects and make the precipitate susceptible to plastic
deformation.Cao  _et al._ (2007) As a model for a softer precipitate, we thus
exchange the B2 crystal phase for fcc copper.

For precipitates that are small relative to the shear-band width, the
nanocrystals undergo mechanical dissolution.Lund and Schuh (2007); Albe  _et
al._ (2013) This is also observed in our setup with $3$-$\mathrm{nm}$
particles as shown in Fig. 10. In samples with larger diameters, the
precipitates do not deform plastically but instead show the same wrapping and
blocking interactions as described before for the “hard” precipitates. Even if
the $(111)$ glide plane is oriented parallel to the shear-band direction to
maximize resolved shear stress on the preferred fcc slip plane, we do not
observe slip transfer into the nanoprecipitate. The corresponding stress-
strain curves are shown in Fig. 11(a). To explain this, we estimate the
critical stress for heterogeneous dislocation nucleation in fcc copper by
shearing a nanowire on the $(111)$ plane in the $[110]$ direction [see Fig.
11(c) for the simulation setup]. For this, we hold the lower layer of atoms
fixed and move the top layer of atoms with a constant velocity in the $[110]$
direction to achieve volume-conserving shear. The diameter of the nanowire is
$10\,\mathrm{nm}$, i.e., on the order of the smaller precipitates to maximize
surface effects. The resulting shear stress over shear curve is also plotted
in Fig. 11(a). The yield stress of the nanowire, $\tau_{\mathrm{crit}}$, is an
estimate for the stress needed for heterogeneous dislocation nucleation at the
glass-precipitate interface. For comparing the shear stress in the nanowire
with the tensile stress in the composite, we assume a Schmid factor of $0.5$
and scale the tensile axis by a factor $0.5$ compared to the shear stress
axis. This graphically estimates an upper bound of the shear stress $\tau_{m}$
that arrives at the precipitate, corresponding values are given in Table 1.
The upper bound of the estimated maximum resolved shear stress in the
composite is comparable to the lower bound for heterogeneous dislocation
nucleation. While this suggests that dislocation nucleation may be possible,
it is important to keep in mind that the nanowire shear test provides only a
lower bound in an idealized case and that the actual shear stress available to
nucleate a dislocation may be lower than $0.5\sigma_{z}$, which is why no
plastic deformation of the precipitate is observed. Evidently, the value of
$\tau_{\mathrm{crit}}$ in the Mendelev potential is much too high, and
therefore the Cu precipitates are “harder” than expected, which is a
deficiency of the potential model for pure Cu.

Table 1: Comparison of maximum stress $\sigma_{m}$ and the corresponding resolved shear stress $\tau_{m}$ with the critical shear stress $\tau_{\text{crit}}$ for heterogeneous dislocation nucleation in copper in the Mendelev and Ward potentials. The calculation of the resolved shear stress assumes a Schmid factor of $0.5$. |  $\sigma_{m}$ (GPa) |  $\tau_{m}$ (GPa) |  $\tau_{\text{crit}}$ (GPa)
---|---|---|---
Mendelev | $3.2$ | $1.6$ | $1.55$
Ward | $2.9$ | $1.45$ | $0.75$

Because of this, we switch to a different potential, which provides a better
description of crystalline Cu. The Finnis–Sinclair-type potential by Ward et
al. Ward  _et al._ (2012) is created by using preexisting potentials for the
elemental phases Zhou  _et al._ (2004) and fitting the cross terms to the
intermetallic phases. As shown in the Appendix, this potential has a more
realistic unstable stacking-fault energy and critical stress for homogeneous
dislocation nucleation than the Mendelev potential. As shown in Fig. 11(b) and
Table 1, the critical stress for heterogeneous nucleation is much lower than
even the steady-state stress in the composite, easily allowing plastic
deformation of the particle.

