# Excellent glass former Ni62Nb38 crystallizing under combined shear and
ultra-high pressure

Bulat N. Galimzyanov<EMAIL_ADDRESS>Maria A. Doronina
<EMAIL_ADDRESS>Anatolii V. Mokshin<EMAIL_ADDRESS>Kazan Federal University, 420008 Kazan, Russia  Udmurt Federal Research Center
of the Ural Branch of the Russian Academy of Sciences, 426067 Izhevsk, Russia

###### Abstract

Study of condensed matter in certain extreme conditions allows one to better
understand the mechanisms of microscopic structural transformations and to
develop materials with completely new mechanical properties. In the present
work, we study the structure and crystallization kinetics of bulk metallic
glass (BMG) Ni62Nb38. We show that this BMG is rapidly crystallized under the
combined effect of shear deformation and ultrahigh pressure. A threshold
pressure required to initiate the formation of a stable crystalline phase is
revealed. Shear deformation and pressure lead to the phase separation: two
high-density inhomogeneous crystal structures are formed. We find that the
crystallization of this BMG occurs in two stages due to the significant
difference in the growth rates of Ni and Nb crystalline phases. The results of
the present study make a significant contribution to understanding the
crystallization and amorphization features of Ni-based BMG’s.

###### keywords:

Crystal growth, Crystallisation, Bulk metallic glass, Shear, Molecular
dynamics

##  1 Introduction

The study of crystallization mechanisms of bulk metallic glasses (BMG’s) with
outstanding glass-forming ability becomes more available to the improvement of
high-pressure devices [1, 2, 3, 4]. Modern experimental methods make it
possible to obtain static pressures more than $1000$ GPa, for example, in two-
stage high-pressure devices with diamond anvils [5, 6, 7]. Such terapascal
pressures comparable with pressures in Jupiter’s interior are usually achieved
for micron-size samples. Moreover, external pressure is an important factor
for controlling the microscopic structure of BMG’s that is confirmed
experimentally [9, 8] and based on the results of molecular dynamics
simulations [10, 11]: ultrahigh pressure can both promote the crystallization
of BMG’s [11, 12, 13, 14] and suppress this process [15, 16, 17].

The impact of pressure on BMG structure strongly depends on the alloy
composition and on the size of a sample. Most BMG’s consist of two or more
chemical elements. The difference in the atomic radii of these elements leads
to the formation of a relatively stable amorphous structure [18, 19]. Among
binary BMG’s, the best glass-forming ability is observed in Ni-Nb system with
composition Ni62Nb38 [20, 21, 22, 23]. Xia and co-workers have shown that
samples of this BMG practically do not crystallize under normal conditions if
their linear size is less than $2$ mm [20]. This feature distinguishes
Ni62Nb38 from other binary Ni-based BMG’s.

The mechanisms of microscopic structural transformation in BMG Ni62Nb38 are
poorly studied experimentally due to difficulties associated with monitoring
the nucleation and growth of nanosized crystallites inside a bulk sample [14,
24, 25, 26]. The initial stage of crystallization is difficult to track using
X-ray and neutron spectroscopy data since these methods, as a rule, provide
averaged information about the structure [27, 28]. On the other hand, until
recently, detailed molecular dynamics simulations of the crystallization
process in Ni62Nb38 were not performed due to the absence of interparticle
interaction potentials capable of correctly reproducing the amorphous
structure and physical properties of this system for a wide range of
temperatures and pressures. A better understanding of structural ordering
mechanisms on the atomistic scale becomes more available after the development
of a semiempirical potential by Zhang and co-workers [29]. This potential
accurately reproduces the bond angle distribution in Ni62Nb38 and allows one
to obtain BMG with a relatively stable amorphous structure. Note that the
bond-angle is the angle between the vectors joining a central particle with
two other atoms. The bond-angle distributions are used to represent the local
angular correlations in disordered media [30].

