Atomic vibration as an indicator of the propensity for

This journal is © The Royal Society of Chemistry 2021 Mater. Horiz., 2021, 8, 2359–2372 | 2359

Cite this: Mater. Horiz., 2021,

8, 2359

Atomic vibration as an indicator of the propensity forconfigurational rearrangements in metallic glasses

Huiyang Fan,ab Zhao Fan, b Xiongjun Liu, *a Zhaoping Lua and En Ma*c

In a metallic glass (MG), the propensity for atomic rearrangements varies spatially from location to location in the

amorphous solid, making the prediction of their likelihood a major challenge. One can attack this problem from

the ‘‘structure controls properties’’ standpoint. But all the current structure-centric parameters are mostly based on

local atomic packing information limited to short-range order, hence falling short in reliably forecasting how the

local region would respond to external stimuli (e.g., temperature and/or stress). Alternatively, one can use

indicators informed by physical properties to bridge the static structure on the one hand, and the response of the

local configuration on the other. A sub-group of such physics-informed quantities consists of atomic vibration

parameters, which will be singled out as the focus of this article. Here we use the Cu64Zr36 alloy to systematically

demonstrate the following two points, all using a single model MG. First, we show in a comprehensive manner

the interrelation among common vibrational parameters characterizing the atomic vibrational amplitude and

frequency, including the atomic mean square displacement, flexibility volume, participation fraction in the low-

frequency vibrational modes and boson peak intensity. Second, we demonstrate that these vibrational parameters

fare much better than purely static structural parameters based on local geometrical packing in providing

correlation with the propensity for local configurational transitions. These vibrational parameters also share a

correlation length similar to that in structural rearrangements induced by external stimuli. This success, however,

also poses a challenge, as it remains to be elucidated as to why short-time dynamical (vibrational) behavior at the

bottom of the energy basin can be exploited to project the height of the energy barrier for cross-basin activities

and in turn the propensity for locally collective atomic rearrangements.

1. Introduction

In contrast to conventional alloys which solidify into crystallinestructures upon cooling from their liquid counterparts, metal-lic glasses (MGs) result from the frustration of the crystal-lization process to form a metastable non-crystalline solid,

a State Key Laboratory for Advanced Metals and Materials, University of Science

and Technology Beijing, Beijing, 100083, China. E-mail: [email protected] Department of Materials Science and Engineering, Johns Hopkins University,

Baltimore, MD 21218, USAc Center for Alloy Innovation and Design (CAID), State Key Laboratory for

Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi’an 710049, China.

E-mail: [email protected]

Huiyang Fan

Huiyang Fan is currently a PhDcandidate at University of Scienceand Technology Beijing (USTB). Shereceived her BS degree from USTBin 2013. Her current researchfocuses on metallic glasses andcomputational materials science.

Zhao Fan

Zhao Fan is currently apostdoctoral fellow at theDepartment of Materials Sci-ence and Engineering, JohnsHopkins University. He receivedhis PhD degree in MaterialsScience and Engineering fromJohns Hopkins University in 2020.His current research focuses onestablishing structure-propertyrelationships in various materialsvia atomistic simulations and deeplearning. He will move toUniversity of California, Irvine, asa postdoctoral fellow soon.

Received 22nd March 2021,Accepted 12th July 2021

DOI: 10.1039/d1mh00491c

rsc.li/materials-horizons

MaterialsHorizons

FOCUS

Publ

ishe

d on

12

July

202

1. D

ownl

oade

d on

12/

10/2

021

1:44

:40

PM.

View Article OnlineView Journal | View Issue

http://orcid.org/0000-0002-2523-2532

http://orcid.org/0000-0002-1663-4636

http://crossmark.crossref.org/dialog/?doi=10.1039/d1mh00491c&domain=pdf&date_stamp=2021-07-31

http://rsc.li/materials-horizons

https://doi.org/10.1039/d1mh00491c

https://pubs.rsc.org/en/journals/journal/MH

https://pubs.rsc.org/en/journals/journal/MH?issueid=MH008009

2360 | Mater. Horiz., 2021, 8, 2359–2372 This journal is © The Royal Society of Chemistry 2021

which is often considered as a frozen-in state of liquids.1

Early MGs were typically obtained by rapid quenching ofmelts.2 In the 1990s, bulk MGs with unusually high glass-forming ability were developed,3,4 which can be produced viacopper-mold casting and have inspired extensive researchactivities. Over the past decades, numerous bulk MGswith interesting physical, chemical and mechanical proper-ties have been reported, such as Zr-, Fe-, Co-, Mg-, and Pd-based alloys.5 Meanwhile, intensive research has beenongoing to explore the structural origin of their unusualproperties. Distinctly different from conventional alloys,MGs are amorphous solids, with no long-range order norwell-defined defects. The local structures inside MGs areinherently inhomogeneous, and depend strongly on thecomposition and the processing history of the alloy.6–9 Con-sequently, under externally imposed stimuli (e.g., tempera-ture and stress) the responses of local regions are variableand influenced by the degree and length scale of theheterogeneities.8,10

It has been generally believed that there are liquid-likeregions inside MGs, which play the role of defects in crystalline

metals to carry the relaxation and plastic flow. As a result, muchattention has been focused on detecting ‘‘defects’’ inside MGs,by specifying the local environment of the atoms. For example,some local regions may contain more ‘‘free volume’’ and hencebe more prone to local plasticity or relaxation.11,12 Later workhas probed the local environment from the atomic packingperspective, in particular the short-range order (SRO) surround-ing an atom. In this regard, the coordination polyhedra13,14 arethe most elementary structural motifs.15 For example, in someMG systems icosahedra are found to be preferable and affectthe glass transition dynamics significantly.16 These motifs arerelatively stable and rigid, and their connection in space mayform skeletons that help resist deformation.17 On the oppositeside, there are also irregularly shaped coordination polydedrawith excess volume and low symmetry, termed ‘geometricallyunfavored motifs’ (GUMs).18 These GUMs are unstable andmore likely to evolve under applied stimuli.18 In lieu of inspect-ing the coordination polyhedra, one can also monitor thedegree of local ordering by investigating the bond orientationorder,19 local five-fold symmetry (LFFS),20 and spherical peri-odic order or local transitional symmetry,21,22 even though sucha simple index may not be able to tell apart different localatomic packing environments.23,24 Beyond SROs, attempts havealso been made to describe the structural correlations on amedium-range length scale and beyond.25,26 This has beendone mostly for the connectivity of representative polyhedra/clusters, sharing vertices, edges or faces, to explain the struc-tural stability during the glass transition26,27 or rigidity of thebackbone.28 However, a simple and yet robust indicator toquantify the medium-range order (MRO) remains sorelymissing.

Despite all these efforts, it is now generally realized that thecorrelation between any of these static structural indicators andlocal properties is unlikely to be satisfactorily strong. This ispartly because the property variations are not controlled by SROalone, and the degree of order and coordination in the mediumrange (e.g., up to 2 nm) and beyond is difficult to decipher and

Zhaoping Lu

Dr Zhaoping Lu is currently a fullprofessor at University of Scienceand Technology Beijing (USTB). Hereceived his PhD degree fromNational University of Singapore in2001, and worked as a post-doctoralfellow for three years at Oak RidgeNational Laboratory, from 2001 to2004. His research interests mainlyfocus on metallic glasses, high-entropy materials, advanced steelsand computational materialsscience. En Ma

Dr Evan Ma is a professor ofMaterials Science and Engi-neering at Xi’an JiaotongUniversity. His research areasinclude metastable alloys such asmetallic glasses, high-entropyalloys, nanocrystalline metals,and phase-change memory alloys.He has published over 370 papers,with over 45 000 citations and anh index of 108. Dr Ma was therecipient of the 2004 ASMMaterials Research Silver Medal.Prof. Ma was named an ASM

Fellow in 2009, elected APS Fellow in 2010, and MRS Fellowin 2015.

