# Yielding in multi-component metallic glasses: Universal signatures of
elastic modulus heterogeneities

Kamran Karimi1 Corresponding author<EMAIL_ADDRESS>Mikko J.
Alava1,2 Stefanos Papanikolaou1 1 NOMATEN Centre of Excellence, National
Center for Nuclear Research, ul. A. Sołtana 7, 05-400 Swierk/Otwock, Poland
2 Aalto University, Department of Applied Physics, PO Box 11000, 00076 Aalto,
Espoo, Finland

###### Abstract

Sheared multi-component bulk metallic glasses are characterized by both
chemical and structural disorder that define their properties. We investigate
the behavior of the local, microstructural elastic modulus across the plastic
yielding transition in six Ni-based multi-component glasses, that are
characterized by compositional features commonly associated with solid
solution formability. We find that elastic modulus fluctuations display
consistent percolation characteristics pointing towards universal behavior
across chemical compositions and overall yielding sharpness characteristics.
Elastic heterogeneity grows upon shearing via the percolation of elastically
soft clusters within an otherwise rigid amorphous matrix, confirming prior
investigations in granular media and colloidal glasses. We find clear
signatures of percolation transition with spanning clusters that are
universally characterized by scale-free characteristics and critical scaling
exponents. The spatial correlation length and mean cluster size tend to
diverge prior to yielding, with associated critical exponents that exhibit
fairly weak dependence on compositional variations as well as macroscopic
stress-strain curve details.

In realistic studies of bulk metallic glasses, the associated yielding
properties and shear banding behavior [1] have important industrial and
technological implications in terms of the _ductility_ of the glass. The
ongoing eagerness to promote the latter (in combination with hardening
properties) have already shifted the focus to the development of _multi-
component_ glasses (also known as amorphous alloys and/or high-entropy
metallic glasses) with the highly tunable microstructural/compositional
complexity [2, 3]. The atomistic origin of this new design paradigm is based
on the common observation that glasses with low atomic disparity limit (where
atomic radii of constituent elements are considered to be very similar) have a
high capacity to form localized deformation patterns whereas those at the
opposite limit have a tendency to delocalize strain [4] and, therefore, deform
in a more ductile way. In this context, the shear band structure and design-
level ductility appear to be highly dependent on inherent heterogeneity as an
essential elemental/compositional feature. However, microstructural origins
and elemental dependence of such inhomogeneities and, more importantly,
associated spatial-dynamical evolution upon shearing have yet to be explored.

Despite certain commonalities of failure properties across disordered solids
[5], the _sharpness_ of yielding transition, as a common signature of strain
localization (and ductility), may show strong variations by altering thermal
treatments and chemical compositions in bulk metallic glasses, ranging from
uniformly distributed patterns to system-spanning crack-like features [2, 4,
6, 7]. Owing to the presence of quenched chemical/structural disorder, glassy
metals may accommodate a distributed plastic flow with a significant
contribution to ductility [8, 9, 10]. Certain (aged) glasses that lack this
heterogeneity element [11, 12, 13] (or associated lengths don’t exceed
interatomic scales) tend to localize strain within a single dominant band
before shear instablility results in a catastrophic brittle-type fracture. In
this context, tailoring _elastic_ heterogeneity has recently emerged as a
novel design framework to build more ductile metallic glasses [8, 14]. Under
special thermal treatments and variations of chemical elements, quenched
metallic glasses may nucleate elastically soft clusters that become structured
and _span_ the entire system upon failure, leading to the enhanced plastic
flow and, hence, ductility. This is illustrated in Fig. 1 where yielding
transition, fine-tuned by the chemical compositions, is accompanied by the
percolating networks of _softness_ (shown in brown) in a driven CoNiFe and
CoNiCrFe bulk metallic glass. The relevance of such a percolation transition
of softness within the context of amorphous plasticity has been already
established in (soft) model glasses [15, 16, 17] as well as colloidal systems
[17, 18] under shear. These studies have mainly focused on certain (implicit)
notions of softness based on the degree of deformation non-affinity or shear-
induced mobility that exhibit percolating features near the criticality. Using
a model metallic glass in [19], the glass transition was accompanied by the
percolation threshold of mechanically unstable regions on approach to the
transition temperature $T_{g}$. However, realistic studies of bulk metallic
glasses driven out of equilibrium, with substantial industrial and
technological implications, and associated composition-dependence has been
more challenging.