The results for a $30$-$\mathrm{nm}$ copper precipitate are shown in Fig. 12
and in Video 3. The precipitate is _cut_ by the shear band, and slip transfer
through the particle can be observed. This mechanism replaces the previously
discussed blocking of the shear band if the nanoprecipitates are “soft”: For
the crystal to partake in the plastic deformation, dislocation nucleation must
be possible at shear stresses below the highest resolved shear stress in the
metallic glass at yield. Despite the participation of the crystalline phase in
the plastic deformation, the stress-strain curve in Fig. 11(b) shows the
distinctive stress drop connected with a single critical shear band and no
strain hardening. The reason is that in this setup the crystalline phase
accounts only for roughly $5\,\mathrm{vol\%}$ of the sample. This means that
the macroscopic mechanical properties are still dominated by the metallic
glass. As the shear band can simply cut through the crystal, the precipitate
poses no obstacle to the percolation of the critical shear band. For a larger
crystalline volume, a ductile crystalline phase could possibly also constrain
the shear bands.Hofmann  _et al._ (2008)

##  IV Discussion

Using TEM imaging, we observe shear-band bending around or close to
precipitates, an attraction of shear bands to the precipitates, and shear
bands being blocked by precipitates. In our MD simulations, we find four
mechanisms of interaction between shear bands and precipitates:

  1. (i)

precipitates that are small relative to the shear-band width dissolve
mechanically,

  2. (ii)

shear bands can wrap around precipitates,

  3. (iii)

shear bands are blocked by precipitates, and

  4. (iv)

shear bands cut through precipitates, and slip transfer into the crystalline
phase takes place.

Which of these mechanisms is active for a given precipitate depends on the
competition between the propagation of the existing shear band, the
heterogeneous nucleation of a new shear band, and the heterogeneous
dislocation nucleation in the precipitate. The wrapping-to-blocking transition
can be quantified by the parameter $\Lambda=A/l\propto A\sqrt{f}/d$. Below
$\Lambda=\Lambda_{\mathrm{crit}}$, the wrapping mechanism is favored. This
value can be be explained by the following simple argument. When the shear
band reaches a precipitate which does not deform, the shearing of the sample
momentarily stops. The stress $\tau_{\mathrm{SB}}$ in the shear band resulting
from the externally applied tensile stress $\sigma_{\mathrm{ext}}$ amounts to

$\tau_{\mathrm{SB}}=\frac{1}{2}\sigma_{\mathrm{ext}}.$ (7)

This stress acts mainly on the shear-band front, allowing us to write

$F_{\mathrm{ext}}\approx\tau_{\mathrm{SB}}\times l\times
h_{\mathrm{SB}}=\frac{\sigma_{\mathrm{ext}}}{2}\,l\,h_{\mathrm{SB}},$ (8)

where $h_{\mathrm{SB}}$ is the width of the shear band. At the moment that the
shear band hits the precipitate, the force $F_{\mathrm{ext}}$ must be equal to
a reaction force $F_{\mathrm{back}}$ from the precipitate (actio est reactio).
Using the projected precipitate area $A$ (cf. Fig. 13), we can convert that
force into a normal stress:

$\sigma_{A}^{n}=\frac{F_{\mathrm{back}}}{A/2}.$ (9)

Figure 13: Projection of the forces and stresses acting around the precipitate
(blue circle) onto the $xz$ plane. Shear bands are shown as hatched areas,
where yellow signifies the arriving shear band, green the path for wrapping,
and red the site for the nucleation of a new shear band. Figure 14: Schematic
view of different mechanisms for the interaction of a shear band with a
crystalline precipitate. With increasing precipitate sizes, the dissolution of
the precipitate is first replaced by the wrapping mechanism. Depending on the
critical stress for dislocation nucleation, wrapping is replaced by blocking
or cutting. Wrapping can be favored by increasing the precipitate distance.

