Evidence for a simple monatomic ideal glass former: The thermodynamicglass transition from a stable liquid phaseMåns Elenius, Tomas Oppelstrup, and Mikhail Dzugutov Citation: J. Chem. Phys. 133, 174502 (2010); doi: 10.1063/1.3493456 View online: http://dx.doi.org/10.1063/1.3493456 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v133/i17 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Evidence for a simple monatomic ideal glass former: The thermodynamicglass transition from a stable liquid phase

Måns Elenius,1 Tomas Oppelstrup,1,2,a� and Mikhail Dzugutov3

1Department of Numerical Analysis, Royal Institute of Technology, 100 44 Stockholm, Sweden2Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, California 94551, USA3Department of Materials Science and Engineering, Royal Institute of Technology, 100 44 Stockholm, Sweden

�Received 20 May 2010; accepted 5 September 2010; published online 1 November 2010�

Under cooling, a liquid can undergo a transition to the glassy state either as a result of a continuousslowing down or by a first-order polyamorphous phase transition. The second scenario has so faralways been observed in a metastable liquid domain below the melting point where crystallinenucleation interfered with the glass formation. We report the first observation of the liquid-glasstransition by a first-order polyamorphous phase transition from the equilibrium stable liquid phase.The observation was made in a molecular dynamics simulation of a one-component system with amodel metallic pair potential. In this way, the model, demonstrating the thermodynamic glasstransition from a stable liquid phase, may be regarded as a candidate for a simple monatomic idealglass former. This observation is of conceptual importance in the context of continuing attempts toresolve the long-standing Kauzmann paradox. The possibility of a thermodynamic glass transitionfrom an equilibrium melt in a metallic system also indicates a new strategy for the development ofbulk metallic glass-forming alloys. © 2010 American Institute of Physics. �doi:10.1063/1.3493456�

I. INTRODUCTION

Avoiding crystallization when cooling a liquid towardthe glass transition remains a central problem of glass sci-ence. Periodic phases are commonly thought to be thermo-dynamically favored at sufficiently low temperatures by allsubstances. The relaxation time of a liquid within its domainof thermodynamic stability above the melting temperatureTm is usually much smaller than the value of 102 s that de-fines the glass transition temperature Tg.1 The glass-formingability of a liquid is measured in terms of the minimumcooling rate that avoids the interfering crystallization. Be-sides the thermodynamic factors, viscosity is the key param-eter that determines the rate of crystal nucleation and growthwithin the supercooled liquid domain Tg�T�Tm. The glass-forming ability of a liquid is therefore determined by itsviscosity at Tm.2 According to a popular empirical rule,3

Tg /Tm�2 /3. This implies that the highest viscosity at Tm isexpected in liquids with Arrhenius temperature dependenceof viscosity, labeled strong in the strong-fragileclassification.4 Indeed, good glass formers are commonlyfound to be strong liquids with high viscosity at Tm, and thecurrent strategy in the design of bulk metallic glass formers5

is focused on the alloys closely approximating the Arrheniusbehavior.

Another possible approach to the problem of avoidingcrystallization in the glass transition is to defy the “2/3 rule”and attempt to reduce Tm relative to Tg. For that purpose, thefree energy of the equilibrium liquid needs to be reducedrelative to that of the respective crystal. In nonpolymericsystems, this can be achieved by tuning anisotropic

interaction,6 by a judiciously designed pair potential,7 or bycomposing multicomponent mixtures with strong chemicalorder and deep eutectic minima.9,10 The ideal glass former11

is defined as a liquid that remains in a stable thermodynamicequilibrium when attaining Tg upon cooling, which entirelyexcludes the possibility of its crystallization regardless of therate of cooling. This class of glass formers has so far beenfound to include atactic polymers12 and some aqueous solu-tions of electrolytes.8 A single-component nonpolymeric liq-uid that does not crystallize upon cooling has so far neverbeen found, and the question of its existence is both intellec-tually challenging and of a significant technological interest,particularly for the area of bulk metallic glasses.

Crystallization of a liquid upon cooling can be precludedby a direct first-order polyamorphous phase transition to aglassy state. A system that performs such a transition fromthe stable equilibrium liquid state would be an ideal glassformer. A possible thermodynamic singularity, transformingthe supercooled liquid into a distinct low-entropy glassyphase, was discussed by Kauzmann13 as a feasible resolutionof his paradox. The Gaussian model14 conjectures that aliquid-liquid first-order phase transition, accompanied by acrossover from fragile to strong behavior, is a generic featureof fragile liquids. This transition is expected to occur bothbelow and above Tg.15 Formation of a glassy phase as aresult of a thermodynamic transition has so far always beenobserved below Tm,16,17 where the presence of critical fluc-tuations due to the liquid-liquid instability significantly en-hances the rate of crystalline nucleation.18,19 Moreover, afirst-order polyamorphous transition from the equilibriumliquid state has never been observed in a monatomic systemupon cooling.20 The Jagla soft-core pair potential21 producestwo stable liquid phases. However, because of a significant

a�Author to whom correspondence should be addressed. Electronic mail:tomaso@nada.kth.se.