Xiongjun Liu

Dr Xiongjun Liu is currently a fullprofessor at University of Scienceand Technology Beijing (USTB). Hereceived his PhD degree from USTBin 2008, and worked as a post-doctoral fellow for two and halfyears at the Hong Kong PolytechnicUniversity, from 2009 to 2012. Hisresearch interests mainly focus onmetallic glasses, high-entropy mat-erials, nanoporous metals andcomputational materials science.

Focus Materials Horizons

Publ

ishe

d on

12

July

202

1. D

ownl

oade

d on

12/

10/2

021

1:44

:40

PM.

View Article Online



quantify.29,30 For example, the applicability of free volume isquestionable, as the atomic relaxation involves not only excessvolume, but also compressed ones.15,31 Similarly, shear trans-formations undergo both volume reduction and dilation duringdeformation. In general, the structural features influence theproperties in a rather complex way, one that is difficult toquantify using a single hand-crafted indicator. In other words,while it is desirable to be able to predict properties solely fromthe (local) atomic configuration, current simple descriptions ofthe packing environment appear to be too simplified to beadequate.

Research has also been progressing along a different line ofthinking, to develop parameters that are informed by physicaland mechanical responses, in addition to being related to thelocal packing structure. Such indicators include atomic-levelstresses,32,33 the configurational potential energy,34–36 localelastic moduli,37 the local yield stress,38,39 thermal energyfluctuation40 and the thermal activation energy barrier.41,42 Inother words, these parameters are physics-oriented, mainlybased on the knowledge of interparticle interactions (oftenthrough atomic potentials and extensive computer simula-tions). In particular, recent experiments and simulations havedeveloped several ways to take dynamics information intoaccount. For example, Harrowell et al. demonstrated that thelong-time dynamical behavior of a particle can be well inferredfrom its short-time dynamics, while it is not readily accessiblefrom its geometric free volume.43,44 Liu et al. defined theparticles with faster dynamics as flow units, where the localregion is more likely to yield under a small cyclic loadingstress.45 The Debye–Waller factor (DWF) is often used tomeasure the degree to which the local structure is constrainedby its surroundings, i.e., the cage dynamics of structuralrelaxation.46 Note that within a short timescale when topologi-cal rearrangement is infrequent, the local Debye–Waller factoris primarily determined by the local configurations. A similarindicator is the atomic mean-square displacement (MSD),which measures the displacement of an atom from its initialposition. The glassy state exhibits a plateau of the MSD,before the atom breaks the cage of local atomic vibration andenters the diffusive regime. In terms of short-time dynamics,several unusual vibrational properties have been noticed forglasses. Among them, an anomalous enhancement at lowfrequencies on the vibrational density of states (VDOS)spectrum has been widely observed in experiments; this isthe so-called boson peak (BP). Also, soft modes (quasi-localized low-frequency vibrational modes)18,47 have beenidentified and are believed to contribute to the BP: theseadditional modes are localized at groups of atoms that movecollectively. Meanwhile, the local regions where particleshave large MSD and low shear modulus are soft domainsthat participate more in the low-frequency vibrationalmodes.48–50 These ‘soft spots’ have been observed to fre-quently coincide with local regions where shear transforma-tions take place.18 Since the quasi-localized modes areresonantly coupled with transverse phonons, Yang et al.proposed a structural parameter termed the orientational

order to describe the most probable direction of the trans-verse vibration; the magnitude of this parameter scaleslinearly with the BP intensity.51 Also, it is tempting to spec-ulate that the BP intensity is closely related to the degree ofdefectiveness. Experimentally, it has been proven that severeplastic deformation would lead to an obvious enhancementof the BP, due to the formation of shear bands. Aftersubsequent annealing, the BP would weaken.52,53 Luo et al.recorded the BP fluctuation depending on the thermal his-tory of glass.54 All these suggest that the BP intensity mayalso correlate well with the local properties.

These ideas in recent years, as well as the hints alreadyin the literature for the various correlations, bring forth twopressing needs. First, it will be useful to interrogate andsort out how these dynamics parameters are interrelated. Inthe following, we will assess for each particle the participationin the low-frequency vibrational modes and the contributionto the BP intensity, and link them with the vibrational ampli-tude (MSD) and flexibility of the atom. Second, it will be ofinterest to systematically evaluate how good these vibration-dynamics indicators are, relative to purely structural para-meters, in connection with configurational rearrangements inrelaxation and shear transformations. To this end, we willcompare the power of the various vibrational parametersfor predicting the local properties, including the local shearmodulus, atomic non-affine displacements as a measureof the propensity for shear transformations, and the localactivation energy as a metric for the tendency towards ther-mally induced relaxation. In other words, this focus articleaims to provide a timely resolution of the two needsabove. We intentionally make our summary easy to follow, byexamining each and every quantity of interest all in a singleCu64Zr36 MG model.

2. Atomic vibration characteristics

We chose the Cu64Zr36 alloy as the model MG for the followingreasons: first, it is a typical binary alloy with good glass-formingability,55 which has been widely investigated in both experi-mental and theoretical studies.56,57 Second, the empiricalembedded atom potentials for this alloy have been widely usedfor studying atomic structures, and dynamic and mechanicalproperties.16,29,58 All the simulations in our work were per-formed using the LAMMPS package.59 The system containing10 000 atoms was melted and fully equilibrated at 2500 K usinga Nose–Hoover thermostat to get the liquid state, and thenquenched to 300 K at a cooling rate of 1 � 109 K s�1 with a timestep of 2 fs to get the glass state. The external pressure was heldat zero during the quenching process, and periodic boundaryconditions were applied in all three dimensions.

2.1 Atomic vibration amplitude and flexibility volume

The mean-square displacement (MSD) is a well-known measureof the deviation of the position of a particle with respect to itsreference position over time. The MSD of the ith particle in a

Materials Horizons Focus

Publ

ishe

d on

12

July

202

1. D

ownl

oade

d on

12/

10/2

021

1:44

:40

PM.

View Article Online



system is given by ri2 = h(xi(t) � xi(0))2i, where xi(t) is the

position of the ith particle at time t, xi(0) the reference position,and h� � �i represents an ensemble average. A larger MSD valuereflects a larger amplitude of displacement and faster dynamicsof the particle. In the MG state the MSD exhibits a plateau overa short time period before leaving the cage to enter the diffusiveregime. This constrained oscillation of the particle is taken tobe its vibrational MSD. We calculated the vibrational MSD ofeach particle inside the sample equilibrated at 300 K under amicrocanonical ensemble. The calculated ri

2 was then averagedover 100 independent runs, all starting from the same configu-ration but with different momenta assigned randomly from theappropriate Maxwell–Boltzmann distribution.