We assume that $F_{\mathrm{ext}}$ predominantly acts on one half of the
obstacle (area $A/2$), which is supported by the deformation pattern of the
plastically deformed particle (Fig. 12 and Video 12). This back stress results
in a shear stress $\tau_{\mathrm{wrap}}\approx 0.5\sigma_{A}^{n}$ in the plane
of the wrapping shear band (green shear band in Fig. 13). With
$F_{\mathrm{back}}=F_{\mathrm{ext}}$, it is

$\sigma_{A}^{n}=\frac{F_{\mathrm{ext}}}{A/2}=\frac{\sigma_{\mathrm{ext}}\,l\,h_{\mathrm{SB}}}{A}=2\tau_{\mathrm{wrap}}.$
(10)

$\tau_{\mathrm{wrap}}$ must surpass a critical value
$\tilde{\tau}_{\mathrm{wrap}}$ to allow the initiation of the wrapping
mechanism; otherwise the shear band simply stops propagating. This is not
observed in our simulations, suggesting that $\sigma_{\mathrm{ext}}$ at yield
is greater than

$\tilde{\sigma}_{\mathrm{ext}}=\frac{2A\tilde{\tau}_{\mathrm{wrap}}}{l\,h_{\mathrm{SB}}}.$
(11)

The competing mechanism, blocking the shear band and nucleating a new one (red
shear band in Fig. 13), can simply be expressed by a critical shear stress
$\tilde{\tau}_{\mathrm{nucl}}$. Because of the low temperature in the
simulation and a stress close to the yield stress, we consider only the
athermal case and do not invoke a nucleation term which takes into account the
relative volume of the interface. With
$\tau_{\mathrm{nucl}}=0.5\sigma_{\mathrm{ext}}$, the transition from wrapping
to blocking takes place where

$\displaystyle
2\tilde{\tau}_{\mathrm{nucl}}=\tilde{\sigma}_{\mathrm{ext}}=\frac{2A\tilde{\tau}_{\mathrm{wrap}}}{l\,h_{\mathrm{SB}}},\quad\text{giving}$
(12)
$\displaystyle\frac{A}{l}=\Lambda_{\mathrm{crit}}=\frac{\tilde{\tau}_{\mathrm{nucl}}}{\tilde{\tau}_{\mathrm{wrap}}}h_{\mathrm{SB}}.$
(13)

This derivation also works in the case of externally applied shear stress, by
replacing $\sigma_{\mathrm{ext}}$ with $2\tau_{\mathrm{ext}}$.

Assuming that nucleating a new shear band at the interface and propagating the
wrapping shear band along the interface have similar critical stresses, we can
simplify Eq. 13 to

$\Lambda_{\mathrm{crit}}\approx h_{\mathrm{SB}}.$ (14)

Shear bands in Cu-Zr-based glasses have widths of around
$10\,\mathrm{nm}$,Ritter and Albe (2011) which fits to the
$\Lambda_{\mathrm{crit}}=12.65\,\mathrm{nm}$ observed in our model systems. As
stated earlier, the wrapping mechanism becomes more favorable again if the
shear band is longer before it hits the precipitate. The reason is that this
gives the shear band time to deviate slightly from its path, so that it does
not hit the precipitate centrally, thus effectively reducing $A$. In practice,
this is not a big problem, as the free shear-band length is constrained to
approximately $l$ anyway due to the distribution of precipitates in the
sample. While this derivation is only approximate, it seems to be sufficient
to explain the observed phenomena and guide future efforts in tuning the
mechanical behavior of crystal–glass composites. Furthermore, it is easily
possible to explain the fourth mechanism, a slip transfer into the crystalline
phase. The critical stress for heterogeneous nucleation of a dislocation in
the precipitate $\tilde{\tau}_{\mathrm{disl}}$ must be provided by the shear
band via $\tau_{\mathrm{SB}}=0.5\sigma_{\mathrm{ext}}$. If
$\tilde{\tau}_{\mathrm{disl}}<\tilde{\tau}_{\mathrm{nucl}}$, we can simply
replace $\tilde{\tau}_{\mathrm{nucl}}$ by $\tilde{\tau}_{\mathrm{disl}}$ in
Eqs. 12 and 13, thereby replacing the blocking mechanism with the plastic
deformation of the precipitate. A lowered $\tilde{\tau}_{\mathrm{disl}}$ also
lowers $\Lambda_{\mathrm{crit}}$.