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Figure 1: Three dimensional softness maps of the b) CoNiFe c) CoNiCrFe
compositions at applied shear strain $\gamma_{xy}\simeq 0.1$ and the
corresponding stress-strain curves $\sigma_{xy}$ versus $\gamma_{xy}$ in a).
The colors in b) and c) indicate different clusters including elastically
unstable regions with $\mu\leq 0$. The slope in a) indicates the softening
modulus $h_{\text{min}}$ . Upon shearing, negative-$\mu$ clusters form
percolating networks (in brown) on approach to failure that tend to correlate
with the (post-failure) macroscopic properties.

Here, using atomistic simulations, we investigate six multi-component bulk
metallic glasses including Co, Ni, Fe, Cr, and Mn at compositions that have
been the focus of recent experimental investigations. The chosen compositions
are commonly believed to optimize solid solution formability, characterized by
low misfit coefficients $\delta_{a}$, and not thought to promote glass forming
ability typically observed for $\delta_{a}>6\%$ [20]. Nonetheless, there is a
way to generate a glassy environment in the opposite limit of
$\delta_{a}\rightarrow 0$ but maintaining a larger number of elements, as in
medium/high entropy metallic glasses (see [3] and references therein).

By probing local elasticity, in particular shear modulus $\mu$ and its
evolution toward failure, we present direct evidence that plastic yielding in
driven metallic glasses takes place _universally_ through a percolation
transition of softness, characterized by regions of negative $\mu\leq 0$ (as
shown in Fig. 1). The sharpness of yielding transition, fine-tuned by chemical
compositions and inferred from the rate of spontaneous drop $h_{\text{min}}$
in the stress response, exhibits _consistent_ correlation features with
softness and its percolation properties. Our cluster analysis associated with
local elasticity maps features critical scaling properties, i.e. scale-free
statistics, divergence of correlation lengths, and critical exponents that are
reminiscent of a non-equilibrium phase transition.

From a broader perspective, our methodology offers a robust indicator of
nonaffinity, e.g. the disorder-induced breakdown of homogeneous response, by
probing elastic heterogeneity in amorphous materials. Conceptually, our
approach is similar to investigations of low energy quasi-localized modes that
are spatially distinguishable from long wave-length phonon modes in elastic
crystalline solids (see [21] and references therein). Predictive plasticity
models _solely_ based on the notion of shear transformation zones (STZs) might
fail to capture such correlations because the latter are generally believed to
be (micro)structural defects but not necessarily indicative of mechanical
instability. Another complication arises from a lack of robust topological
signatures that can identify STZs from their parent liquid-like microstructure
within a glassy matrix. There have been related efforts based upon the machine
learning-rooted concept of “structural” softness but with very limited
predictive capabilities in terms of glass failure and deformation [22, 23,
24]. More focused studies of metallic glasses made an attempt to associate
soft spots to the presence/absence of geometrically unfavored motifs and/or
short/medium range ordering [25, 26, 27] but theses local motifs on their own
often fail to describe the collective nature of plasticity and shear banding
in a broader context of amorphous plasticity. Our study aims to augment such
efforts, directed mainly towards the notion of structural heterogeneity, by
adding the elastic heterogeneity picture to the scene which is more conforming
with the long-established notion of structure-property correlations in
(metallic) glasses.

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Figure 2: Cluster size statistics corresponding to the CoNiFe and CoNiCrFe
glasses. a,c) Cluster size distribution $n_{s}$ b,d) scatter plot of cluster
size $s$ and associated radius of gyration $r_{s}$ at different srains
$\gamma_{xy}=0.03$ (  ), $0.07$ (  ), $0.1$ (  ). The dashdotted lines denote
power laws $n_{s}\propto s^{-\tau}$ with a) $\tau=2.0$ c) $\tau=2.1$ and
$s\propto r_{s}^{d_{f}}$ with b) $d_{f}=2.3$ d) $d_{f}=2.6$.

