Shear Banding in Binary Cu-Zr Metallic
Glass: Comparison of the G-Phase
With L-Phase
Yidi Shen
1
, William L. Johnson
2
*, Konrad Samwer
2
,
3
, Sydney L. Corona
2
,
William A. Goddard
4
*
†
and Qi An
1
*
1
Department of Chemical and Materials Engineering, University of Nevada-Reno, Reno, NV, United States,
2
Department of
Materials Science, California Institute of Technology, Pasadena, CA, United States,
3
I. Physikalisches Institut, University of
Goettingen, Goettingen, Germany,
4
Materials and Process Simulation Center, California Institute of Technology, Pasadena, CA,
United States
We identi
fi
ed two glass phases formed in three undercooled liquids of elemental Ag, binary
Cu-Ag, and binary Cu-Zr alloys using molecular dynamics (MD) simulations: 1) The
homogeneous L-phase arises from quenching quickly from high temperature liquid. 2)
The heterogeneous solid-like G-phase arises from the isothermal equilibration at
temperatures below the melting point. The G-phase exhibits a core-shell structure with
the ordered cores surrounded by percolating liquid-like shells. The distinguishable
structures between these two phases are expected to different mechanical behavior.
The present study reports MD simulations to compare the shear deformation of these two
phases in binary Cu
2
Zr system. At room temperature, the G-phase exhibits a higher critical
stress, a higher critical strain, and higher shear modulus than the L-phase, suggesting that
the G-phase has improved strength and rigidity compared to the homogeneous L-phase.
The plastic yielding mechanism of both the G-phase and L-phase is accompanied by shear
band formation. However, the formation of shear band in G-phase is con
fi
ned by the cores
to a highly localized region and characterized by local mechanical melting. In contrast, the
shear band in L-phase exhibits greater width and much more homogenous character. We
conclude that the mechanical properties of a metallic glass will vary signi
fi
cantly according
to the type of glassy phase formed during processing.
Keywords: metallic glass, plastic
fl
ow, shear deformation, MD, shear band, bauschinger effect
INTRODUCTION
The mechanical properties of bulk metallic glasses (BMGs) have stimulated much interest due to
such outstanding properties as high strength, large elastic strain limit, and good fracture toughness.
However, the BMGs normally show low plasticity at room temperature, which limits extended
applications (
Gu et al., 2007
;
Li et al., 2013
;
Yang et al., 2019
;
Wu et al., 2021
). The low plasticity
originates from the unique deformation mechanism under loading, which is different from typical
crystalline metals. BMGs tend to form highly localized shear bands due to the lack of dislocation
systems and grain boundary structures (
Pekarskaya et al., 2001
;
Yang et al., 2005
;
Schuh et al., 2007
;
Greer et al., 2013
). These shear bands are much softer and more susceptible to subsequent
fl
ow than
surrounding regions (
Greer et al., 2013
). Thus, plastic strains are localized within the shear bands
while regions outside the shear bands deform elastically, which leads to the low global plasticity of
BMGs (
Klaumünzer et al., 2011
;
Tang et al., 2016
). The soft shear bands have been widely observed in
Edited by:
Yue Fan,
University of Michigan, United States
Reviewed by:
Bin Wu,
Beijing Normal University, China
Kostya Trachenko,
Queen Mary University of London,
United Kingdom
*Correspondence:
William L. Johnson
wlj@caltech.edu
William A. Goddard
wag@caltech.edu
Qi An
qia@unr.edu
†
ORCID:
William A. Goddard
orcid.org/0000-0003-0097-5716
Specialty section:
This article was submitted to
Computational Materials Science,
a section of the journal
Frontiers in Materials
Received:
28 February 2022
Accepted:
31 March 2022
Published:
14 April 2022
Citation:
Shen Y, Johnson WL, Samwer K,
Corona SL, Goddard WA and An Q
(2022) Shear Banding in Binary Cu-Zr
Metallic Glass: Comparison of the G-
Phase With L-Phase.
