Understanding The Maximum Dynamical Heterogeneity During The Unfreezing Process in Metallic Glasses
Understanding The Maximum Dynamical Heterogeneity During The Unfreezing Process in Metallic Glasses
Understanding The Maximum Dynamical Heterogeneity During The Unfreezing Process in Metallic Glasses
metallic glasses
B. Wang, L. J. Wang, W. H. Wang, H. Y. Bai, X. Q. Gao, M. X. Pan, and P. F. Guan
In-situ atomic force microscopy observation revealing gel-like plasticity on a metallic glass surface
Journal of Applied Physics 121, 095304 (2017); 10.1063/1.4977856
* +
preparation, the system was first allowed to reach equilibrium 1 X
N
at 1900 K under an external pressure of zero with the constant Gs ðr; tÞ ¼ d r þ ri ð0Þ ri ðtÞ ; (1)
N i¼1
number, pressure, and temperature (NPT) ensemble for 10 ns.
Following this, the sample was quenched to 50 K step by step
where dðrÞ is the d function and r is the distance traveled by
at a rate of 1011 K/s with the NPT ensemble. The samples in a particle in the time t.8,30,31 The physical meaning of
Ref. 23 were also employed to ensure the validity of the results
Gs ðr; tÞ is the probability of finding a particle in the distance
(for more details, see supplementary material). During the
of “r” at some other time “t” and the volume integral of it is
cooling process, the configurations of each sample at the tem-
a conserved quantity and equals unity. Hence, 4pr2 Gs ðr; tÞ
peratures of interest were collected for further dynamical
reflects the probability distribution density of the displace-
mechanical spectroscopy. For all simulations, periodic bound-
ment within time interval t and has the same physical mean-
ary conditions were applied in three directions, the temperature
ing of p(u), which has been introduced in previous
is maintained by the Nose-Hoover thermostat,26 and the time
studies.23,27,32 Obviously, the shape of 4pr 2 Gs ðr; tÞ reflecting
step is 2 fs. In order to obtain the relaxation dynamics of glass,
the dynamic behavior is dependent on the observation time33
the extensive molecular dynamics (MD) simulation method
or interval time t. Since we are trying to understand the
was employed, which combines DMS and the isoconfigura-
dynamic behaviors during the unfreezing process by cycle
tional ensemble.27 The details of the MD-DMS method can be
loading, we set the interval time t as tp ¼ 100 ps, the period
found in the supplementary material.
of cycle loading, for all investigated temperatures to avoid
the influence of the observation time. The calculated
III. RESULTS AND DISCUSSION
4pr2 Gs ðr; t ¼ 100 psÞ for different temperatures is shown in
Figure 1(a) shows the temperature dependence of the loss Figure 1(b). As expected, this probability distribution is
modulus E00 , based on the systematic MD-DMS simulations, found to broaden with increasing temperature, which is con-
which is analogous to the previous experimental and numerical sistent with previous studies.32 If there is no local atomic
results.28,29 The curve presents the excess wing for the low rearrangement, Gs ðr; tÞ should have a standard Gaussian
temperature relaxation region and exhibits a peak at Ta that form with
corresponds to a relaxation. This peak signals the model sys-
3
tem transform from the glassy state to the supercooled liquid Ggs ðr; tÞ ¼ 3=2phr 2 ðtÞi 2 exp 3r2 =2hr 2 ðtÞi ; (2)
state, which means that the unfreezing of metallic glass can be
achieved through cycle loading. This provides the opportunity according to the thermal vibration effect. Here, we assume
to investigate the evolution of the dynamic properties of MGs that the atomic displacements within the time interval t ¼
with increasing temperature and thereby reveal insight into the 100 ps at low temperatures are mainly caused by thermal
atomic-level mechanism. To study the atomic dynamic behav- vibration and 4pr 2 Ggs ðr; t ¼ 100 psÞ should have the same
ior and the presence of dynamical heterogeneities, we calcu- mode as the calculated 4pr2 Gs ðr; t ¼ 100 psÞ. The curves of
lated the self-part Gs ðr; tÞ of the van Hove correlation 4pr2 Gs ðr; t ¼ 100 psÞ and 4pr 2 Ggs ðr; t ¼ 100 psÞ at 750 K are
function, which is defined by shown in the inset of Figure 1(b). Obviously, the difference
between 4pr 2 Ggs ðr; t ¼ 100 psÞ and the calculated 4pr 2 Gs various temperatures. The relative difference between Ggs