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Figure 3: Mean cluster size $S$ plotted against $p$ associated with the a)
CoNiFe and c) CoNiCrFe glasses. The graphs in b) and d) are the same as a) and
c) but plot $S$ as a function of $1-p/p_{\text{max}}$ with a)
$p_{\text{max}}=0.14$ and c) $p_{\text{max}}=0.20$. The (red) curves indicate
binning averaged data. The dashdotted lines denote power laws
$S\propto(p_{\text{max}}-p)^{-\gamma}$ with b) $\gamma=2.89$ and d)
$\gamma=2.20$.

_Simulations & Protocols—_ Details of molecular dynamics simulations are given
in [4] including relevant units, initial configurations, interatomic forces,
and deformation parameters of FeNi , CoNiFe , CoNiCrFe , CoCrFeMn , CoNiCrFeMn
, and Co5Cr2Fe40Mn27Ni26 model metallic glasses. In order to compute the
elasticity tensor locally for simulated glasses, simple shear tests were
performed on the $xy$ plane at a fixed strain rate
$\dot{\gamma}_{xy}=10^{-5}\leavevmode\nobreak\ \text{ps}^{-1}$ and temperature
$T=300$ K up to a prescribed (pre-)strain $\gamma_{xy}\leq 0.2$. In line with
[28, 29], we subsequently perturbed the simulation cell through six
deformation modes in Cartesian directions $xyz$ and evaluated resulting
differences in atom-wise stresses to construct the local elasticity tensor for
each atom (see the appendix). Relevant atom-based quantities were interpolated
on a fine cubic grid to be used as input for our three dimensional cluster
processing. This methodology allows us to probe the spatial-dynamical
evolution of the local shear modulus $\mu=c_{xyxy}$ and associated percolation
features near failure transition in sheared glasses. We further measure the
softening modulus $h_{\text{min}}$ defined as the maximum _rate_ of the
_macroscopic_ stress drop for all the different compositions. The stress drop,
typically defined as the difference between the overshoot stress and the
subsequent flow stress, is associated with the initiation of a catastrophic
shear band and has been used as an appropriate order parameter in model glass
studies [30, 31] showing meaningful variations with glass compositions and
processing parameters [11]. We note that in metallic glass simulations and/or
experiments a robust measurement of the macroscopic drop is not always
feasible due to the lack of a well-defined steady flow regime that is expected
to follow the stress overshoot. In a recent work [4], we established
$h_{\text{min}}$ as a more robust experimentally-relevant indicator of shear
banding and associated structural features. Here, we show that the variations
in $h_{\text{min}}$ tend to correlate with the softness properties as inferred
from local elasticity.

_Results_ — Figure 1 displays results of the shear tests performed on the
quenched CoNiFe and CoNiCrFe glasses. The cluster maps based on unstable
regions with $\mu\leq 0$ are visualized in Fig. 1(b) and (c). We denote the
fraction of unstable sites by (probability) $p$ which appears to evolve in a
non-monotonic fashion, Fig. LABEL:fig:statistics_gamma(b) and (c), with a peak
value $p_{\text{max}}$ that almost coincides with that of the (bulk) shear
stress $\sigma_{xy}$ (at $\gamma_{xy}\simeq 0.1$) as in Fig. 1(a). Apart from
variations in $p_{\text{max}}$, Fig. LABEL:fig:statistics_gamma(a-f) shows
similar trends for the evolution of $p$ with $\gamma_{xy}$ corresponding to
the other compositions. We notice $p>0$ even in unstrained quenched glasses
(at $\gamma_{xy}=0$) owing to the distributed (but disconnected) networks of
soft (Eshelby-like [32]) inclusions within the glassy matrix.