Front. Mater. 9:886788.
doi: 10.3389/fmats.2022.886788
Frontiers in Materials | www.frontiersin.org
April 2022 | Volume 9 | Article 886788
1
ORIGINAL RESEARCH
published: 14 April 2022
doi: 10.3389/fmats.2022.886788
previous studies. For example, a single shear band in Zr-based
BMGs has been isolated in a recent experimental study (
Maaß
et al., 2014
). Even though the width of the shear band is of the
order of some nanometers, the region showing structural
softening due to the shear band has micrometer width. Thus,
to understand and improve the mechanical properties for future
engineering applications, it is essential to understand the
mechanism of shear band formation.
Many studies have focused on understanding the failure
mechanism and improving the mechanical properties of
various BMG systems. Regarding the failure mechanism, it
was proposed that the shear bands arise from the generation
of free volume around the atoms to facilitate atomic motion and
nucleating shear bands (
Yang et al., 2005
;
Spaepen, 2006
;
Dai
and Bai, 2008
;
Tang et al., 2016
). In addition, adiabatic heating
and softening is reported as another factor for localization of
plasticity in BMGs. Local heating beyond glass transition
temperature produces thermal s
oftening, resulting in rapid
shear band propagation and structural failure in BMGs
(
Lewandowski and Greer, 2006
;
Yang et al., 2006
;
Dai and
Bai, 2008
;
Wang et al., 2015
). Regarding the improvement of
mechanical properties, several approaches have been developed
to improve the global plasticity of BMGs at room temperature.
This includes structure modi
fi
cations (
Hays et al., 2000
;
Szuecs
et al., 2001
;
Schroers and Johnson, 2004
;
Lee et al., 2006
;
Liu
et al., 2007
;
Chen et al., 2008
), elastostatic compression (
Lee
et al., 2008
), and ion irradiation (
Ke et al., 2011
). For example, a
family of BMGs consisting of ZrCuNiAl were synthesized by
choosing the composition to maximize the Poisson ratio (
Liu
et al., 2007
). The ZrCuNiAl BMGs can undergo a true strain in
compression as large as 160% without failure. TEM
investigations on the microstructure of BMG composites
indicate that these super plastic BMGs are composed of
alternating hard regions and soft regions. Consequently, the
shear transformation zones (STZ) prefer to activate in soft
regions leading to numerous shear bands under loading.
Propagation of the shear bands is arrested by hard regions,
which encourages shear-band multiplication. Multiplication of
the shear bands assists the enhanced global plasticity in the
BMG composites.
The mechanical properties of BMGs are expected to be
related to the local atomistic structure. In our recent MD
simulations, we identi
fi
ed two glass phases in both
elementary Ag and binary (Ag-Cu and Zr-Cu) liquids (
An
et al., 2020a
;
2020b
,
2021
,
2022
).Whenthemetallicliquids
are ultrafast quenched to room
temperature (quench rate:
~10
12
K/s), the homogeneous liquid-like L-phase glass forms.
In contrast, if the metallic liquid is quenched to intermediate
temperatures below melting temperature, it evolves
isothermally to the heterogen
eous solid-like G-phase glass
through a
fi
rst order freezing transition. Unlike the
homogenous L-phase, the G-phase possesses local core
regions with crystalline-like short-range order surrounded by
a liquid-like matrix. In particular, for Ag and Cu-Ag alloys, the
core regions of the G-phase are mainly composed of crystalline-
like FCC + HCP short range order with a heterogeneity
correlation length scale
Λ
~4 nm in pure Ag metals
decreasing to
Λ
~2 nm in the eutectic Ag-Cu system (
An
et al., 2020b
). For Cu-Zr alloy, the core regions of G-phase
show icosahedral short-range order, with correlation length
Λ
in
the range of 1.5
–
1.8 nm(
An et al., 2022
). We also calculated the
shear modulus and viscosity of both G-phase and L-phases from
MD simulations. The G-phase has much higher elastic modulus
and viscosity compared to the L-phase, suggesting the G phase
shows greater elastic rigidity (
An et al., 2020a
;
2020b
,
2022
).
