The role of local-geometrical-orders on the growth of dynamic-length-scales in glass-forming liquids

The precise nature of complex structural relaxation as well as an explanation for the precipitous growth of relaxation time in cooling glass-forming liquids are essential to the understanding of vitrification of liquids. The dramatic increase of relaxation time is believed to be caused by the growth of one or more correlation lengths, which has received much attention recently. Here, we report a direct link between the growth of a specific local-geometrical-order and an increase of dynamic-length-scale as the atomic dynamics in metallic glass-forming liquids slow down. Although several types of local geometrical-orders are present in these metallic liquids, the growth of icosahedral ordering is found to be directly related to the increase of the dynamic-length-scale. This finding suggests an intriguing scenario that the transient icosahedral connectivity could be the origin of the dynamic-length-scale in metallic glass-forming liquids.

that it is of the order of interparticle distance and grows by a factor of 2 to 9 as the glass-forming liquids cool towards a mode-coupling critical temperature, T c 4,22,23 . Studies on polydispersed glass-forming liquids 24 and a two-diemnsional metallic liquids 25 have domenstrated that the static and dynamic are coupled 26,27 . However, some other studies have shown that the static length scale increases at a slower pace than the dynamic-length-scale in decreasing the temperature of glass-forming liquids 5,28 . The growth of local geometrical-orders (LGOs) is also argued to be the cause of the rapid rise in the relaxation time of cooling glass-forming liquids [29][30][31] . In three dimensional systems of monodispersed hard-spheres, the icosahedron is the most locally-preferred structure and increasing its number while cooling is believed to be linked directly to vitrification 32,33 . In multicomponent, polydispersed metallic systems, an increasing number of five-fold symmetry clusters is reported to be the reason for their better glass-forming ability (GFA) 34,35 . However, the role of specific LGO on the growth of dynamic-length-scale is not yet undersood, although numerous studies have shown rapid growth of dynamic-length-scale and LGOs on cooling glass-forming liquids towards the glass-transition temperature T g . In this article, we studied a ternary metallic glass-forming system to understand whether the growth of any specific LGO correlates with the increase of dynamic-length-scales by using quasielastic neutron scattering (QENS) and molecular dynamics (MD) simulation techniques.

Results
The Cu-Zr-Al is a well-known glass-forming metallic system with a distinct GFA 36 . For this study, we have chosen the following composition: (Cu 50 Zr 50 ) 100−x Al x (x = 2, 4, 8, and 10). The GFA of the system is found to increase with the addition of Al in Cu 50 Zr 50 36 . The QENS experiments were conducted on a time-of-flight neutron scattering instrument, Pelican, at Bragg Institute in Sydney, Australia 37 where f q is the Debye-Waller factor, τ α is the relaxation time, and β is the stretching exponent. The value of the stretching exponent was found to be β < 1 and temperature dependent, but composition-and Q-independent. At the lowest measured temperature, we obtained a value of β = 0.6 ± 01, and at the highest temperature β = 0.8 ± 0.1. The stretching of Φ(q, t) indicates the multiple relaxation processes and the presence of heterogeneous dynamics in CuZrAl liquids. As we mentioned earlier, the DH can be estimated by calculating the four-point correlation function χ 4 (t), but this quantity cannot be evaluated from a QENS experiment. However, recent theoretical advances have shown that the χ 4 (t) is related to the dynamic susceptibility, χ T (Q, t) by the fluctuation-dissipation theorem 38 . The χ T (Q, t) can be evaluated from the Φ(Q, t) which is readily obtained from QENS experiments. The dynamic susceptibilities were obtained by χ = ∂Φ ∂ Q t ( , ) T Q t T ( , ) . Figure 1b shows the χ T (Q, t) of the (Cu 50 Zr 50 ) 94 Al 4 liquid in a semi-logarithmic representation, which shows that the strength of DH increases with cooling. The strength of χ T (Q, t) indicates the extent of spatial correlation in the atomic motion 39 (Fig. 1b). The strength of DH in (Cu 50 Zr 50 ) 100−x Al x liquids is quite similar, but slightly increases with increasing concentration of Al. It has been proved that the onset of cooperative dynamics in metallic glass-forming melts begin at a temperature ~2T g , where T g is the calorimetric glass-transition temperature 40 . Therefore, in these high density liquids our experimental results confirm the presence of dynamic heterogeneities in (Cu 50 Zr 50 ) 100−x Al x well above the T g to the measured maximum temperature and the increase in the length-scale of correlated atomic motion in cooling the liquids.