ðr; t ¼ 100 psÞ highlights the non-Gaussian behavior of and Gs is weak for small and intermediate values of r, but
Gs ðr; t ¼ 100 psÞ. This confirms the theory of local atomic strong for larger r, which means that Ggs underestimates Gs
rearrangements and dynamical heterogeneity in MGs during significantly for larger r. It is evidence that a significant
sinusoidal strain driving at low temperatures. This is consis- number of atoms move farther than expected from the
tent with the concept of structural and dynamical heterogene- Gaussian approximation and the heterogeneity or relaxation
ities in MGs and agrees with the structural model in which is mainly contributed to by fast dynamic atoms, which is
the MGs consist of elastic regions (or solid-like regions) and consistent with previous studies.27 Therefore, we can define
inelastic regions (or liquid-like regions).34,35 The liquid-like a cutoff distance rcut to select the “fast dynamic atoms” as
regions can easily be activated during the loading process atoms that have moved farther than a distance rcut within a
and are responsible for the viscoelastic flow of glass.34,36 To time tp ¼ 100 ps. Here, we define rcut as the larger of the two
quantify the dynamical heterogeneity in the sinusoidal strain values of r for which Ggs ðrcut Þ ¼ Gs ðrcut Þ, shown by the
driven system, the non-Gaussian parameter arrow point in Figure 1(d); i.e., rcut is the value of r at which
the difference between Ggs ðrcut Þ and Gs ðrcut Þ starts to become
a2 ðt ¼ tp Þ ¼ 3hr 4 ðtÞi=5hr2 ðtÞi2 1; (3) positive and very large. The atoms can then be divided into
two parts at each temperature: the slow atoms belonging to
which is a departure from standard Gaussian form, was used
the elastic matrix and the fast atoms that mainly contribute
for this study, with the larger value of a2 reflecting the more
heterogeneous dynamics of the glass.8 The computed a2 ðt ¼ to the dynamic relaxation. The fast atoms can be viewed as
100 psÞ in the temperature range of 200–950 K is shown in “flow units,” considering that flow units are any available
Figure 1(c). It shows that the dynamical heterogeneity appears and thermally or mechanically activated localized rearrange-
to vary non-monotonically with increasing temperature and ments.34,36,37 The calculated fractions of average numbers of
presents a peak at Ta2;max , which is below Ta. It is consistent selected fast atoms are shown in Figure 2(a) (red histogram).
with the recent experimental work in which the non-monotonic The fraction of selected atoms increases as temperature
evolution of dynamical heterogeneity was found with increas- increases and this suggests that more and more flow units are
ing temperature,16 but the precise atomic level features remain activated and then that enhances the dynamic heterogeneity.
unclear. We also measured the temperature dependence of At T ¼ Ta2;max ¼ 750 K, the fraction of selected fast atoms
a2 ðt ¼ 100 psÞ for the various samples employed in Ref. 23 approaches 25% (red histogram in Figure 2(a)), and the value
and all curves of the samples exhibit a peak at a temperature agrees well with the fraction of frozen liquid-like atoms,
below its Ta (see supplementary material). It confirms that (i) 24.3%, in the theory of the glass transition for metallic
the dynamic response of MGs corresponding to sinusoidal glass.38,39 In essence, the dynamical heterogeneity results
strain loading becomes more heterogeneous with increasing from the frozen liquid-like atoms below Tg. When nearly all
temperature; (ii) a maximum value of a2 ðt ¼ 100 psÞ can be the frozen liquid-like atoms are activated, the dynamical het-
observed before the system transforms from the glassy state to erogeneity of the system achieves its maximum. Thus, the
the supercooled liquid state; and (iii) a2(t ¼ 100 ps) decreases crossover of a2 at Ta2;max (see Figure 1(c)) may suggest a dra-
to zero at very high temperatures. This is evidence that there is matical change of relaxation behavior corresponding to the
coupling between dynamical heterogeneity and the relaxation onset temperature Ta2;max .
behavior at low temperatures, but decoupling as the tempera- To investigate the evolution of the relaxation behavior
ture gets close to Ta. with increasing temperature, we analyzed the microscopic
To understand the monotonic increase of a2 ðt ¼ 100 psÞ features of selected fast atoms (or flow units) for T < Ta2;max .