By combining the atomic vibrational MSD with the atomicvolume, Oa (obtained from the Voronoi tessellation60), a newparameter termed the flexibility volume (vflex) can be set up.61

The atomic vflex,i is defined as vflex,i = ri2�ai, where ai ¼

ffiffiffiffiffiffiffiffiOa;i

3p

isthe atomic spacing. The magnitude of vflex,i reflects the flex-ibility in space, or wiggle room of the atom in question. Bothvflex and ri

2 show a non-Gaussian distribution as seen in Fig. 1.The long tail corresponds to the atoms with the largest vibra-tional amplitude or atomic spacing in the system. The simu-lated MSD and flexibility volume are found to be consistentwith reported values in the literature.61,62 Based on Debyetheory, the flexibility volume can be derived from measurablequantities including the vibrational MSD, Debye temperature andshear modulus. The Debye temperature can be expressed as63

yD ¼h

kB

4p9

� ��1=3Oa�13

1

vl3þ 2

vs3

� ��1=3. Here h is the Planck con-

stant, kB is the Boltzmann constant, vl ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBþ 4

3G

r

vuutand vs ¼

ffiffiffiffiG

r

rare the longitudinal and transverse velocities, respectively (B is the

bulk modulus, and G is the shear modulus), and r ¼ m

Oais the

mass density. There is also a scaling relation between the Debye

temperature and vibrational MSD,64 as yD2 ¼ 9�h2T

mkB r2h i, where �h is

the reduced Planck constant, T is the temperature, m is the

averaged atomic weight, and hr2i is the averaged vibrationalMSD. If we take an approximation of vl = 2vs for MGs, a linearrelationship between the averaged vflex and shear modulus G can

be established,61 i.e., G ¼ CkBT

vflex, where C is a universal constant.

Besides, the percent change of vflex caused by structural relaxationor rejuvenation in a given MG is also comparable to that of themodulus.65 This quantitative scaling with the modulus is anadvantage over that with the vibrational MSD, making vflex,i

particularly useful when it comes to characterizing the rigidityof the surroundings61,65,66 and assessing the barrier (which scaleswith G) encountered in shear transformations.

2.2 Low-frequency vibrational modes and boson peak

There are other characteristic vibrational properties that can beobtained from the VDOS spectrum. The frequency (o)-dependent VDOS g(o) of target atomic groups was calculatedfrom the Fourier transform of the velocity auto-correlationfunction (VACF). The system was equilibrated at 300 K andambient pressure. The VACF of the whole system was subse-quently measured at 300 K in the micro-canonical ensemble.100 independent runs were generated and averaged to reducethe noise. To specify the contribution of each atom, we alsocalculated the atomic VACF, each being an average of at least50 000 calculations. After obtaining the VDOS, the BP intensity(IBP) contributed by each particle and the whole system wasobtained by locating the maximum in the plot of the reducedVDOS g(o)/o2. We also confirmed that the weighted average ofthe partial VDOS is equal to the total VDOS of the whole system.

In order to evaluate the atomic participation fraction in thesoft modes, we adopted normal mode analysis of the sample bydiagonalizing the dynamical matrix of the inherent structureobtained using the conjugate-gradient method. The participa-tion fraction of particle i in eigenmode eo is defined bypi = |ei

o|2, where eio is the corresponding polarization vector

of particle i. The participation fraction in the low-frequencyvibrational modes of each atom was calculated by summing pi

of the 1% lowest-frequency modes in the spectrum of the

Fig. 1 Distribution of the (a) vibrational mean square displacement (ri2) and the (b) flexibility volume (vflex,i) in a Cu64Zr36 MG.


Publ

ishe

d on

12

July

202

1. D

ownl

oade

d on

12/

10/2

021

1:44

:40

PM.

View Article Online



dynamical matrix. This summation Pi measures the degree ofinvolvement of each atom in all the softest modes.

3. Correlations between thevibrational parameters

We now illustrate that the vibrational properties introduced in thelast section are closely interrelated. First, the atomic contributionsto the BP intensity and to soft modes are correlated, as shown inFig. 2. By and large, atoms that participate more in the low-frequency vibrational modes also contribute more to the BPintensity. It supports the idea that quasi-localized low-frequencyvibrational modes constitute the BP,67 a notion that has led tointensive research on the structural origin of the BP.68,69 Forexample, density or elastic fluctuations,68,70 excess free volume,52

and locally unfavored motifs associated with larger MSD71 havebeen linked to the domains in a quasi-continuum region wherethe transverse phonons produce additional modes.

Second, the low-frequency vibrational modes go hand inhand with the vibration amplitude. Fig. 3 shows the monotonicincrease of the atomic participation fraction in the low-frequency vibrational modes with increasing atomic MSD orwith the flexibility volume. The most flexible atoms can beeasily separated from the rest by their exceptional contributionto the low-frequency vibrational modes, although the separa-tion is not as obvious for atoms with smaller MSD or flexibility.

We next examine a similar correlation in the vibrationspectra. Fig. 4 shows how the partial reduced VDOS changeswith increasing MSD or flexibility. The atoms in the systemwere sorted into 10 groups by the value of ri

2 or vflex,i. IBP is themost pronounced for the top 10% flexible atoms, but down toalmost the Debye level for the most inflexible group of atoms.The inset in Fig. 4a is the global VDOS of the system, showingthe BP frequency (oBP) at a frequency of 5 THz, where the dashline marks the estimated Debye level. oBP shifts toward lowerfrequency as the coarse-grained MSD or vflex increases, asshown in Fig. 4a and b.

Note here that while the most flexible group of atoms, interms of their MSD or flexibility, contributes the most to the BP

intensity, atoms with lower MSD or flexibility do not necessarilycontribute less to the BP. We noticed that Zr atoms have loweraveraged MSD and vflex when compared with Cu atoms, but giverise to higher partial BP intensity. This observation poses achallenge to the idea of an absolute correlation between theatomic MSD and BP intensity. In fact, it has been suggestedthat the BP intensity depends on not only the vibrational MSD,but also the element characteristics (i.e., atomic mass, radiusand bonding features). Recently, Zhang et al. proposed thatthere is a quantitative correlation between the BP intensity andatomic mass weighted MSD for different MGs, written as

IBP ¼ Cm r2� �kBT

� �3=2

, where C is a universal constant, m is

the atomic mass, hr2i is the averaged vibrational MSD, kB is theBoltzmann constant and T is the temperature.72 Besides, theparameter ‘orientational order’ proposed by Yang et al. alsoachieved a decent correlation linking the BP intensity with boththe vibrational MSD and atomic species: the distance from thecenter atom to the farthest vertex of its coordination polyhedron isproportional to the contribution of local structure to the BPenhancement.51 Moreover, Ding et al. noticed that there is astrong correlation between vflex and the vibrational anisotropy,61

the latter originating from the local structural anisotropy. Actually,our results indicate that any two of the vibrational parameters arestrongly correlated, which will be demonstrated in Fig. 2–4.However, since a causal relationship between the parameters isoften lacking, it is natural to raise the question as to whichparameter can be more robustly correlated with the properties.

4. Predicting stress- and thermal-induced relaxation from vibrationalparameters4.1 Local shear transformation propensity

To gauge the degree/extent of shear transformations upondeformation, the non-affine atomic displacement D2

min for eachatom was calculated after athermal quasi-static shear to a

Fig. 2 The participation fraction in the low-frequency vibrational modescorrelates with the contribution to the boson peak intensity (IBP). The plotincludes all atoms in the Cu64Zr36 system.