Besides, a recent theoretical study shows formation of a partially
crystalized glass phase in Cu
2
Zr metallic liquid when
isothermally holding at 950 K. This ordered state exhibits
icosahedral structural character, which is similar to the
G-phase in our study(
Yue et al., 2020
). Evidence for the
existence of these two glass phases in BMG forming alloys
has also been reported in the recent experimental studies. In
the ultra-fragile Pt
–
Cu
–
P ternary alloy system, the glass
transition evolves to an apparent
fi
rst-order melting
transition with varying composition (
Na et al., 2020
). This
study suggests the existence of two amorphous glass phase: a
high temperature disordered liquid-like glassy L-phase and a
con
fi
gurationally ordered lower temperature solid-like G-phase
glasses. The G-phase like metallic glacial glass (MGG) was
synthesized by a
fi
rst order transformation of an initially
quenched metallic glass phase
(MG) that was reheated into
the supercooled liquid of a La
32.5
Ce
32.5
Co
25
Al
10
alloy (
Shen
et al., 2020
).TheMGG-phaseexhibitsa
fi
rst-order melting-like
transition to the disordered liquid phase upon subsequent rapid
FIGURE 1 |
An example illustrating the model for the shear deformation.
During shear deformation, the atoms in the lower layer are
fi
xed and the atoms
in upper layer have the maximum velocity of 20 m/s along the shear direction.
The velocity of atoms in the mobile layer increases linearly from the
bottom boundary to the upper boundary. In the mobile layer, the Cu and Zr
atoms are represented by blue and green balls.
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April 2022 | Volume 9 | Article 886788
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Shen et al.
Shear Banding in Cu-Zr Glass
heating. The hardness of the MGG phase is 20% higher
compared to the initially quenched metallic glass (MG),
providing further evidence of two distinct glass/liquid phases.
In the present study, we focus on the binary Cu
2
Zr system
and perform MD simulations to examine shear deformation
behavior of both the L-phase and the G-phase at room
temperature. Our simulations show that the critical shear
stress and shear modulus of the G-phase (2.82 GPa for
critical shear stress and 29.97 GPa for shear modulus) are
both much greater than those of the L-phase (1.25 GPa for
critical shear stress and 17.93 GPa for shear modulus). This
suggests that the G-phase is stronger and exhibits greater
rigidity compared to the L-phase. In addition, the G-phase
has a larger critical shear strain (0.15) than that of L-phase
(0.10). Thus, the G-phase is both stiffer and stronger than
L-phase. Further analysis of the yielding mechanism indicates
that shear strain localization (shear band formation) occurs for
both G-phase and L-phases, following initial yielding. Within
the localized shear band, the ordered core regions of the
G-phase appear to lose their icosahedral character during
shear sliding, this suggests local mechanical melting within
the shear band. In contrast, the L-phase exhibits much broader
and more diffuse shear band apparently related to its more
homogenous structural character. After G-phase and L-phase
reaches their steady
fl
ow state, their response under following
cyclic loading exhibits the Bauschinger effect. The reason
might be that the atoms in soft shear band have higher
mobility and less resistance to shear. The plastic
deformation in these structures becomes dominated earlier
compared to unsheared structures.
METHODS
Model Construction and MD Simulations
All MD simulations were performed using the LAMMPS code
(
Plimpton, 1995
). Sheng
’
sCu
–
Zr embedded atom model (EAM)
potential is applied in MD simulations to describe the
interatomic interactions (
Cheng et al., 2009
). The velocity
Verlet algorithm is used for integrating the equations of
motion with a timestep of 1.0 fs in all simulations. Periodic
boundary conditions (PBC) along all three directions are used to
eliminate surface effects in all simulations except for the shear
deformation.