The MD simulations were performed with a system of 100,000 atoms using the LAMMPS software and employing the potential developed by Sheng et al. 41 . To estimate the strength of DH and its temperature dependence, we first evaluated the self-part of the four-point correlation function 23 , 2 are overlap functions that are unity for − ≤ r r a 1 2 and 0 otherwise (Fig. 1c). We chose the distance parameter a = 1, which is the plateau value of the square of mean square displacement 23 . Although the absolute values of the strength of χ T (Q, t) and χ 4 (Q, t) obtained from the QENS and MD simulation cannot be compared, the growth rates with respect to the temperature are very similar (see Fig. 1b,c). Our results indicate that the relation between χ T (Q, t) and χ 4 (Q, t) proposed by Berthier and co-workers  of polyhedrons in these metallic liquids at all temperature ranges 35 . However, the most abundant polyhedron in all four systems, above the melting temperature, was found to be <0,3,6,4> polyhedron, while the next abundant polyhedron was <0,2,8,2>. The population of these two polyhedrons increases with the Al concentration, while the growth of a specific polyhedron in cooling the liquids is dependent on the type of polyhedrons. The growth of the <0,3,6,4> polyhedron, which is most abundant in these liquids, is saturated in the supercooled liquid (see Fig. 2b). The next most abundant <0,2,8,2> polyhedron has grown almost linearly while cooling the liquids (see Fig. 2c). Interestingly, the icosahedra, <0,12,0,0> has grown much faster in the undercooled liquids (see Fig. 2d). In cooling these four alloy liquids, the growth of specific polyhedrons is similar, but the percentage of icosahedrons increased with the concentration of Al at any given temperature (Fig. 2d).
The dynamic-length-scale in these alloy liquids at various temperatures was obtained by the following procedures. First, we calculated the four-point, time-dependent structure factor for self-overlapping particles, S 4 (q, t), which is defined as, , and t is the time at which the maximum of dynamical four-point susceptibility χ 4 (t) 4,42,43 . The DH is transient in time, reaching a maximum value at around the structural relaxation time, which measures the degree of cooperativity of structural relaxation. Second, the S 4 obtained at low-q values at a given temperature were fitted with Ornstein-Zernike form (see Fig. 3b), where ξ 4 is the dynamic correlation length or dynamic-length-scale. In decreasing the temperature, the ξ 4 increases exponentially (see Fig. 3b). However, the growth rate of ξ 4 depends on alloy composition, with a higher amount of the Al content in the liquids a faster growth rate of ξ 4 was observed. In the (Cu 50 Zr 50 )Al 10 , which has the highest Al content, the ξ 4 increasing from 2 to 7.5, while in alloys with 2% Al, ξ 4 increases marginally (see Fig. 3b). As we compare the growth of ξ 4 with different polyhedrons, the growth rate shows two regimes for all types of polyhedron other than the icosahedron. The ξ 4 grew a little with respect to number of <0,3,6,4> or <0,2,8,2> polyhedrons in the melts but grew substantially in the undercooled states. Surprisingly, the growth of ξ 4 shows a direct correlation with the population of icosahedrons in these liquids. This relation holds good in these highly dense alloy liquids in a temperature range as low as 300 K below T m and as high as 300 K above T m .