in the low temperature regime and what happens at the tem- The spatial correlation between selected atoms is shown in
perature Ta2;max , we selected the atoms which contribute to the Figure 2(b), where we compare (cf. inset) gs(r) and gall(r),
value of a2 ðt ¼ 100 psÞ. First, we compared the calculated the radial distribution functions for the fast atoms and for the
Gs ðr; t ¼ 100 psÞ with the distribution Ggs ðr; t ¼ 100 psÞ that bulk, respectively. We find that at the first peak, at lower
is obtained from the Gaussian approximation. Figure 1(d) temperatures the fast atoms are more strongly correlated than
shows the ½Gs ðr; tÞ Ggs ðr; tÞ=Ggs ðr; tÞ with t ¼ 100 ps for the bulk. This is demonstrated more clearly by computing the
ratio gs/gall, which is shown in Figure 2(b) for three different the properties of metallic glasses effectively just above
temperatures. From this figure, we can infer that the rela- Ta2;max , but not necessarily above the glass transition tempera-
tively strong correlation between the fast atoms at 200 K sug- ture Tg. Furthermore, the consequences may suggest that the
gests that the flow units tend to form clusters and the size of low temperature relaxation can act as a precursor of a relaxa-
a cluster is on the order of 5 A. This can be confirmed by the tion and the relation between a and low temperature relaxa-
motif of these atoms (see Figure 3(a)) and the atomic frac- tion will be clarified in the further.
tion of the largest cluster (see Figure 2(a), blue histogram).
As shown in Figure 3(a), the clusters constructed by flow IV. CONCLUSIONS
units disperse homogenously in the system. As the tempera-
In summary, the non-monotonic evolution of dynamical
ture increases, the difference between the gs and gall becomes
heterogeneity was found during the unfreezing process and
weaker and the atomic fraction of the largest cluster becomes
the maximum of dynamical heterogeneity appears at a tem-
bigger. It suggests that more and more flow units are acti-
perature below Ta. The flow unit perspective was applied to
vated and form bigger clusters by interconnecting, which is
account for the phenomenon microscopically. As tempera-
supported by the spatial distribution of selected atoms in
ture increases, more and more flow units are activated and
Figures 3(b) and 3(c). Finally, nearly all the flow units
tend to interconnect into larger clusters during this unfreez-
involve in the largest cluster and the largest cluster, the red
ing process, and then at the temperature Ta2;max the largest
cluster in Figure 3(d), passes through the whole system at the
cluster starts to show the percolation property. Our results
temperature T ¼ Ta2;max ¼ 750 K. Similar results were also
also provide new insights into the correlation between a
obtained when the various samples from Ref. 23 were stud-
relaxation and low temperature relaxation.
ied. It implies that the system at Ta2;max behaving with maxi-
mum dynamical heterogeneity starts to show the percolation
SUPPLEMENTARY MATERIAL
property of the largest activated cluster. The temperature
Ta2;max roughly corresponds to the temperature at which a See supplementary material for the MD-DMS method
relaxation begins in Figure 1(a), and we can speculate that and details about samples employed in Ref. 23. It also pro-
the flow units tend to interconnect with each other and form vides the dynamical heterogeneity evolution of these sam-
extended cooperative flowing regions at this temperature. ples to ensure the validity of the results in the manuscript.
Therefore, the a-process begins to contribute to the relaxa-
tion dynamics. This is consistent with the result that nearly ACKNOWLEDGMENTS
all the frozen liquid-like atoms are activated at Ta2;max , which Insightful discussions with K. L. Ngai, B. S. Shang, P.
implies that glass transition or a relaxation starts to occur Luo, Y. C. Wu, L. J. Wang, and S. Zhang are highly
through the percolation transition of liquid-like states, as pre- acknowledged. We also thank D. W. Ding, D. Q. Zhao, and
dicted by Cohen and Grest.40 It indicates that we could tune B. B. Wang for helpful discussions. The financial support of
the NSF of China (Grant No. 51571011), the NSAF of China
(Grant No. U1530401), and the MOST 973 Program (No.
2015CB856800) is acknowledged. B.W., W.H.W., H.Y.B.,
and M.X.P. are also supported by the NSF of China (Grant
Nos. 51571209, 51461165101, and 51601150).
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