In the present work, we carry out the non-equilibrium molecular dynamics
simulation study of the microscopic structure of metallic glass Ni62Nb38
crystallizing under the impact of external forces. We will show that the
“explosive” crystallization of this BMG is observed only when the system
undergoes a fixed ultrahigh pressure and shear deformation with a fixed shear
rate.

##  2 Computational procedure

The considered system Ni62Nb38 consists of $13\,203$ Ni atoms and $8\,093$ Nb
atoms (the total number of atoms is $N=21\,296$). The atoms are located inside
a triclinic simulation box with the linear size $L_{x}=L_{y}=L_{z}\simeq 67$
Å. The interatomic interaction energies and forces are given by the
semiempirical Finnis-Sinclair potential [29, 31]:

$U(r_{ij})=\sum_{i=1}^{N-1}\sum_{j\neq
i}^{N}\phi_{\alpha_{i}\alpha_{j}}(r_{ij})+\sum_{i=1}^{N}\Phi_{\alpha_{i}}(\rho_{i}),$
(1)

where

$\rho_{i}=\sum_{j\neq i}^{N}\psi_{\alpha_{i}\alpha_{j}}(r_{ij}).$ (2)

Here, $\rho_{i}$ is the local electronic charge density at site $i$, which is
constructed by a rigid superposition of atomic charge densities
$\psi_{\alpha_{i}\alpha_{j}}(r_{ij})$, where $\alpha\in\\{$Ni, Nb$\\}$ is the
type of atom; $r_{ij}$ is the distance between the atoms $i$ and $j$;
$\phi_{\alpha_{i}\alpha_{j}}(r_{ij})$ is the pairwise potential;
$\Phi_{\alpha_{i}}(r_{ij})$ is the embedding energy function. Note that this
type of potential reproduces accurately the structure of Ni62Nb38 in the
liquid and amorphous states [29].

To prepare amorphous samples, liquid samples with the temperature $T=2500$ K
are rapidly cooled to the temperature $T=300$ K at the pressure $p=10^{-4}$
GPa with the cooling rate $10^{13}$ K/s. To reveal the effect of the cooling
procedure on the crystallization kinetics of the system, additional studies
were carried out at the cooling rate $10^{10}$ K/s. Note that the liquidus and
glass transition temperatures of Ni62Nb38 at zero pressure are $T_{l}\simeq
1450$ K (the experimental value is $\approx 1400$ K) and $T_{g}\simeq 1010$ K,
respectively, and these temperatures follow from the results of molecular
dynamics simulations with the Finnis-Sinclair interaction potential [29].
Figure 1(a) shows that the structure factor $S(k)$ of amorphous Ni62Nb38
calculated at the temperature $T=300$ K is in agreement with the
experimentally measured diffraction pattern [32].

Figure 1: (a) Structure factor of the amorphous solid found from the molecular
dynamics simulations (MD) at the temperature $300$ K and at the pressure
$10^{-4}$ GPa is compared with the experimental data (Exp) from Ref. [32]. (b)
Schematic representation of shear deformation of an amorphous sample.

Prepared amorphous samples undergo shear deformation in the $xy$ plane and in
the $x$-direction at the fixed shear rate $\dot{\gamma}=0.01$ ps-1 as shown in
Figure 1(b). This strain rate is typical in molecular dynamics simulations and
this rate is difficult to obtain in experiments [13, 33, 34, 35]. For example,
shear rates from $0.0001$ ps-1 to $0.1$ ps-1 were used by Chang and co-workers
to study the effect of shear strain on the behavior of single-crystal titanium
nanowire [34]. They have shown that the physicomechanical properties of
titanium nanowire are weakly dependent on the strain at shear rates in the
interval $\dot{\gamma}\in[0.0001;\,0.01]$ ps-1. Sha and co-workers are
achieved to the same conclusions in the study of single-layer borophene
mechanical properties at shear rates from $10^{-5}$ ps-1 to $0.01$ ps-1 [35].
Thus, we reasonably expect that structural transformations occurring in
metallic glass Ni62Nb38 at the shear rate $\dot{\gamma}=0.01$ ps-1 will also
be observed in experiments at lower shear rates. Here, it should be noted that
this relationship between simulation and experiment may not be fulfilled in
the case of systems with complex chemical bonds. We perform shear deformation
at the fixed pressures $p=200$, $400$, $600$, $800$, $1000$ GPa and these
pressures are experimentally achievable with high-pressure devices with
diamond anvils [1, 5, 6, 7]. These pressures are applied only along the $y$
direction – the so-called gradient direction [see Figure 1(b)]. The constant
pressure $10^{-4}$ GPa acts only in the $z$-direction.