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Figure 4: Correlation length $\xi$ plotted against $p$ corresponding to the a) CoNiFe and c) CoNiCrFe glasses. The graphs in b) and d) are the same as a) and c) but plot $\xi$ as a function of $1-p/p_{\text{max}}$ with a) $p_{\text{max}}=0.14$ and c) $p_{\text{max}}=0.20$. The (red) curves indicate binning averaged data. The dashdotted lines denote power laws $\xi\propto(p_{\text{max}}-p)^{-\nu}$ with b) $\nu=1.26$ and d) $\nu=0.51$. Chemical Composition |  Cluster Size Distribution $n_{s}\propto s^{-\tau}$ $\tau$ |  Mean Cluster Size $S\propto(p-p_{\text{max}})^{-\gamma}$ $\gamma$ |  Correlation Length $\xi\propto(p-p_{\text{max}})^{-\nu}$ $\nu$ |  Fractal Dimension $s\propto r_{s}^{d_{f}}$ $d_{f}$
---|---|---|---|---
FeNi |  $1.96$ |  $2.23$ |  $0.87$ |  $2.27$
CoNiFe |  $1.98$ |  $2.89$ |  $1.26$ |  $2.25$
CoNiCrFe |  $2.05$ |  $2.20$ |  $0.51$ |  $2.62$
CoCrFeMn |  $2.02$ |  $2.42$ |  $0.95$ |  $2.48$
CoNiCrFeMn |  $2.00$ |  $2.26$ |  $0.62$ |  $2.43$
Co5Cr2Fe40Mn27Ni26 |  $1.93$ |  $2.69$ |  $0.94$ |  $2.28$
CuZr[17] |  $-$ |  $-$ |  $0.85\pm 0.1$ |  $2.0$
Percolation ($\emph{d}=3$) [33] |  $2.18$ |  $1.80$ |  $0.88$ |  $2.53$
Directed Percolation ($\emph{d}=3$) [34] |  $-$ |  $1.25$ |  $0.6$ |  $2.0$
Table 1: Comparison between estimated scaling exponents associated with
different chemical compositions and three dimensional ($d=3$) percolation
theory.

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Figure 5: Scatter plot of softening modulus $h_{\text{min}}$ and scaling
exponents a) $\gamma$ b) $\nu$ corresponding to the FeNi (  ), CoNiFe (  ),
CoNiCrFe (  ), CoCrFeMn (  ), CoNiCrFeMn (  ), and Co5Cr2Fe40Mn27Ni26 (  )
bulk metallic glasses.

The above elasticity maps give a visual impression that the glass failure (at
$\gamma_{\text{max}}\simeq 0.1$) might indeed coincide with a percolation
transition of softness at $p_{\text{max}}$ upon shear loading. In order to
validate this picture, we adopted ideas from the classical percolation theory
[33] including investigations of cluster sizes and their dynamical evolution.
As a basic statistical property, $n_{s}$ denotes the probability distribution
function associated with the number of clusters containing $s$ unstable sites.
Figure 2(a) and (c) plots $n_{s}$ associated with the CoNiFe and CoNiCrFe
glasses at three different strains. The cluster size distributions tend to
develop fairly long tails as $p$ approaches $p_{\text{max}}$ . Our data
suggest a robust power-law decay $n_{s}\propto s^{-\tau}$ with $\tau=2.0$ and
$2.1$, associated with CoNiFe and CoNiCrFe , over at least two decades in $s$.
The estimated range of exponent $\tau$ is fairly robust showing minimal
compositional dependence (as reported in Table 1) and/or variations near
$p_{\text{max}}$ . The radius of gyration associated with a cluster of size
$s$ may be defined as
$r^{2}_{s}=\sum_{i=1}^{s}|\vec{r}_{i}-\vec{r}_{0}|^{2}/s$ with the center of
mass $\vec{r}_{0}=\sum_{i=1}^{s}\vec{r_{i}}/s$. Figure 2(b) and (d)
illustrates that $s\propto r_{s}^{d_{f}}$ with fractal dimension $d_{f}=2.3$,
$2.6$ corresponding to CoNiFe and CoNiCrFe , respectively. This almost agrees
with Fig. 1(b) and (c) in that the soft spots tend to form fairly compact
clusters in the latter glass whereas the former one is associated with more
localized (but still system-spanning) features.