A simple deflection of the shear band is not observed and seems unlikely, as
any deviation of the shear band from its path leads to a reduced resolved
shear stress and thereby to a driving force to put it back “on track.”
Contrary to the experiment, a change of shear-band path towards the
precipitate was also not observed. Because the precipitates in our simulations
are inserted artificially and do not have a large stress field around them,
this seems reasonable. In thermally grown precipitates, a stress field due to
density mismatch between glass and crystal seems likely. Still, due to the
geometry of the MD simulations, the shear band has two equivalent propagation
pathways from the notch but always chooses the one leading towards the
precipitate. This seems to be a weaker form of the attraction observed
experimentally.

With these results, we can attempt an explanation of the experimentally
observed phenomena. First of all, the blocking of the shear band is a one-to-
one correspondence between simulation and experiment. Comparing Fig. 3(e) with
Fig. 7, we can see that the path of the shear band looks identical. The shear
band wraps partly around the precipitate but is then blocked and does not
propagate. Contrary to the simulation, no fully formed shear band but only a
small shear-band-like region appears at the opposite crystal–glass interface.
This may be a result either of the more complex stress state in the experiment
or the fact that other precipitates are available at which the new shear band
may nucleate. For the winding shear-band path, we can now exclude a simple
deflection as discussed above. A possible explanation would be the concurrent
nucleation of nascent shear bands at the crystal–glass interfaces which grow
together into a single mature shear band. While the interfaces are known to be
nucleation sources for shear bands,Albe  _et al._ (2013); Zaheri  _et al._
(2014); Wang  _et al._ (2014a) our simulations show that the nucleation of a
shear band at a stress concentrator like a notch or a crack always takes
precedence to nucleation at interfaces or surfaces. The shear bands shown in
the TEM images all originate from crack tips, making it unlikely that the
shear band shown in Fig. 2 consists of several concurrently nucleated shear
bands. This leaves the explanation that this winding path is a series of
subsequent blocking and renucleation events.

The observed $\Lambda_{\mathrm{crit}}$ corresponds to precipitate diameters
somewhere between $20\,\mathrm{nm}$ and $40\,\mathrm{nm}$, depending on the
interparticle distance. This critical diameter is on the order of magnitude
reported in several experimental studies for twinning in B2 crystals in Cu-Zr-
Al-based metallic glasses of $20\pm 5\,\mathrm{nm}$.Pauly  _et al._ (2010b);
Kuo  _et al._ (2014) It also fits an experimental work on Al-based glasses,
where crystallites growing during deformation are sheared apart when they
reach a critical size of about $10\,\mathrm{nm}$.Hebert  _et al._ (2006)
Cu50Zr45Ti5 metallic glasses exhibit a critical size of about $9\,\mathrm{nm}$
for twinning of B2 precipitates.Wang  _et al._ (2014b) The interparticle
distances in these experiments are on the same order of magnitude as for our
simulations. This supports an explanation of a transition from wrapping to
slip transfer in these systems.

Figure 14 summarizes the competition between the different mechanisms.
Mechanical dissolution of the crystalline particles occurs only if their size
is comparable to the shear-band size.Lund and Schuh (2007) With further
increasing precipitate sizes, the shear band can still wrap around the
obstacle until the size reaches a threshold value. This critical size also
depends on the precipitate distance, as discussed before, expressed in the
parameter $\Lambda_{\mathrm{crit}}(d,l)$. If wrapping is no longer possible, a
precipitate which reacts only elastically to the applied stress will block the
shear band. If the precipitate is susceptible to plastic deformation, slip
transfer into the precipitate will take place.