Bond orientational analysis is performed for the identification of different
crystalline phases and clusters inside amorphous samples. The local $q_{l}$
and the global $Q_{l}$ orientational order parameters (here $l=4,\,6,\,8$) are
computed according to the definitions given in Refs. [36, 37]. To identify the
crystalline phase atoms, we compute the local orientational order parameters
$q_{l}$ separately for Ni and Nb atoms [see Eq. (4) in Appendix]. The
existence of atoms with values of $q_{l}$ close to those of an ideal ordered
fcc, hcp, bcc, simple cubic, and icosahedron structures is evidence of the
presence of ordered clusters [see Table I in Ref. [38]]. To detect a single
crystalline order across the entire sample, we compute the global
orientational order parameters $Q_{l}(\alpha)$ (here
$\alpha=\\{$Ni$,\,$Nb$\\}$) [see Eq. (8) in Appendix]. Note that these order
parameters with indices $l=4,\,6,\,8$ are sufficient to correctly detect
ordered structures.

##  3 Results and Discussions

The impact of high pressure and shear initiates the phase transition in the
amorphous alloy Ni62Nb38. We have found that intense formation and growth of
stable crystalline nuclei occur only at pressures above $p=200$ GPa.
Regardless the magnitude of the strain $\dot{\gamma}t$, the formation of
stable crystallites is not observed at lower pressures. Figure 2(a) shows that
unstable nanocrystallites randomly distributed throughout the amorphous matrix
are formed in the system at $p\leq 200$ GPa. The concentration of these
nanocrystallites does not change with increasing strain up to
$\dot{\gamma}t=10$. Such poor crystallizability may be due to the fact that
the degree of supercooling of the system at the temperature $300$ K is $\Delta
T/T_{l}\simeq 0.79$ (where $\Delta T=T_{l}-T$). At this supercooling, the
mobility of atoms is suppressed by high viscosity [13, 39]. The energy entered
into the glassy system due to applied pressure and shear is insufficient to
relax this system into the equilibrium crystalline phase.

Figure 2: Crystallizing amorphous Ni62Nb38 at the different strains
($\dot{\gamma}t=1$, $5$, and $10$) and at the various pressures: (a) $p=200$
GPa; (b) $p=600$ GPa; (c) $p=1000$ GPa. Crystalline Ni and Nb atoms are marked
by green and blue colors, respectively. The atoms of the parent disordered
phase are colored by light blue.

“Explosive” crystallization of Ni62Nb38 occurs at pressures from $600$ to
$1000$ GPa. At such pressures, the crystallization degree of the system
increases in a direct proportion to the strain $\dot{\gamma}t$ [see Figures
2(b) and 2(c)]. The higher the applied pressure, the less shear strain is
required to initiate the crystallization process. This is clearly seen from
Figure 2(c), where the sample with crystalline domains is formed at the
pressure $p=1000$ GPa and at strains $\dot{\gamma}t\leq 1$, whereas
crystallization is not observed at similar strains and at the pressure $p=600$
GPa [see Figure 2(b)]. This finding is of great practical importance, since
crystalline samples obtained at such relatively small shear strains do not
experience pronounced mechanical injuries [40].