Discussion
Our study experimentally confirmed the existence of the dynamic heterogeneity in the glass-forming (Cu 50 Zr 50 ) 100−x Al x liquids and the corresponding length scale of correlated motion over a temperature range of 600 K. This is visible from the stretching of intermediate scattering function and the strength of dynamic susceptibility, respectively. Both the dynamic heterogeneity and the dynamic length scale were found to increase with Al content and cooling of alloy liquids. At the same time, the atomic mobility was found to decrease, which indicates strong coupling between structural relaxation and the dynamic-length-scale. Using the MD simulation, we quantitatively determined the population of different polyhedrons as a function of temperature in these liquids. Although the population of the icosahedron is very little in high temperature liquids, it increases substantially in an undercooled state. The population of icosahedrons grew exponentially in the range of temperature studied. Similarly, the dynamic length scale evaluated from the four-point dynamic structure factor increases exponentially with a decreasing in temperature. However, the growth of other polyhedrons did not follow specific temperature dependence. In fact, the growth of most abundant <0,3,6,4> polyhedron saturated in undercooled liquids, and the <0,2,8,2> polyhedron grows almost linearly with temperature. This suggests a strong correlation between the increase of the dynamic-length-scale and growth of icosahedrons in liquids and the possible percolation of a string-like icosahedral medium-range order 35 . Additionally, as a function of Al content, the number of icosahedra increases and atomic dynamics slows down. This result suggests that the slowdown of the atomic dynamic in these liquids must have a strong coupling to the growth of icosahedron with a characteristic lengthscale changes as seen also in nonlinear dielectric studies 44 . Entire existing studies in which both static and dynamic-length-scale have been computed for the same glass-forming liquids show that these length scales do not correlated each other [4][5][6] . The value of static-length-scales calculated in various studies generally found to be smaller than the dynamic one and the difference increases with decrease in temperature 45,46 . Therefore, we did not attempt to calculate the static-length-scale in the (Cu 50 Zr 50 ) 100− x Al x liquids. However, dynamic-length-scales obtained in our study exhibit a power law behaviour with relaxation , where a is constant z is the dynamic critical exponent. The value of z varied from 4.37-10.06 (see Fig. 3f). Such a power law behaviour has been reported in many glass-forimg liquids 23,47 . It has also been shown that the icosahedral clusters have a strong tendency for connectivity and form a string-like network [48][49][50] . Therefore, we explored the possibility of a correlation between the increase of dynamic-length-scale and growth of icosahedra in these alloy liquids. We scaled the population of the icosahedron with dynamic-length-scale in the four alloy liquids investigated. Surprisingly, a linear relationship was observed in the alloy liquids (Fig. 3e)   a Q range of 0.2 Å −1 -1.9 Å −1 and an energy resolution of 70 µeV for the experimental setup. Since the maximum of static structure factors of the alloy melts are around 2.8 Å −151 , the scattered signals from our experiments are mainly due to the incoherent scattering from the Cu atoms of the samples. Therefore, the dynamics that we observed from the QENS data are of the self-dynamics of Cu atoms in the respective alloy liquids. For the high temperature measurements, we used an ILL-design vacuum furnace. The QENS data were collected from high temperature down to the melting temperature of each alloy (1573 K, 1473 K, 1373 K, 1273 K, and 1223 K). The QENS data were collected over a duration of 4 hours at each temperature. An empty Al 2 O 3 sample holder was also measured at each temperature for background subtraction and self-absorption correction. For correcting the detector efficiency, a vanadium crucible with a similar sample geometry was also measured at room temperature for 4 hours.
QENS data analysis. The raw data was normalized to the vanadium data, and converted to the dynamic structure factor using LAMP (Large Array Manipulation Program). The self-absorption of the sample S in a container C was corrected using where S c and S S+C is the scattering from the container, and from both sample and container respectively.
is the correction factor for scattering and self-absorption of the container, is the correction factor for scattering due to the container and absorption in both sample and container and θ ω is the correction factor for scattering due to the sample and absorption in both sample and container. At last, the intermediate scattering functions Φ(Q, t) were obtained by Fourier transforming the dynamic structure factor. Molecular Dynamics Simulation. The molecular dynamics (MD) simulations were done for the same compositions measured in QENS experiments using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS); a free software obtained from the Sandia National Laboratories, USA. The embedded-atom method (EAM) potential was used to describe the interatomic interactions. The time step used in the simulation was 2 fs and periodic boundary conditions were applied. The Nose-Hoover thermostat was used to control the temperature. For each system, the initial configuration containing 100,000 atoms was equilibrated at 2000 K for 5 ns followed by rapid quenching to the desired temperatures with the rate of 2 × 10 11 Ks −1 in NPT ensemble. The volume of the system was adjusted to give zero pressure during cooling. Before taking the structural configurations, the systems were relaxed for extra 1 ns by the NVT ensemble. Voronoi tessellation calculates the polyhedral cells which have planar faces and completely fill the space by constructing bisecting planes along the lines connecting the target atom and its neighbors. The polyhedrons surrounding the central atom are described by the Voronoi index <n 3 ,n 4 ,n 5 ,n 6 …>, where n i is the number of i-edged faces of the polyhedron. A cutoff distance of 5 Å was used in this study.