The L-phase is obtained from ultrafast quenching, which is
similar to our previous studies (
An et al., 2021
,
2022
). Firstly,
the liquid with 41,472 atoms is equilibrated at 2000 K using
the NPT ensemble for 1ns. Then, the liquid is quenched from
2000 K to room temperature with a high cooling rate of 2.125
×10
12
K/s. In contrast, the G-phase is produced by
isothermal aging the supercooled liquid at 900 K for
750 ns, as discussed in our previous work (
An et al., 2022
).
Then, the G-phase is quenched to room temperature using
thesamecoolingrateasfortheL-phase(2.125×10
12
K/s).
Next, both L-phase and G-phase are relaxed at room
temperature for 16 and 1 ns using NPT dynamics until
reaching equilibrium, respectively.
FIGURE 2 |
The L-phase and G-phase at room temperature characterized using OP-8 bond-orientational order parameters analysis:
(A)
G-phase and
(B)
L-phase.
FIGURE 3 |
The shear-stress-shear-strain relationships of both G-phase
and L-phase under shear deformation at room temperature.
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Shen et al.
Shear Banding in Cu-Zr Glass
After obtaining the equilibrium L-phase and G-phase at room
temperature, we apply
fi
nite shear rate deformations to examine
their shear-induced failure processes. Here, our model is
partitioned into three layers: the 0.5 nm thick immobile lower
layer, the 0.5 nm thick immobile upper layer, and the mobile layer
between these two layers with thickness of 7.6 nm for L-phase and
8.5 nm for G-phase, as shown in
Figure 1
. Then, a
fi
xed velocity
of 20 m/s along the shear direction is assigned to the atoms in the
upper layer while the atoms in the lower layer are
fi
xed. The
velocities of atoms in the mobile layer increases from 0 to 20 m/s
on top of their thermal velocities at 300 K. The corresponding
engineering strain rate for all simulations is ~0.002 ps
−
1
. We also
consider lower strain rates of 0.0002 ps
−
1
and 0.00002 ps
−
1
to
investigate the effects of strain rate on shear deformation of both
G-phase and L-phase. During the shear deformation, the NVE
ensemble is applied to for the mobile atoms. Meanwhile, the
velocities of two non-sheared directions for the atoms in the
mobile layer is rescaled every 100fs to maintain 300 K for the
system.
Plastic Yielding and Flow Mechanism
Analysis: Atomic Structure Analysis and
Bond-Orientational Order Parameter
To analyze the structural changes during shear band formation
and steady state shear, we calculated the atomic von Mises strains
FIGURE 4 |
The atomic von-Mises strain map of G-phase under the shear deformation.
(A)
The intact structure,
(B)
the structure at 0.15 shear strain before the
initiation of shear band, and
(C)
the structure at 0.22 shear strain after forming the shear band.
FIGURE 5 |
The structural changes of G-phase under shear deformation characterized using OP-8 order parameter.
(A)
The intact structure,
(B)
the structure at
0.15 shear strain before the initiation of shear band, and
(C)
the structure at 0.22 strain after forming the shear band. The shear band is displayed between two solid red
lines at
(C)
.
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April 2022 | Volume 9 | Article 886788
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Shen et al.
Shear Banding in Cu-Zr Glass
(
Shimizu et al., 2007
) using the software OVITO (
Stukowski,
2010
). The atomic von Mises strains are examined based on the
particle displacement vectors of current atomistic con
fi
guration
referenced to the initial structure (
Shimizu et al., 2007
;
Tucker
et al., 2011
). A large von Mises strain suggests a large local atomic
strain.
The local ordered structures during the shear deformation are
analyzed using the bond-orientational order parameters (BOOP),
q
l
(l = 2,4,6,8,
...