Crystalline nuclei with a non-spherical shape and consisting of several tens
of atoms are formed at the initial stage of BMG crystallization. The non-
spherical shape of these nuclei is due to the impact of shear strain as it was
shown earlier, for example, in Refs [33, 41, 42]. The formation of these
crystal nuclei obeys the Poisson distribution, and this observation is in good
agreement with the main postulates of the classical theory of homogeneous
stationary nucleation presented in Refs. [43, 44, 45, 46]. However, further
growth of crystallites does not follow the classical scenario [47]. We have
found that shear strain leads to crystal separation and formation of two
fractions with different composition and structure. One of the fractions
consists only of Ni atoms, whereas the second one contains of Nb atoms. Figure
2 shows that this separation is clearly manifested at high pressures (over
$400$ GPa) and at deformations $\dot{\gamma}t>1$. The growth of each fraction
occurs both due to the attachment of individual Ni or Nb atoms and due to the
coalescence of small crystallites. The coalescence of nuclei leads to the
formation of the inhomogeneous crystal structure typical for phase separation
[48, 49]. Note that a similar diffusive redistribution of components with a
change in the crystal structure is observed in many multicomponent alloys
undergoing phase transformations [50, 51, 52]. For example, Bachmaier et al.
have found the spinodal-type decomposition and formation of a nanoscale
composite structure in the binary alloy Cu-Co under high-pressure torsion
deformation [52]. Yang and Ming have observed surface-localized phase
separation behavior of LiFePO4 (cathode material for Li-ion batteries) induced
by elastic deformation and lattice expansion due to Li insertion [50].

Figure 3: Global orientational order parameters $Q_{4}$, $Q_{6}$, $Q_{8}$ at
different pressures and at the strain $\dot{\gamma}t=5$: (a) for the crystal
structure of Ni; (b) for the crystal structure of Nb. Note that the values of
these parameters were calculated without applying the correlation condition
(6). 3D-distribution of the atoms $P(q_{6})$ over the value of the order
parameter $q_{6}$ is presented as a function of the strain $\dot{\gamma}t$ at
$p=600$ GPa: (c) for Ni atoms; (d) for Nb atoms. Arrows indicate the type of
crystal lattice: fcc (face-centered cubic), hcp (hexagonal closest packed),
and bcc (body-centered cubic).

From the found values of the global orientational order parameters $Q_{4}$,
$Q_{6}$ and $Q_{8}$ [see. Figures 3(a) and 3(b)] it follows that the
nanocrystallites of Ni have the fcc lattice at the pressure $p=200$ GPa and at
the strain $\dot{\gamma}t=5$, while Nb forms the bcc phase under these
conditions. Note that at low pressures the fcc and bcc phases are typical for
Ni and Nb, respectively [53, 54]. The system forms a high-density crystalline
phase close to hcp at pressures above $400$ GPa. In the case of Ni, the mixing
of the hcp and fcc structures is observed, whereas Nb forms the mixed hcp-bcc
phase. As an example, Figures 3(c) and 3(d) show the distribution $P(q_{6})$
of the order parameter $q_{6}$ calculated for Ni and Nb crystallites at
different strains and at the pressure $p=600$ GPa. The shape of the
distribution $P(q_{6})$ directly indicates the predominance of the number of
atoms forming the hcp structure over the atoms entering the fcc and bcc
structures. As can be seen from Figures 3(c) and 3(d), the shape of the
3D-distributions varies weakly with the degree of strain at $\dot{\gamma}t>1$.
This finding indicates that the pressure (or density of the system) plays a
crucial role in the formation of a type of crystal lattice symmetry during the
crystallization of amorphous Ni62Nb38. It should be noted that the hcp, fcc,
and bcc crystal lattices are distorted as a result of applied shear strain.
Therefore, the calculated values of the parameters $Q_{l}$ ($l=4,\,6,\,8$) and
the distribution $P(q_{6})$ [see Figure 3] may differ from the known
literature data obtained for systems with an ideal crystal structure [38].