Concerning the mechanical performance of such in situ composites with
crystalline precipitates that originate from nucleation and growth within the
glass, the current results suggest that the discussed geometrical effects
serve to improve the macroscopic mechanical performance. None of the presented
mechanisms seem to inhibit the percolation of a critical shear band, yet,
catastrophic slip along a shear band leading to complete failure is delayed in
the case of winding shear bands due to the increase of the shear-band path
length as well as the raised activation barriers for slip along shear bands
that have a more complex topology. The wrapping mechanism does not pose an
obstacle to shear-band propagation, but can be avoided by appropriate
adjustment of the crystalline volume fraction and precipitate diameter. “Soft”
precipitates additionally open possibilities to adjust the plastic deformation
by participating in it. Consistently, by increasing the volume fraction of the
ductile crystalline phase, the constraints on shear-band propagation can be
increased, immediate failure can be prevented (cf. Ref. Hofmann  _et al._ ,
2008), and the composite displays macroscopic mechanical behavior according to
a mixing rule, further allowing one to tailor the properties.

##  V Conclusions

Using TEM imaging, we observe shear-band bending around or close to
precipitates, an attraction of shear bands to the precipitates, and shear
bands being blocked by precipitates. MD simulations reveal that the shear-band
bending is most likely the result of the subsequent blocking and renucleation
of shear bands. Moreover, we identify shear bands wrapping around precipitates
and slip transfer into the crystalline phase. By describing the competition
between the critical stress for wrapping, the nucleation of a new shear band,
and the nucleation of dislocations in the crystal, we could derive a mechanism
map for metallic glasses with nanocrystalline precipitates. This detailed
description of shear-band propagation not only helps to understand the
mechanical failure of these composites but also aids in tuning them.

Figure 15: Generalized stacking-fault energies (top) and the resulting shear
stresses (bottom) for fcc copper. The DFT values for the stress curve are from
Ogata et al.,Ogata  _et al._ (2002) and the corresponding stacking-fault
energies are approximated by using a numerical integration of the stress data.

###### Acknowledgements.

The authors thank J. Bünz and M. Gerlitz for help with sample preparation and
DSC characterization. Financial support by the Deutsche Forschungsgemeinschaft
(DFG) through project grants nos. AL 578/13-1, AL 578/6-2, and WI 1899/12-1 is
gratefully acknowledged. Computing time was granted by the John von Neumann
Institute for Computing (NIC) and provided on the supercomputer JUROPA at
Jülich Supercomputing Centre (JSC), as well as by the Gauss Centre for
Supercomputing (GCS) through the NIC on the GCS share of the supercomputer
JUQUEEN at JSC. GCS is the alliance of the three national supercomputing
centers HLRS (Universität Stuttgart), JSC (Forschungszentrum Jülich), and LRZ
(Bayerische Akademie der Wissenschaften), funded by the German Federal
Ministry of Education and Research and the German State Ministries for
Research of Baden-Württemberg, Bayern, and Nordrhein-Westfalen. Additional
computing time was made available by the Technische Universität Darmstadt on
the Lichtenberg cluster.

*

##  Appendix A Generalized stacking-fault energy in fcc copper

The generalized stacking-fault energy in fcc copper is calculated by using
both Mendelev Mendelev  _et al._ (2009) and Ward Zhou  _et al._ (2004); Ward
_et al._ (2012) potentials. Density-functional theory (DFT) calculations from
Ogata et al. Ogata  _et al._ (2002) using the same method are used for
comparison. This is plotted in Fig. 15. The Ward potential much more
accurately describes the stacking-fault energy than the Mendelev potential,
which has a critical stress for homogeneous dislocation nucleation which is
more than two times too high.

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