Fig. 3 All the atoms are sorted and coarse-grained by the values of ri2 and

vflex,i, each bin containing 2% of the atoms. The corresponding averagedparticipation fraction in the low-frequency vibrational modes is calculated,and found to monotonically increase with increasing ri

2 and vflex,i.


Publ

ishe

d on

12

July

202

1. D

ownl

oade

d on

12/

10/2

021

1:44

:40

PM.

View Article Online



global strain of 5%. It is determined by accumulating DD2min

with a step of shear strain Dg = 0.1%. The shear was appliedalong six different loading directions, to average out variationsdue to direction-dependent loading. The colored contour mapsshown in Fig. 5a and b present the spatial distribution of thenormalized atomic displacement D2

min (by dividing it with a

threshold, above which the atom is regarded as having partici-pated obviously in a local shear transformation), each slabhaving a thickness of 3 Å (roughly the average atomic spacing).The sites with large D2

min percolate in space. The white spotssuperimposed onto Fig. 5a correspond to atoms with the largest10% of each vibrational indicator. These white spots overlap

Fig. 4 All the atoms are sorted and coarse-grained by the magnitude of ri2 and vflex,i, each group containing 10% of the atoms. The reduced g(o)/o2 of

each group changes with the magnitude of the (a) MSD and (b) flexibility volume. The inset in (a) is the global reduced VDOS of the Cu64Zr36 MG.

Fig. 5 The contour maps in (a) and (b) show the spatial distribution of normalized D2min (scale in the sidebar, normalized relative to a threshold value) in

the Cu64Zr36 MG with a cooling rate of 109 K s�1. The slab, randomly chosen in the simulation box, has a thickness of 3 Å. The white spots superimposedin (a) mark the atoms with the largest 10% ri

2, vflex,i, Pi and IBP,i; the white spots in (b) mark the atoms with the lowest 10% ri2, vflex,i, Pi and IBP,i. (c) The two

groups of atoms, those of the largest 10% and smallest 10% nonaffine displacement, respectively, are compared by their distribution in terms of ri2, vflex,i,

Pi and IBP,i.


Publ

ishe

d on

12

July

202

1. D

ownl

oade

d on

12/

10/2

021

1:44

:40

PM.

View Article Online



well with the regions most prone to shear transformations. Thepanels in Fig. 5b, on the other hand, display the locations ofatoms with the lowest 10% of each indicator as the white spots.They obviously tend to distribute in the least shear-transformedregions. Taking Fig. 5a and b together, the correlation is very clear,regardless of which of the four vibrational parameters is used.This is not surprising, as we have already demonstrated in thepreceding section that these vibration parameters themselves arestrongly interrelated (for example, it was shown before that theBP49 and soft modes73,74 are associated with local regions whereatoms have high flexibility61). As another way to demonstrate thecontrast, Fig. 5c uses bar charts to plot the population distributionas a function of the magnitude of each vibrational indicator,contrasting the atoms with the highest versus those with thelowest 10% D2

min. These two contrasting groups are obviously wellseparated by any of the four indicators.

4.2 Local shear modulus

We also calculated the shear modulus (G) at ambient tempera-ture by using the fluctuation method. The simulation box wassubdivided into 1000 small cubes, to calculate the local G. For asystem at equilibrium, the relation between the stress fluctua-tions and the local elastic modulus tensor Cijkl is obtained fromthe second derivative of the free energy with respect to strain,Cijkl = CS

ijkl + CBijkl + CK

ijkl, where CSijkl is the stress fluctuation

term, CBijkl is the Born term and CK

ijkl is the kineticcontribution. The subscripts i, j, k, l indicate the Cartesian

components. CSijkl ¼ �

V

kBTPijPkl

� �� Pij

� �Pklh i

� �, CB

ijkl ¼1

VPmo n

@2U

@rmn2� 1

rmn

@U

@rmn

� �rmni rmn

j rmnk rmn

l

rmn2

, and CK

ijkl ¼2NkBT

V

ðdikdjl þ dildjkÞ. m, n are particle indices, V is the volume of

interest, rmn is the separation between two interacting particlesm and n, and P denotes the local stress tensor. The local shearmodulus of a selected region was evaluated as

G ¼ C44 þ C55 þ C66

3. For MG systems, C44 = C55 = C66. The

heterogeneous distribution of local G is shown in Fig. 6. Thewhite spots are atoms with the largest 10% (Fig. 6a), orthe lowest 10% (Fig. 6b), of each vibrational indicator. InFig. 6a, the white spots tend to avoid areas of high G,preferentially residing in regions of lower G. In Fig. 6b, thewhite spots tend to be in the regions of lower G. Actually, thefindings are expected. For example, it is known that the DWFreflects the stiffness: the high-frequency plateau shear mod-ulus can be directly related to the DWF through the Langevinmodel for Brownian motion.75 Also, because vflex directlyscales with G, as discussed earlier, it is not surprising toobserve that vflex fares better than the other indicators in thecorrespondence with the local G. It should be pointed outthat the local shear modulus here was calculated from agroup of atoms (B10 atoms) within a local region (i.e., it is alocally averaged quantity), whereas the vibrational para-meters and local properties were all calculated for each atom.As such, it is reasonable to see a slightly weaker correlationbetween such a shear modulus and the atomic-level vibra-tional parameters, when compared with other correlationsthat are on the atomic scale.

4.3 Activation energy for thermally induced relaxation

A schematic picture of the potential energy landscape (PEL) isoften used to help understand the transition between config-urational states. From the PEL perspective, thermallyactivated b relaxations are identified as hopping betweenneighboring sub-basins confined within a meta-basin, whereas

Fig. 6 The contour maps in (a) and (b) show the spatial distribution of local G (see the sidebar for the scale in GPa) in the Cu64Zr36 MG with a cooling rateof 109 K s�1. The random slab has a thickness of 3 Å. The white spots superimposed in (a) mark the atoms with the largest 10% ri

2, vflex,i, Pi and IBP,i; thewhite spots in (b) mark the atoms with the lowest 10% ri

2, vflex,i, Pi and IBP,i.


Publ

ishe

d on

12

July

202

1. D

ownl

oade

d on

12/

10/2

021

1:44

:40

PM.

View Article Online



the transitions between meta-basins are known as thepercolation of b relaxations, to form the global a relaxation.Here we employed the activation–relaxation technique(ART)76 for searching for the saddle points and relaxationpathways of local regions, each process corresponding toa b event. The initial perturbations in the ART wereintroduced by applying random displacement of a smallgroup of atoms (a specified atom and its nearest neighbors).The magnitude of the displacement was fixed, while thedisplacement direction was randomly chosen. When thecurvature of the PEL surpasses the pre-set threshold, thesystem is pushed towards the saddle point using the Lanczosalgorithm. The saddle point is accepted if the overall force ofthe total system is below 0.01 eV Å�1. The correspondingactivation energy Eact is thus the difference between thesaddle point energy and the initial state energy. For eachatom at the center, we employed 100 ART searches withdifferent random perturbations. The contours in Fig. 7aand b are colored by the value of Eact. The white spots inFig. 7 are defined in the same way as those in Fig. 5 and 6.The unstable (in terms of the tendency for vibrationalexcursion) sites predicted by the white spots in Fig. 7a tendto locate in areas of the lowest Eact, while the stable sites in

Fig. 7b overlap well with the areas of the highest Eact. InFig. 7c, we compare atoms with the highest and the lowest10% Eact: they can be separated by each of the four vibra-tional parameters.