It should be noted that the final crystal structure weakly depends on the
cooling rate with which the sample has been quenched. For example, Figure 4
shows the distribution of the order parameter $q_{6}$ calculated at the strain
$\dot{\gamma}=5$ and at the pressure $p=600$ GPa. The results are given for
glassy samples prepared at the cooling rates $10^{10}$ K/s and $10^{13}$ K/s.
Note that the realization of the cooling procedure with rates less than
$10^{10}$ K/s in molecular dynamics simulations can take several months even
when using supercomputers. Despite the significant difference in cooling
rates, the shape of the $P(q_{q})$-distribution changes insignificantly and
retains the positions of the main maxima and peaks corresponding to the fcc,
hcp and bcc crystalline phases. On the other hand, the inset in Figure 4 shows
that the dependence of the global orientational order parameter $Q_{6}$ on the
strain $\dot{\gamma}t$ calculated at the pressure $p=600$ GPa saturates faster
at the cooling rate $10^{10}$ K/s. This result shows that the crystallization
kinetics of the considered system depends on the cooling history: a slow
cooling rate leads to a faster relaxation of the system into the crystalline
phase. This result is in agreement with previous experimental and simulation
studies of other BMG’s [55, 56, 57, 58], where it was shown the positive
effect of decreasing cooling rate on the crystallization kinetics as well as a
weak dependence of the structural characteristics on the cooling procedure.

Figure 4: Distributions $P(q_{6})$ of crystalline Ni and Nb atoms obtained for
the samples of considered BMG, which are prepared at different cooling rates.
The curves are obtained at the shear strain $\dot{\gamma}t=5$ and at the
pressure $p=600$ GPa. Inset: global orientational order parameter $Q_{6}$ as a
function of the strain $\dot{\gamma}t$ calculated for the samples, which were
cooled with different rates (for the case of $p=600$ GPa).

Figure 5 shows the relationship between the global order parameter $Q_{6}$ and
the strain $\dot{\gamma}t$ calculated at different pressures and for samples
prepared at the cooling rate $10^{13}$ K/s. Note that the $Q_{6}$ is the most
sensitive to the processes of structural ordering in comparison with other
order parameters [38]. As seen from Figure 5, the growth kinetics of the Ni
and Nb crystalline phases are very different. Namely, Ni atoms tend to order
much rapidly than Nb atoms, and this tendency becomes more pronounced with
increasing pressure. In the case of Ni, this is indicated by the rapid
saturation of the $Q_{6}(\dot{\gamma}t)$-curves at pressures $p\geq 600$ GPa,
at which the parameter $Q_{6}$ takes the value $\approx 0.4$ corresponding to
the fcc/hcp phase. In the case of Nb, the value of the parameter $Q_{6}$
demonstrates a slower growth in comparison with the case of Ni atoms at the
same pressures. Here, the $Q_{6}(\dot{\gamma}t)$-curves do not form a
pronounced stable plateau, which indicates a slow ordering of Nb atoms.

Figure 5: Global orientational order parameter $Q_{6}$ as a function of the
strain $\dot{\gamma}t$: (a) for the crystal structure of Ni; (b) for the
crystal structure of Nb.

Figures 6(a) and 6(b) show the partial fraction of crystalline atoms $X\equiv
n/N$ (where $n$ is the number of crystalline atoms, $N$ is the number of all
atoms in the system) as a function of the time $t$ at different pressures. The
fraction of the crystalline phase increases both with the increasing pressure
$p$ and with the increasing strain $\dot{\gamma}t$. The results presented in
Figures 6(a) and 6(b) also confirm that the crystalline Ni phase grows rapidly
than the fraction of crystalline Nb atoms. This is mainly due to the low
concentration of Nb atoms in the considered system. As a consequence, the
formation of stable Nb crystallites is slowed down due to the relatively large
distances between Nb atoms. External high pressure increases the local partial
density of each component, which in combination with the shear initiates the
crystallization.

Figure 6: Fraction of the crystalline phase atoms $X$ as a function of the
time $t$ computed for Ni and Nb atoms at different pressures (for samples
cooled with the rate $10^{13}$ K/s): (a)–(b) at the shear rate
$\dot{\gamma}=0.01$ ps-1 and (c)–(d) at zero shear rate $\dot{\gamma}=0$ ps-1.
Solid lines are the results of Eq. (3). Insets in panels (c) and (d) show the
result of separation of Ni and Nb atoms in the system under the pressure
$p=1000$ GPa.