)(
Steinhardt et al., 1983
;
Mickel et al., 2013
). In
BOOP, the q
l
are sensitive to local atomic symmetries in the bond
angles, which are compared to values for ideal lattices. This allows
analysis of the atomic arrangements on the level of individual
constituents. In this study, q8 parameters were adopted to
identify the structural change because they are sensitive to the
icosahedral structure in the G-phase and the disordered structure
in the L-phase (
An et al., 2022
). Note that the cut-off range of each
atomic site relative to its neighboring atoms for both the atomic
strain and the BOOP analysis is determined using the
fi
rst
minimum value following the
fi
rst maximum of the radial
distribution function.
RESULTS AND DISCUSSION
The Flow Mechanism of G-phase Under
Shear Deformation
The local atomic structure of the G-phase at room temperature is
characterized using BOOP analysis, as shown in
Figure 2A
. This
structure shows a heterogenous character being composed of two
different con
fi
gurational regions: a icosahedral-ordered core
region with low OP-8 surrounded by a percolating liquid-like
disordered matrix with high OP-8 (
An et al., 2022
).
We performed the shear deformation of G-phase at room
temperature. The predicted shear-stress-shear-strain relationship
of the G-phase is shown in
Figure 3
. The predicted critical shear
stress is 2.82 GPa at a shear strain of 0.15. Before the shear stress
reaches its maximum value, the G-phase
fi
rst undergoes an elastic
deformation at a calculated shear modulus of 29.97 GPa. Then the
shear stress reaches a peak followed by a continuous decrease
from the maximum shear stress to ~1.0 GPa with an increase of
shear strain up to 0.22, suggesting plastic
fl
ow in the G-phase. The
fl
ow behavior of the G-phase under shear deformation is shown
in
Figure 4
. To analyze the plasticity mechanism, we display the
atomic von Mises shear strain maps during shear deformation.
Figure 4A
shows the intact structure before shear. As the shear
strain increases to 0.15, which corresponds to the peak shear
stress of 2.82 GPa, the structure deforms elastically, as shown in
Figure 4B
. Note that the atoms in the core regions experience
relatively smaller shear strain compared to the atoms in the
surrounding disordered region up to 0.15 shear strain. As the
shear strain increases further continuously, a shear band
nucleates and then propagates along the shear direction,
leading to relaxation of the overall shear stress as well as
localization of the shear strain into a shear band. In particular,
at 0.22 shear strain, a shear band having a width of ~3 nm
propagates across the entire G-phase MD cell, which correlates
with the stress drop to ~1.0 GPa, as shown in
Figure 4C
.
Interestingly, the width of the formed shear band varies along
the shear direction while a region with low atomic strain exists
inside the shear band near the sample boundaries along the
x
-axis. This suggests that this shear band may actually be
bifurcated.
To understand the plasticity mechanism of shear band
sliding in the G-phase, the BOOP analysis was performed to
investigate the evolution of loc
ally ordered structures during
shear band sliding. The OP-8 order parameters are adopted to
examine the local structures dur
ing the shear deformation. The
intact structure before shear is shown in
Figure 5A
in which the
FIGURE 6 |
The Bauschinger effect in G-phase after reaching to the state of plastic
fl
ow.
(A)
the shear-stress-shear-strain relationships of G-phase under cyclic
shear deformation,
(B)
the
fi
nal structure characterized using OP-8 order parameter when shear along opposite direction, and
(C)
the
fi
nal structure characterized using
OP-8 order parameter when shear along initial direction.
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Shen et al.
Shear Banding in Cu-Zr Glass
core regions and surrounding di
sordered matrix regions can be
clearly delineated. At 0.15 shear strain, the OP-8 values in the
core regions begin to increase compared to the intact structure
due to the evolving plastic shear strain, as shown in
Figure 5B
.
However, not all core regions are deconstructed at this shear
strain. As the global shear strain increases to 0.22, a shear band
core with high OP-8 forms, which agrees with the atomic strain
map, as shown in
Figure 5C
. More important, as strain
increases, part of the core regions lying within the shear
band transform from the ordered structure (low OP-8) to the
disordered structure (high OP-8), suggesting that the shear
band sliding in the G-phase arises from mechanical melting
of these core regions. In addition, the plasticity is clearly
localized and correlated with the results of OP-8 and is
consistent with the results from atomic von-Mises strain. The
nearby core region with a crystal-like local structure appears to
block the propagation of the shear band.