5. Spatial heterogeneities and theircorrelations

Each of the vibrational properties, and the local D2min and Eact

responses, shows spatial heterogeneity. Atoms correspondingto different magnitudes of the indicator tend to aggregate andconnect in 3D space, as demonstrated in the 3D colored mapsin Fig. 8 for the MG simulation box. The higher the magnitudeof the vibrational parameters, the brighter the color in thediagram. For easy comparison, we encircle the most flexibleregion identified by the highest values of ri

2, vflex,i, IBP,i and Pi,using dotted lines. These lines are then superimposed onto the3D box showing D2

min and Eact. We observe that the regionsenclosed by these lines coincide very well with those having thehighest D2

min and Eact.Following the definition of the spatial autocorre-

lation function, the correlation coefficient r of each

Fig. 7 The contour maps in (a) and (b) show the spatial distribution of Eact (see the sidebar for the scale in eV) in the Cu64Zr36 MG with a cooling rate of109 K s�1. The slab has a thickness of 3 Å. The white spots superimposed in (a) mark the atoms with the largest 10% ri

2, vflex,i, Pi and IBP,i; the white spots in(b) mark the atoms with the lowest 10% ri

2, vflex,i, Pi and IBP,i. (c) Atoms of the largest 10% and smallest 10% local activation energy are compared by thedistribution of the corresponding ri

2, vflex,i, Pi and IBP,i.


Publ

ishe

d on

12

July

202

1. D

ownl

oade

d on

12/

10/2

021

1:44

:40

PM.

View Article Online



parameter (or local property) was calculated via rðrÞ ¼Pr0þrr0

xr0 � �x� �

xr0þr � �x� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPr0þrr0

xr0 � �x� �2s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPr0þr

r0

xr0þr � �x� �2s , where %x denotes the averaged

value of each indicator, xr0is the value at the reference position

and xr0+r is the value at the position a distance r away from thereference. r is calculated for all appropriate pairs that aredistance r apart from each other. As shown in Fig. 9, r decaysexponentially with respect to r. An exponential covariancefunction r(r) = exp(�3r/e) is used to describe the decay (solidline), where e is the correlation length defined to be thesmallest distance beyond which r is less than 0.05, as listedin Table 1. It ranges from 6.4 to 7.2 Å, falling within the lengthof the 3rd coordination shell as marked by the gray shade onthe pair distribution function curve. The correlation length forEact is 6.0 Å, around the position of the 2nd valley in the pairdistribution function, while D2

min has the largest correlationlength of 8.5 Å, which goes beyond the 3rd coordination shell.

To quantify the size of the local region participating instructural rearrangement, we roughly estimated the activationvolume Vact involved in thermal activation or shear transforma-tion via Vact = e3. As such, the number of atoms involved in Vact

is around 14 and 38 for the thermal activation (Eact) and sheartransformation events (D2

min), respectively. As reported by Fanet al.,77 thermal activation only involves a small number ofatoms, typically less than 10 atoms, while a single sheartransformation zone (STZ, defined as a localized group ofatoms in the noncrystalline solid that shoulder almost all ofthe shear strains in response to applied stresses)78 usually

contains more atoms, typically around 20 or 25–33 atoms.79,80

It is generally accepted that a single b event that represents

Fig. 8 Spatial distribution of atoms in terms of the magnitude of their vibrational parameters, ri2, vflex,i, Pi and IBP,i, displayed side by side, and also aside of

the spatial distribution of the responses D2min and Eact. See the color sidebar for the relative magnitude of each. The high-magnitude region identified by

each vibrational parameter is encircled using the dotted lines; these regions apparently resemble those of the highest D2min or Eact, correlating with one

another very well in 3D space.

Fig. 9 Upper panel: the spatial autocorrelation coefficient of variousvibrational parameters and local properties as a function of distance, fittedusing an exponential covariance function (solid lines). The correlationlength is defined as the distance at which the value of the correlationcoefficient decreases to 0.05, and compared with (lower panel) the pairdistribution function of the Cu64Zr36 MG.

Table 1 Correlation lengths of the vibrational parameters and localproperties

ri2 vflex Pi IBP D2

min Eact

e (Å) 7.0 7.2 6.9 6.4 8.4 6.0


Publ

ishe

d on

12

July

202

1. D

ownl

oade

d on

12/

10/2

021

1:44

:40

PM.

View Article Online



hopping between sub-basins on the PEL is akin to an STZ indeformation. The correlation length of Eact can be smaller thanthat of a shear transformation (D2

min measures the local irrever-sible shear transformations induced by the triggers), because thelatter inevitably involves the surroundings and possibly initiatesan autocatalytic process. Therefore, it is reasonable to observe alarger correlation length of D2

min than that for Eact.To quantify the accuracy of using vibrational parameters to

predict local atomic rearrangement, we employed Spearman’srank correlation coefficient Cs to specify the monotonic rela-tionship between each two variables.81 A monotonicallyincreasing correlation between two quantities gives Cs = 1and anti-correlation gives Cs = �1; Cs = 0 indicates the absenceof correlation. In this comparison, we also include severalpreviously used structural parameters. For example, we includelocal five-fold symmetry (LFFS), which was used to correlate thestructure with deformation and liquid dynamics.23,82,83 LFFS

can be evaluated as W ¼Pi

n5iPk¼3;4;5;6

nki� Pi

0B@

1CA, where n5

i is the

number of 5-edged polygons in Voronoi polyhedron type i, nki is

the number of k-edged polygons in Voronoi polyhedron type iand Pi is the fraction of polyhedron type i. Another usefulstructural indicator is the icosahedra (ICO) density Cico,28 asfull icosahedra are the dominant motif in Cu64Zr36. Here, the

ICO density is defined as Cico ¼nMC

nM, where nMC is the number

of M-centered icosahedra and nM is the total number of Matoms in the Voronoi cluster of the atom at the center. As such,Cico is a structural parameter covering information aboutmedium-range atomic packing around the atom of interest,including the environment of the coordinated atoms in termsof their first-nearest neighbor coordination (extending up tothe 2nd shell of the center atom) and reflecting how likely theICOs are to connect/interpenetrate. The next structural para-meter is the atomic volume (Voronoi volume), or atomic pack-

ing efficiency Z ¼ Va

Vu, where Va and Vu denote the volume

occupied by the atoms inside a cluster and the total volumeof the cluster,84 respectively.

Fig. 10 is the matrix that displays the Spearman’s correlationbetween any two of the variables. We observe from this matrix thatthe predictive power of structural parameters is generally weakerthan that of vibrational parameters. The correlation betweenstructural and vibrational parameters is weak, stressing againthat structural descriptors based on short-range informationwould not be sufficiently robust to predict the dynamics.

vflex gives the highest Cs for correlating with D2min, as well as

the most negative Cs with Eact (high flexibility corresponding toa low activation barrier), exhibiting the best predictive powerfor the responses of the local structure under applied stress ortemperature. This is not surprising as vflex was designed tocapture both the atomic packing and dynamical information.From its definition, vflex can be taken as a volume-scaledvibrational MSD. The vibrational MSD term is similar to theDWF that evaluates the flexibility of local structures in

supercooled liquids, corresponding to the curvature of thebasin on the PEL. Although there is an overall trend of MGswith larger Oa giving smaller elastic modulus, there is no one-to-one correspondence between the local structure and itselastic modulus. As known from the Lindemann criterion ofmelting, the mobility of atoms not only depends on the traveldistance of the particle, but also the atomic spacing betweenthe rearranging particles. vflex combines these two together, andhas an explicit physical relation with G, which is widely regardedas a key baseline property correlating with many properties forMGs.58,85,86 This lays the foundation for a credible forecast ofproperties using vflex. Another point worth noting is that vflex canbe calculated from the MSD, Debye temperature or shear mod-ulus, which are experimentally measurable.