For quantitative evaluation of phase transformation kinetics, we fit the
calculated $X(t)$-curves with the exponential function

$X_{\alpha}(t)=X_{m}^{(\alpha)}\left[1-e^{-K_{\alpha}t^{\xi}}\right],\,\,\alpha=\\{\text{Ni},\,\text{Nb}\\},$
(3)

which converts into the well-known Kolmogorov-Johnson-Mehl-Avrami-Evans
(KJMAE) expression at $X_{m}^{(\alpha)}=1$ [43, 59, 60, 61]. The parameter
$X_{m}^{(\alpha)}$ defines the position of the plateau in the $t$-dependencies
of the quantity $X_{\alpha}$. Note that this quantity is $X_{m}^{(\alpha)}=1$
in the case of crystallization of single-component systems and in the case of
complete crystallization of a system [60]. The kinetic rate factor
$K_{\alpha}$ [in units of $1/$psξ] depends on the nucleation and crystal
growth rates; $\xi$ is the so-called Avrami exponent. According to the KJMAE-
theory, this exponent is $\xi\simeq 3$ for the case of three-dimensional
crystal growth. If the phase transformation occurs with a constant nucleation
rate and the crystal growth is controlled by diffusion, then we have
$\xi\simeq 2.5$. When the crystal growth is controlled only by the diffusion,
the exponent $\xi$ takes the value close to $1.5$. Expression (3) correctly
reproduces the $X(t)$-curves calculated at the pressures $p=600$, $800$ and
$1000$ GPa [see Figures 6(a) and 6(b)]. The found values of the parameters of
Eq. (3) are given in Table I.

Table 1: The values of the parameters of Eq. (3) obtained by fitting the computed $X_{\alpha}(t)$-curves, where $\alpha=\\{$Ni, Nb$\\}$. The values of the parameters $X_{m}^{(\alpha,\,\dot{\gamma}=0)}$, $K_{\alpha}^{(\dot{\gamma}=0)}$ and $\xi^{(\dot{\gamma}=0)}$ computed for the case of crystallization without shear, $\dot{\gamma}=0$. $\alpha$ | p, GPa | $X_{m}^{(\alpha)}$ |  $K_{\alpha}$, $\times 10^{-4}$ps-ξ | $\xi$ | $X_{m}^{(\alpha,\,\dot{\gamma}=0)}$ |  $K_{\alpha}^{(\dot{\gamma}=0)}$, $\times 10^{-4}$ps-ξ | $\xi^{(\dot{\gamma}=0)}$
---|---|---|---|---|---|---|---
| $1000$ | $0.54\pm 0.02$ | $46.1\pm 3.5$ | $1.1\pm 0.1$ | $0.16\pm 0.03$ | $9.8\pm 0.7$ | $1.6\pm 0.1$
Ni | $800$ | $0.5\pm 0.02$ | $5.12\pm 0.4$ | $1.4\pm 0.1$ | – | – | –
| $600$ | $0.33\pm 0.02$ | $0.07\pm 0.01$ | $2.1\pm 0.2$ | – | – | –
| $1000$ | $0.14\pm 0.02$ | $7.96\pm 0.6$ | $1.2\pm 0.1$ | $0.018\pm 0.002$ | $4.5\pm 0.3$ | $1.8\pm 0.1$
Nb | $800$ | $0.09\pm 0.01$ | $1.07\pm 0.08$ | $1.6\pm 0.1$ | – | – | –
| $600$ | $0.03\pm 0.01$ | $0.02\pm 0.01$ | $1.9\pm 0.1$ | – | – | –