After the G-phase reaches plastic-
fl
ow state at 1.0 shear strain,
we then applied reversed shear strain to investigate its response
under opposite loading condition. The obtained shear-stress-
shear-strain relationship is shown in
Figure 6A
. During the
shear deformation along reversed direction, the stress quickly
decreases to 0 and then increases to the same absolute value of
steady-state
fl
ow stress (~1.0 GPa). After G-phase reaches the
plastic
fl
ow state again, we switched the shear direction one more
time to the initial direction and further explored the plastic
evolution of G-phase under cyclic loading. Similar to the
response during the shear deformation along reversed
direction, the shear stress quickly
fl
ips back and then shows a
steady
fl
ow with the value of ~1.0 GPa, as shown in
Figure 6A
.
This stress reversal under cyclic loading indicates the Bauschinger
effect (
Karmakar et al., 2010
;
Frahsa et al., 2013
;
Urata and Li,
2018
;
Patinet et al., 2020
) in G-phase under pure shear
deformation. In order to understand the origin of this
FIGURE 7 |
The shear-stress-shear-strain relationships and the atomic von-Mises strain map of G-phase with various strain rates ranging from 0.00002 to
0.002 ps
−
1
.
(A)
The shear-stress-shear-strain relationships,
(B
–
D)
the von-Mises strain maps after forming shear band under different strain rates of
(B)
0.002 ps
−
1
,
(C)
0.0002 ps
−
1
, and
(D)
0.00002 ps
−
1
.
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Shen et al.
Shear Banding in Cu-Zr Glass
phenomenon, we then used OP-8 order parameters to examine
the evolution of local structures. The structures at 0 and 1 shear
strains, which correspond to the
fi
nal structures when shear along
both opposite and initial directions, are shown in
Figures 6B,C
.
Compared to the structure after forming the shear band
(
Figure 5C
) during the initial shear deformation, no obvious
structural change is observed in these two structures. Thus, the
Bauschinger effect for G-phase might be explained by the
composite model (
Pedersen et al., 1981
;
Sun et al., 2014
).
Under the initial loading, the formation of shear band suggests
the existence of both the softer region (shear band) and the harder
region (surrounding undeformed region) in plastic deformed
G-phase (
Greer et al., 2013
). When loading is reversed, the
atoms in the shear band have higher mobility and less
resistance to shear than those in the surrounding regions (
Sun
et al., 2014
). Therefore, the formed shear band can plastically
deform earlier than the initially unsheared structure, causing the
Bauschinger effect.
From the results of both atomic strain and BOOP analyses, we
conclude that the plastic
fl
ow mechanism of the G-phase under
shear deformation arises from local mechanical melting of core
regions within the operating of a shear band. In addition, the
nearby icosahedral-ordered core regions apparently obstruct and
de
fl
ect propagation of the shear band, causing additional
localization of the shear band between surrounding core
regions. After G-phase reaches plastic-
fl
ow state, it follows the
plastic manner of Bauschinger effect under cyclic loading. This
might result from the formed soft shear band that has less
resistance to shear and can plastically deform earlier than
initially unsheared structure.
Strain rate is an important factor for the shear deformation. To
illustrate the strain rate effects, we considered two lower strain
FIGURE 8 |
The shear band formed in the G-phase characterized using OP-8 order parameter under various strain rates of
(A)
0.002 ps
−
1
,
(B)
0.0002 ps
−
1
,and
(C)
0.00002 ps
−
1
.
FIGURE 9 |
The atomic von-Mises shear strain map of L-phase under the shear deformation.
(A)
The intact structure,
(B)
the structure at 0.1 shear strain before the
initiation of shear band, and
(C)
the structure at 0.24 strain after forming the shear band.
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Shear Banding in Cu-Zr Glass