6. A comment on the connection withthe atomic packing structure

The atomic vibrational behavior is rooted in the glass structure,and the readiness for atomic rearrangement is also influencedby the local environment. It is therefore expected that acommon structural link can be found. To this end, the atomswere divided into 10 groups according to the difference invibrational or mechanical/thermal activated properties.Fig. 11 compares the contents of the 5 major Cu- andZr-centered polyhedra of the grouped atoms. Here we use theVoronoi index hn3, n4, n5, n6i to describe the geometric structurefeature, with n3, n4, n5, and n6 specifying the number oftriangles, rectangles, pentagons, and hexagons, respectively,in the tessellated Voronoi polyhedron. For an MG of thisCu-rich Cu–Zr composition, Cu-centered h0, 0, 12, 0i andZr-centered h0, 0, 12, 4i clusters are energetically preferable.The most stable 10% of atoms contain most of these two

Fig. 10 Matrix of Spearman’s correlation coefficient between any two ofthe variables of the 10 indicators, including structural parameters W, Oa,i,and Cico, vibrational parameters ri

2, vflex,i, Pi, and IBP,i, and local responseproperties D2

min and Eact. See the colored sidebar for the value of thecoefficient. The radius (size) of the circle also shows the absolute value ofthe correlation coefficient.


Publ

ishe

d on

12

July

202

1. D

ownl

oade

d on

12/

10/2

021

1:44

:40

PM.

View Article Online



dominant types of motifs. With increasing flexibility, the con-tent of these stabilizing motifs decreases significantly. Note,however, that the atomic response can vary under differentstimulus conditions (e.g. local stress field, loading direction).For example, Eact is obtained from local excursions in randomdirections, and these events do not necessarily get activatedunder imposed stresses.

We now briefly comment on the structural parameters.Because of the dominance of icosahedra in this MG composi-tion, structural parameters such as the LFFS and ICO densityexhibit a fair predictive capability. According to the matrix inFig. 10, the ICO density shows a better correlation with D2

min

and Eact, compared with the LFFS. The ICO density depends onnot only the degree of LFFS, but also the local aggregation oficosahedral motifs. It characterizes the structural feature of alength scale larger than nearest neighbors, i.e., the medium-range connection of organized SROs. The atomic volume andatomic packing efficiency, in comparison, are less effective inpredicting the propensity for local deformation or relaxation,even though they are often used to reflect the free volumecontent.87,88 Specifically, a denser atomic packing efficiencywas considered to cause the dynamic slowdown during theglass transition,84,89 and the dense structure tends to be morestable in glassy states. However, it has long been argued thatthe atomic packing density is not ideal for determining thepropensity for atomic rearrangement, one reason being thatboth excess free volume and anti-free volume can be involved indeformation and relaxation.90,91 Also, for complex MGs, thecritical structural motif is influenced by chemical SROs, whichcan be diverse.92,93 Besides, evidence is mounting that short-range geometrical configurations may not be sufficient todecisively govern the response, as discussed in the Introductionsection.

7. Concluding remarks and pendingissues

In this work, we have used a model MG alloy to establish directinterrelations among the various vibrational properties.Furthermore, we have shown that, by and large, the vibrationalparameters have the ability to predict the local structuralrearrangements induced by applied stress or temperature, ata level obviously more reliable than static structural para-meters. In general, it is clear that atoms with large flexibilityamplitude and significant participation in low-frequency vibra-tions are prone to local relaxation, softening and non-affinedisplacement. Of the several vibrational parameters, vflex is theoptimal, in terms of predictive power and accessibility.

Finally, a few remaining issues come to mind, as an outlook.First, from what has been summarized above, correlating thestructure/dynamics in MGs with the properties has so far reliedheavily on atomistic simulations. There is a pressing need tobridge the gap between simulations and experiments. Second,while Fan et al. have observed similar correspondence betweenthe flexibility volume and properties in covalent networkglasses,65 it remains to be seen whether vflex maintains itspredictive power in all kinds of glasses. Third, the dynamicsswitch-over underpinning the glass transition is one of themost important issues in the glass field, which needs to beunderstood. Fourth, the vibrational dynamics as a function oftemperature, and the correlation between vibrational para-meters and the glass forming ability, are also worth exploring.Fifth, it is a bit surprising that short-time dynamical behavior,which only samples the short-time vibrational dynamics nearthe bottom of the sub-basin in the PEL, seems to reflect in someway the energy barrier for cross-basin inelastic relaxation vialocal atomic rearrangements. While this is a convenient scaling

Fig. 11 The atoms at the center of different types of (a) Cu-centered and (b) Zr-centered coordination polyhedra contribute differently to (from left toright) the flexibility volume, BP intensity, local nonaffine displacement and thermal activation barrier. In each panel, each bar contains 10% of all the Cu (orZr) atoms, sorted by (from left to right, as indicated by the red arrow) increasing magnitude of the indicator in question. The fractions of the 5 mostpopulous Cu- or Zr-centered polyhedra are represented by the size of the corresponding bar segment.


Publ

ishe

d on

12

July

202

1. D

ownl

oade

d on

12/

10/2

021

1:44

:40

PM.

View Article Online



as we showed in this article, why it is so remains to be betterunderstood. Sixth, with regard to the structural underpinningof the vibrational and relaxation properties, although the para-meters evaluated in this study are for individual atoms, theirmagnitude is actually controlled by the local environmentsurrounding the center atom. One may set the goal to basethe predictions on the overall static structure. In this regard, sofar the structural input is from the nearest neighbors in theshort-to-medium range (up to a couple of nanometers).94 Itappears necessary to include information from length scales atand beyond medium range, as well as subtle features not easilycaptured by simple structural indicators. Recently, machinelearning methods have been emerging as a promising tool inconquering this problem: by summing up the intersticefeatures from SROs to MROs,95,96 or by utilizing local radialdensity functions to wrap longer-range structural informationinto the picture.97 But it still remains difficult to translate theinsight gained into a single and user-friendly parameter/indi-cator. It is also a grand challenge to understand how/why suchdata-driven models were able to achieve those predictive per-formances, when it comes to using machine learning to predictfertile sites from the (relative) positions of the atoms insideamorphous matter. Interestingly, progress is being made veryrecently to derive physical insight into the structural featuresgoverning the local responses, building upon the impressivepredictive power generated via deep learning.98

Abbreviations and symbols

ART Activation–relaxation techniqueBP Boson peakDWF Debye–Waller factorGUM Geometrically unfavored motifICO IcosahedronLFFS Local five-fold symmetryMG Metallic glassMRO Medium-range orderMSD Mean-square displacementPEL Potential energy landscapeSRO Short-range orderSTZ Shear transformation zoneVACF Velocity auto-correlation functionVDOS Vibrational density of statesg(o) Vibrational density of statesG Shear modulusIBP Boson peak intensityPi Participation fraction of the ith atom in the soft

modesr2 Vibrational mean-square displacementvflex Flexibility volumeyD Debye temperatureo FrequencyOa Atomic volumeD2

min Non-affine atomic displacementEact Activation energy

Vact Activation volumer Correlation coefficientr Atomic distancee Correlation lengthCs Spearman’s rank correlation coefficientW Local five-fold symmetry indexCico Icosahedra densityZ Atomic packing efficiency

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This research was supported by the National Natural ScienceFoundation of China (No. 52071024, 11790293, 51871016, and51671021), the Funds for Creative Research Groups of China(51921001), Project of International Cooperation andExchanges NSFC (51961160729), 111 Project (B07003), and theFundamental Research Fund for the Central Universities ofChina. H. Y. F. acknowledges the financial support from theChina Scholarship Council (No. 201806460074) for her study atthe Johns Hopkins University. E. M. is grateful to XJTU forsupporting his research at CAID.