It was found that the exponent $\xi$ weakly depends on the crystalline phase
composition and takes values from the range $\xi\in[1.1;\,2.1]$. Such small
values confirm that the crystal growth is not uniform. The value of the
exponent $\xi$ increases from ($1.1\pm 0.1$) to ($2.1\pm 0.1$) with decreasing
pressure that indicates a possible transition to the classical nucleation
scenario at low pressures. Moreover, the value of the kinetic rate factor
$K_{\alpha}$ increases by several orders of magnitude with increasing pressure
[see Table I] that confirms the conclusion about “explosive” crystallization
of the considered BMG at high pressures. For example, the growth of Ni
crystals at the pressure $p=1000$ GPa occurs with the kinetic rate factor
$K_{\text{Ni}}\simeq 46\times 10^{-4}$ ps-1.1, while the crystalline Nb phase
grows with the rate factor $K_{\text{Nb}}\simeq 7.96\times 10^{-4}$ ps-1.2 at
the same pressure. This difference in the values of the kinetic rate factor
indicates that the crystallization of Ni62Nb38 occurs in two stages. At the
first stage, the system with the crystalline Ni phase is formed intensively
with a small amount of Nb crystals. The second stage is accompanied by an
increasing proportion of Nb crystals, whereas the growth of crystalline Ni
phase is slowing down. These conclusions are also confirmed by the results
shown in Figures 2 and 6.

To estimate the contribution of a pressure on the crystallization of the
considered BMG, we have performed calculations without applying a shear,
$\dot{\gamma}=0$. Figures 6(c) and 6(d) show a stable increase in the fraction
of the crystalline phases Ni and Nb only at pressures above $600$ GPa. For
example, at the pressure $p=1000$ GPa, the fraction of crystalline Ni and Nb
atoms rapidly reaches a saturation that is accompanied by the formation of a
pronounced plateau in the $X(t)$-curves. It is noteworthy that the formation
of this plateau is preceded by the separation of Ni and Nb atoms to branched
structures consisting of a mixture of crystalline and disordered parent phases
[see insets in Figures 6(c) and 6(d)]. It follows from the position of the
plateau that only $15$% of Ni atoms and $1.8$% of Nb atoms form the
crystalline phase. For comparison, under shear strain and at the pressure
$p=1000$ GPa, the plateau is reached at the fractions of crystalline Ni and Nb
atoms $50$% and $10$%, respectively. This indicates the failure of the system
to complete crystallization at the considered pressures and time scale.
Approximation of the $X(t)$-curve by Eq. (3) gives the kinetic velocity factor
$K_{\alpha}^{(\dot{\gamma}=0)}=9.8\times 10^{-4}$ ps-ξ (where $\xi=1.6$) for
Ni atoms that is $\approx 4.7$ times less than the case of crystallization
under shear. In the case of Nb atoms, the kinetic velocity factor decreases
only $\approx 1.7$ times. In the absence of shear, the value of the parameter
$\xi$ is greater than in the case of shear strain [see Table 1]. This is due
to the fact that the growth of nuclei at zero shear rate occurs more uniformly
in all directions. It should be noted that Eq. (3) misdescribes the
$X(t)$-curves at other pressures due to the poor crystallizability of the
system. Thus, the “explosive” and complete crystallization of BMG in the
considered time scale can be provided by applying shear deformation.

##  4 Conclusions

The results of the present work show that ultrahigh pressure and shear
deformation promote the crystallization of Ni62Nb38, whereas this BMG forms a
relatively stable amorphous structure under normal conditions. We reveal the
presence of a threshold pressure, the excess of which triggers the structural
ordering of the glassy system. Shear deformation does not lead to
crystallization at pressures below the threshold. In this case, the
crystallization of Ni62Nb38 proceeds through the separation of the system into
two crystalline fractions. The kinetic rate factor of Ni crystallites rapidly
increases with pressure, and this rate factor is several times higher than the
growth rate factor of Nb crystals. On the one hand, these results show that
the pressure $p$ is a key factor controlling BMG crystallization. On the other
hand, it follows from the obtained results that the useful functional
properties of BMG (for example, corrosion resistance, strength, hardness, and
weak dependence of electrical resistance on temperature) directly related to
the presence of a homogeneous amorphous structure can be lost under extreme
conditions. The applied pressure and shear strain do not lead to the formation
of BMG with higher-energy metastable state, which is also called rejuvenation
effect [62]. The reason for this can be the peculiarities of the interatomic
interactions, which do not allow the formation of a high local energy barrier
around the atom at high pressures. Note that the rejuvenation can lead to
strain-hardening of some metallic glasses. This was demonstrated earlier by
the example of ternary and quaternary systems based on zirconium and palladium
[63, 64].