Figure 3(a) and (c) displays the mean cluster size
$S={\sum_{s}n_{s}s^{2}}/{\sum_{s}n_{s}s}$ and its evolution with $p$. The
average size reveals a certain algebraic divergence of $S$ on approach to the
maximum fraction $p\rightarrow p_{\text{max}}$. This divergence is
demonstrated in Fig. 3(b) and (d) with the mean size scaling as
$S\propto(p_{\text{max}}-p)^{-\gamma}$ and $\gamma=2.89$, $2.20$ corresponding
to the CoNiFe and CoNiCrFe compositions, respectively. The proposed scaling is
valid for a little less than a decade in $1-p/p_{\text{max}}$ down to a roll-
off at small arguments potentially due to finite size effects. The (squared)
correlation length
$\xi^{2}={2\sum_{s}r^{2}_{s}s^{2}n_{s}}/{\sum_{s}s^{2}n_{s}}$ is defined based
on a weighted average associated with the radius of gyration
$r^{2}_{s}=\sum_{i=1}^{s}|\vec{r}_{i}-\vec{r}_{0}|^{2}/s$ of a cluster of size
$s$, as shown in Fig. 4(a) and (c). Here the center of mass is
$\vec{r}_{0}=\sum_{i=1}^{s}\vec{r_{i}}/s$. As $p$ increase toward
$p_{\text{max}}$ in Fig. 4(b) and (d), the correlation length scales like
$\xi\propto(p_{\text{max}}-p)^{-\nu}$ with $\nu=1.26$ corresponding to CoNiFe
and $\nu=0.51$ for CoNiCrFe . The center and right panels of Fig.
LABEL:fig:statistics_gamma show the mean cluster size $S$ and correlation
length $\xi$ as a function of applied strain $\gamma_{xy}$ associated with
different compositions.

We repeated the above analysis to infer critical exponents corresponding to
the other chemical compositions (see Fig. LABEL:fig:statistics_p and Table 1)
and sought for potential connections with the softening modulus
$h_{\text{min}}$ . Figure 5 plots the latter and estimated critical exponents
$\tau$, $\gamma$, $\nu$, and $d_{f}$ for the FeNi , CoNiFe , CoNiCrFe ,
CoCrFeMn , CoNiCrFeMn , Co5Cr2Fe40Mn27Ni26 metallic glasses. As previously
discussed, the cluster size exponent $\tau$ in Fig. 5(a) shows insignificant
variations with chemical compositions. The scatter plot of $h_{\text{min}}$
and mean cluster size exponent $\gamma$ in Fig. 5(b) suggests noticeable anti-
correlations between the two observables. Likewise, the correlation length
exponent $\nu$ seems to be statistically (anti-)correlated with
$h_{\text{min}}$ as in Fig. 5(c). The increasing trend in the fractal
dimension $d_{f}$ might be indicative of localized (system-spanning) soft
spots in _good_ glasses that become increasingly packed in space as
compositional variations lead to a smoother yielding transition. Given that
there are only two independent exponents in standard percolation theory, the
behavior of $\nu$ and $\gamma$ should be largely controlled by variabilities
in $d_{f}$.

_Conclusions & Discussions—_ We have presented direct evidence that yielding
transition of bulk metallic glasses is is accompanied with percolation
threshold of softness upon failure across different chemical compositions. The
former has been quantified by the softening modulus $h_{\text{min}}$ ,
indicative of the sharpness of plastic yielding transition, and the latter was
characterized by analyzing connected networks of mechanically soft regions
that grow under application of external stress. Our analysis further indicates
critical scaling features associated with the dynamics and topology of soft
clusters suggesting a close relevance of the percolation transition. Relevant
scaling exponents and potential correlations with the macroscopic stress-
strain curve have been considered from the compositional-dependence
perspective. Table 1 features the range of estimated exponents corresponding
to different multi-component metallic glasses probed in this work and the
study by Schall et al. [17] as well as those inferred from the percolation
theory. In this framework, we find a fairly robust range (across different
systems) associated with the cluster size exponent $1.9\leq\tau\leq 2.1$ and
comparable with theoretical predictions [33]. On the other hand, $\gamma$,
$\nu$, and $d_{f}$, inferred from the scaling of the correlation lengths and
mean cluster sizes, somewhat correlate with softening modulus $h_{\text{min}}$
suggesting that the softness percolation might influence macroscopic yielding
properties through compositional dependence. Such correlations may also
suggest that the notion of _universality_ in deformation and flow properties
of multicomponent glasses could be limited owing to
compositional/microstructural associations. Nevertheless, given a relatively
narrow range of the softness exponents across different glasses, one might
still consider the same universality class for the observed transition despite
the fact that our results show substantial composition-based variabilities in
yielding properties.

###### Acknowledgements.

This research was funded by the European Union Horizon 2020 research and
innovation program under grant agreement no. 857470 and from the European
Regional Development Fund via Foundation for Polish Science International
Research Agenda PLUS program grant no. MAB PLUS/2018/8. We wish to thank A.
Esfandiarpour and R. Alvarez for providing the data sets.

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