References

1 M. Telford, Mater. Today, 2004, 7, 36–43.2 H. S. Chen and C. E. Miller, Rev. Sci. Instrum., 1970, 41,

1237–1238.3 A. L. Greer, Nature, 1993, 366, 303–304.4 A. Inoue, Acta Mater., 2000, 48, 279–306.5 A. L. Greer and G. Editors, MRS Bull., 2007, 32, 611–619.6 G. Kumar, T. Ohkubo, T. Mukai and K. Hono, Scr. Mater.,

2007, 57, 173–176.7 L. S. Luo, B. B. Wang, F. Y. Dong, Y. Q. Su, E. Y. Guo, Y. J. Xu,

M. Y. Wang, L. Wang, J. X. Yu, R. O. Ritchie, J. J. Guo andH. Z. Fu, Acta Mater., 2019, 171, 216–230.

8 W. Li, H. Bei, Y. Tong, W. Dmowski and Y. F. Gao, Appl.Phys. Lett., 2013, 103, 171910.

9 H. B. Yu, M. Tylinski, A. Guiseppi-Elie, M. D. Ediger andR. Richert, Phys. Rev. Lett., 2015, 115, 1–5.

10 F. Zhu, S. Song, K. M. Reddy, A. Hirata and M. Chen, Nat.Commun., 2018, 9, 3965.

11 F. Spaepen, Acta Metall., 1977, 25, 407–415.12 M. H. Cohen and G. S. Grest, Phys. Rev. B: Condens. Matter

Mater. Phys., 1979, 20, 1077–1098.13 Y. Shi and M. L. Falk, Phys. Rev. Lett., 2005, 95, 1–4.14 Y. Shi and M. L. Falk, Phys. Rev. B: Condens. Matter, Mater.

Phys., 2006, 73, 1–10.15 T. Egami, K. Maed, D. Srolovitz and V. Vitek, Le J. Phys.,

Colloq., 1980, 41, C8-272–C8-275.16 J. Ding, Y. Q. Cheng and E. Ma, Acta Mater., 2014, 69,

343–354.


Publ

ishe

d on

12

July

202

1. D

ownl

oade

d on

12/

10/2

021

1:44

:40

PM.

View Article Online



17 R. Soklaski, Z. Nussinov, Z. Markow, K. F. Kelton andL. Yang, Phys. Rev. B: Condens. Matter, Mater. Phys., 2013,87, 1–8.

18 J. Ding, S. Patinet, M. L. Falk, Y. Cheng and E. Ma,Proc. Natl. Acad. Sci. U. S. A., 2014, 111, 14052–14056.

19 P. J. Steinhardt, D. R. Nelson and M. Ronchetti, Phys. Rev. B:Condens. Matter Mater. Phys., 1983, 28, 784–805.

20 H. L. Peng, M. Z. Li and W. H. Wang, Phys. Rev. Lett., 2011,106, 1–4.

21 X. J. Liu, Y. Xu, X. Hui, Z. P. Lu, F. Li, G. L. Chen, J. Lu andC. T. Liu, Phys. Rev. Lett., 2010, 105, 1–4.

22 X. J. Liu, S. D. Wang, H. Wang, Y. Wu, C. T. Liu, M. Li andZ. P. Lu, Phys. Rev. B, 2018, 97, 134107.

23 M. Z. Li, J. Mater. Sci. Technol., 2014, 30, 551–559.24 W. Mickel, S. C. Kapfer, G. E. Schroder-Turk and K. Mecke,

J. Chem. Phys., 2013, 138, 1–8.25 D. B. Miracle, Nat. Mater., 2004, 3, 697–702.26 Z. W. Wu, M. Z. Li, W. H. Wang and K. X. Liu, Phys. Rev. B:

Condens. Matter, Mater. Phys, 2013, 88, 1–5.27 H. Tanaka, T. Kawasaki, H. Shintani and K. Watanabe, Nat.

Mater., 2010, 9, 324–331.28 M. Lee, C. M. Lee, K. R. Lee, E. Ma and J. C. Lee, Acta Mater.,

2011, 59, 159–170.29 Y. Q. Cheng and E. Ma, Prog. Mater. Sci., 2011, 56,

379–473.30 J. Ding and E. Ma, npj Comput. Mater., 2017, 3, 1–12.31 L. Li, E. R. Homer and C. A. Schuh, Acta Mater., 2013, 61,

3347–3359.32 D. Srolovitz, K. Maeda, V. Vitek and T. Egami, Philos. Mag. A

Phys. Condens. Matter, Struct. Defects Mech. Prop., 1981, 44,847–866.

33 T. Egami, Prog. Mater. Sci., 2011, 56, 637–653.34 W. L. Johnson and K. Samwer, Phys. Rev. Lett., 2005, 95, 2–5.35 M. D. Demetriou, J. S. Harmon, M. Tao, G. Duan, K. Samwer

and W. L. Johnson, Phys. Rev. Lett., 2006, 97, 41–44.36 W. L. Johnson, M. D. Demetriou, J. S. Harmon, M. L. Lind

and K. Samwer, MRS Bull., 2007, 32, 644–650.37 M. Tsamados, A. Tanguy, C. Goldenberg and J. L. Barrat,

Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2009, 80,1–17.

38 S. Patinet, D. Vandembroucq and M. L. Falk, Phys. Rev. Lett.,2016, 117, 1–5.

39 A. Barbot, M. Lerbinger, A. Hernandez-Garcia, R. Garcıa-Garcıa, M. L. Falk, D. Vandembroucq and S. Patinet, Phys.Rev. E, 2018, 97, 1–13.

40 J. Zylberg, E. Lerner, Y. Bar-Sinai, E. Bouchbinder andJ. S. Langer, Proc. Natl. Acad. Sci. U. S. A., 2017, 114,7289–7294.

41 B. Xu, M. L. Falk, J. F. Li and L. T. Kong, Phys. Rev. Lett.,2018, 120, 125503.

42 C. Liu, P. Guan and Y. Fan, Acta Mater., 2018, 161, 295–301.43 A. Widmer-Cooper, P. Harrowell, P. H. Asaph Widmer-

Cooper, A. Widmer-Cooper and P. Harrowell, J. Non-Cryst.Solids, 2006, 1–18.

44 A. Widmer-Cooper and P. Harrowell, Phys. Rev. Lett., 2006,96, 2–5.

45 S. T. Liu, F. X. Li, M. Z. Li and W. H. Wang, Sci. Rep., 2017,7, 11558.

46 A. Ottochian and D. Leporini, J. Non-Cryst. Solids, 2011, 357,298–301.

47 M. L. Manning and A. J. Liu, Phys. Rev. Lett., 2011, 107, 2–5.48 H. Shintani and H. Tanaka, Nat. Mater., 2008, 7, 870–877.49 N. Jakse, A. Nassour and A. Pasturel, Phys. Rev. B: Condens.