In this regard, it seems interesting to develop the results of the present
study to solve the following problems:

(i) Crystalline separation and differences in the kinetic rate factors of the
crystalline phases of Ni and Nb should impact the values of the main
mechanical characteristics of Ni62Nb38. It would be interesting to test or
predict the mechanical properties of this BMG under extreme conditions.

(ii) To study the phase diagram of Ni62Nb38 at high pressures.

(iii) Verification of the applicability of the crystallization conditions and
the observed scenario of the phase transition (crystal separation, difference
in crystal growth kinetics) in the case of other binary and ternary BMG’s.

## Acknowledgement

This study is supported by the Russian Science Foundation (project No.
19-12-00022).

## Appendix: Calculation of orientational order parameters

According to Refs. [36, 37], the local orientational order parameters $q_{l}$
($l=4,\,6,\,8$) are defined as follows

$q_{l}(\alpha,i)=\left[\frac{4\pi}{2l+1}\sum_{m=-l}^{l}\left|\bar{q}_{lm}(\alpha,i)\right|^{2}\right]^{1/2},\,\,i=1,2,...,N_{\alpha},$
(4)

where local structure around $i$th atom characterizes by

$\bar{q}_{lm}(\alpha,i)=\frac{1}{n_{\alpha,i}}\sum_{j=1}^{n_{\alpha,i}}Y_{lm}(\theta_{\alpha_{i}\alpha_{j}},\phi_{\alpha_{i}\alpha_{j}}).$
(5)

Here, $Y_{lm}(\theta_{\alpha_{i}\alpha_{j}},\phi_{\alpha_{i}\alpha_{j}})$ are
the spherical harmonics ($\alpha=\\{$Ni$,\,$Nb$\\}$); the quantities
$\theta_{\alpha_{i}\alpha_{j}}$ and $\phi_{\alpha_{i}\alpha_{j}}$ determine
the polar and azimuthal angles, respectively; $n_{\alpha,i}$ is the number of
nearest neighbors of $i$th atom. An atom $i$ is considered entering the
crystalline phase if it forms an ordered structure with four or more $j$th
atoms from the nearest environment. In this case, the atoms $i$ and $j$ belong
to a common crystal cluster with the following correlation condition [37]:

$\left|\sum_{m=-6}^{6}\bar{q}_{6m}(\alpha,i)\bar{q}_{6m}^{*}(\alpha,j)\right|>0.5,$
(6)

where

$\bar{q}_{6m}^{*}(\alpha,j)=\frac{\bar{q}_{6m}(\alpha,j)}{\sqrt{\sum_{m=-6}^{6}\left|\bar{q}_{6m}(\alpha,j)\right|^{2}}}.$
(7)

By calculating the local invariants $\bar{q}_{lm}(\alpha,i)$, we define the
global orientational order parameters $Q_{l}(\alpha)$

$Q_{l}(\alpha)=\left(\frac{4\pi}{2l+1}\sum_{m=-l}^{l}\left|\bar{Q}_{lm}(\alpha)\right|^{2}\right)^{1/2},$
(8)

where

$\bar{Q}_{lm}(\alpha)=\frac{\sum_{i=1}^{N_{\alpha}}n_{\alpha,i}\cdot\bar{q}_{lm}(\alpha,i)}{\sum_{i=1}^{N_{\alpha}}n_{\alpha,i}}.$
(9)

In the case of liquids and amorphous solids, the order parameters $Q_{l}$
($l=4,\,6,\,8$) take values close to zero. For the crystalline phase, these
parameters take specific values that characterize the type of crystal lattices
[38].

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