Matter, Mater. Phys., 2012, 85, 1–6.50 P. M. Derlet, R. Maaß and J. F. Loffler, Eur. Phys. J. B, 2012,

85, 1–20.51 J. Yang, Y. J. Wang, E. Ma, A. Zaccone, L. H. Dai and

M. Q. Jiang, Phys. Rev. Lett., 2019, 122, 1–6.52 J. Bunz, T. Brink, K. Tsuchiya, F. Meng, G. Wilde and

K. Albe, Phys. Rev. Lett., 2014, 112, 1–5.53 Y. P. Mitrofanov, M. Peterlechner, S. V. Divinski and

G. Wilde, Phys. Rev. Lett., 2014, 112, 1–5.54 P. Luo, Y. Z. Li, H. Y. Bai, P. Wen and W. H. Wang, Phys. Rev.

Lett., 2016, 116, 1–5.55 D. Xu, B. Lohwongwatana, G. Duan, W. L. Johnson and

C. Garland, Acta Mater., 2004, 52, 2621–2624.56 D. Wang, Y. Li, B. B. Sun, M. L. Sui, K. Lu and E. Ma, Appl.

Phys. Lett., 2004, 84, 4029–4031.57 Y. Q. Cheng, H. W. Sheng and E. Ma, Phys. Rev. B: Condens.

Matter, Mater. Phys., 2008, 78, 1–7.58 Y. Q. Cheng, A. J. Cao and E. Ma, Acta Mater., 2009, 57,

3253–3267.59 S. Plimpton, J. Comput. Phys., 1995, 117, 1–19.60 V. A. Borodin, Philos. Mag. A Phys. Condens. Matter, Struct.

Defects Mech. Prop., 1999, 79, 1887–1907.61 J. Ding, Y.-Q. Cheng, H. Sheng, M. Asta, R. O. Ritchie and

E. Ma, Nat. Commun., 2016, 7, 13733.62 B. Schonfeld, J. Zemp and U. Stuhr, J. Phys.: Condens. Matter,

2016, 29, 015401.63 C. Kittel, P. McEuen and P. McEuen, Introduction to solid

state physics, Wiley, New York, 1996, vol. 8.64 J. A. Reissland, The physics of phonons, Wiley-Interscience,

1973.65 Z. Fan, J. Ding and E. Ma, Mater. Res. Lett., 2018, 6, 570–583.66 Z. Fan, J. Ding, Q. J. Li and E. Ma, Phys. Rev. B, 2017, 95,

1–10.67 H. Tanaka, J. Phys. Soc. Jpn., 2001, 70, 1178–1181.68 T. Brink, L. Koch and K. Albe, Phys. Rev. B, 2016, 94, 1–9.69 R. Milkus and A. Zaccone, Phys. Rev. B, 2016, 93, 1–10.70 H. Mizuno, S. Mossa and J. L. Barrat, Proc. Natl. Acad. Sci.

U. S. A., 2014, 111, 11949–11954.71 T. S. Grigera, V. Martın-Mayor, G. Parisi and P. Verrocchio,

Nature, 2003, 422, 289–292.72 H. P. Zhang, B. B. Fan, J. Q. Wu, W. H. Wang and M. Z. Li,

Phys. Rev. Mater., 2020, 4, 1–6.73 A. Widmer-Cooper, H. Perry, P. Harrowell and

D. R. Reichman, Nat. Phys., 2008, 4, 711–715.74 M. Mosayebi, P. Ilg, A. Widmer-Cooper and E. Del, Gado,

Phys. Rev. Lett., 2014, 112, 1–5.75 B. A. Pazmino Betancourt, P. Z. Hanakata, F. W. Starr and

J. F. Douglas, Proc. Natl. Acad. Sci. U. S. A., 2015, 112,2966–2971.


Publ

ishe

d on

12

July

202

1. D

ownl

oade

d on

12/

10/2

021

1:44

:40

PM.

View Article Online



76 G. T. Barkema and N. Mousseau, Comput. Mater. Sci., 2001,20, 285–292.

77 Y. Fan, T. Iwashita and T. Egami, Nat. Commun., 2014,5, 5083.

78 J. S. Langer, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys.,2008, 77, 1–14.

79 I. C. Choi, Y. Zhao, B. G. Yoo, Y. J. Kim, J. Y. Suh,U. Ramamurty and J. Il Jang, Scr. Mater., 2012, 66, 923–926.

80 J. D. Ju and M. Atzmon, Acta Mater., 2014, 74, 183–188.81 W. H. Press, S. A. Teukolsky, W. T. Vetterling and

B. P. Flannery, Numerical Recipes 3rd Edition: The Art ofScientific Computing, Cambridge University Press, USA,3rd edn, 2007.

82 Y. C. Hu, F. X. Li, M. Z. Li, H. Y. Bai and W. H. Wang, Nat.Commun., 2015, 6, 8310.

83 Y. C. Hu, F. X. Li, M. Z. Li, H. Y. Bai and W. H. Wang, J. Appl.Phys., 2016, 119, 205108.

84 L. Yang, G. Q. Guo, L. Y. Chen, C. L. Huang, T. Ge, D. Chen,P. K. Liaw, K. Saksl, Y. Ren, Q. S. Zeng, B. Laqua, F. G. Chenand J. Z. Jiang, Phys. Rev. Lett., 2012, 109, 1–5.

85 L. S. Huo, J. F. Zeng, W. H. Wang, C. T. Liu and Y. Yang,Acta Mater., 2013, 61, 4329–4338.

86 W. H. Wang, Prog. Mater. Sci., 2012, 57, 487–656.87 Q.-K. Li and M. Li, Mater. Trans., 2007, 48, 1816–1821.88 H. W. Sheng, E. Ma and M. J. Kramer, JOM, 2012, 64,

856–881.89 L. Ward, D. Miracle, W. Windl, O. N. Senkov and K. Flores,

Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 88, 1–10.90 V. A. Khonik and N. P. Kobelev, J. Appl. Phys., 2014, 115,

2014–2017.91 A. Foroughi, H. Ashuri, R. Tavakoli, M. Stoica, D. Azopu and

J. Eckert, J. Appl. Phys., 2017, 122, 215106.92 P. F. Guan, T. Fujita, A. Hirata, Y. H. Liu and M. W. Chen,

Phys. Rev. Lett., 2012, 108, 1–5.93 Q. Yu, X. D. Wang, H. B. Lou, Q. P. Cao and J. Z. Jiang,

Acta Mater., 2016, 102, 116–124.94 B. Wang, L. Luo, E. Guo, Y. Su, M. Wang, R. O. Ritchie,

F. Dong, L. Wang, J. Guo and H. Fu, npj Comput. Mater.,2018, 4, 1–11.

95 Q. Wang and A. Jain, Nat. Commun., 2019, 10, 1–11.96 Q. Wang, J. Ding and E. Ma, npj Comput. Mater., 2020, 6,

1–12.97 Z. Fan, J. Ding and E. Ma, Mater. Today, 2021, 40, 48–62.98 Z. Fan and E. Ma, Nat. Commun., 2021, 12, 1506.


Publ

ishe

d on

12

July

202

1. D

ownl

oade

d on

12/

10/2

021

1:44

:40

PM.

View Article Online


Documents

Atomic vibration as an indicator of the propensity for