THE JOURNAL OF CHEMICAL PHYSICS 133, 174502 �2010�

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difference in densities, the transition between these phaseshas only been found under compression and not uponcooling.22,23

Here, we report a molecular dynamics simulation thatprovides evidence for the existence of a simple monatomicideal glass former. The simulation explores a one-componentmodel liquid with a prototype metallic pair potential. It isfound that when cooled, the liquid performs a first-orderpolyamorphous phase transition to a low-entropy state, whileremaining stable with respect to crystallization. The low-entropy state has a mesoscopic-range order and the rate ofstructural relaxation characteristic of the glassy state. Thephase transition is found to be accompanied by the fragile tostrong crossover. We discuss some conceptual aspects of thisfinding and its possible implications for the technology ofbulk metallic glasses.

II. COMPUTER MODEL

The results we report here were produced in a moleculardynamics simulation of a simple one-component system. Thesimulation utilized an earlier reported pair potential24 �namedZ2 in that reference�; it is shown in Fig. 1. The functionalform of the potential energy for two atoms separated by thedistance r is

V�r� = ae�r

r3 cos�2kfr� + b��

r�n

+ V0

for 0�r�rc and 0 otherwise. The parameters are as follows:

a = 1.04, � = 0.348,

� = 0.33, n = 14.5,

kf = 4.139, V0 = 0.133 915 . . . ,

b = 4.2 · 106 rc = 2.644 877 . . . .

The values of V0 and rc are chosen so that U�rc�=U��rc�=0. In Ref. 24, b is reported as 4.2·107. This is a typo. Thecorrect value is b=4.2·106, as displayed in the table above.

The potential was designed to imitate effective interionicinteraction potentials in liquid metals with characteristicFriedel oscillations.25,26 Earlier investigations of the energyminima configurations of this model24 revealed its strongpredisposition for tetrahedral ordering. When simulated inthe liquid state under cooling, the model demonstrated a pro-nounced super-Arrhenius behavior of the diffusivity24 as wellas other dynamical anomalies characteristic of fragile glassformers.27,28

The main part of the simulation was performed using asystem of 128 000 particles. In some simulation runs, wehave also used a smaller system of 3456 particles, in order totest possible size-dependence of the observed phase behav-ior. All the quantities we report here are expressed in termsof reduced simulation units of length and time �energy�which are defined by the pair potential, with the particlemass assumed to be unity. We note that the main repulsivepart of the present pair potential closely approximates that ofthe Lennard-Jones potential. Considering the latter as amodel potential of argon,29 and the mass of the argon atom asa unit of mass, our reduced units can be interpreted in termsof standard macroscopic units. In this interpretation, the re-duced units of length and time can be estimated as 0.34 nmand 2.16 ps, respectively.

III. RESULTS

To investigate the glass-forming behavior of this model,we cooled it, using the large system of 128 000 particles,from the high-temperature thermodynamically stable liquidstate. The cooling was performed isochorically at the numberdensity �=0.85. The liquid’s temperature T was reduced in aconsecutive stepwise manner, with a comprehensive equili-bration at each temperature step. The equilibration run-timesincreased with cooling ranging from 103 to 104, in reducedunits. The variation of the system’s enthalpy upon cooling,presented in Fig. 2�a�, exhibits a discontinuous drop at T=0.72. The resulting thermodynamic state was reheated inthe same consecutive stepwise as cooling with a comprehen-sive equilibration at each temperature point. This was foundto produce a pronounced hysteresis in the enthalpy variationconfined between T=0.72 and T=0.78. In both points of thethermodynamic discontinuity, the system was fully equili-brated within the run-time of about 106, in reduced units.That equilibration time-scale was used in cooling the systembelow T=0.72 to T=0.68.

Figure 2�b� compares the structures of the two phasesseparated by the observed phase transition. These are pre-sented in terms of the respective structure factors S�Q�= ��Q���−Q�, where ��Q� is the Q-component of theFourier-transform of the system’s number density. The appar-ent close proximity of the two structures indicates that theobserved first-order transition is a polyamorphous transitionconnecting two liquid phases with a large degree of similar-ity in the local order. These phases will be referred to as thehigh temperature liquid �HTL� and the low temperature liq-uid �LTL�. We also note that the pronounced split in secondpeak of S�Q� featured by both liquid phases indicates that thelocal order is predominantly tetrahedral.30

0 0.5 1 1.5 2 2.5−1

−0.5

0

0.5

1

1.5

Distance

Energy

Z2L−J

FIG. 1. Pair potential used in the present simulation �Z2� �Ref. 24�, com-pared with the Lennard-Jones �LJ� potential.

174502-2 Elenius, Oppelstrup, and Dzugutov J. Chem. Phys. 133, 174502 �2010�

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We now explore the transition domain in the P−� phasediagram at constant T. Figure 2�c� shows the isotherms pro-duced by heating, compression, and/or expansion of theoriginal thermodynamic states produced by the phase transi-tion from the HTL phase as a result of its isochoric cooling at�=0.85. All the data shown in the plots represent compre-hensively equilibrated states. At each point, the system wasobserved in equilibrium within run-times ranging from 105 to106, in reduced units, which significantly exceeded the sys-tem’s relaxation time as assessed from the decay of thedensity-density correlations. That the system attained equi-librium is also confirmed by the reversibility of the isother-mal compression-expansion that can be observed in Fig.2�c�. The isotherms exhibit density regions of infinite com-pressibility, indicating that there exists a domain of spinodalinstability interposed between the two liquid phases of dis-tinctly different densities. We also tested the size-dependenceof the observed phase behavior by exploring the small sys-tem of 3456 particles along the T=0.75 isotherm. No differ-ence in the general pattern of phase behavior was observed.At the same time, the isotherm for the smaller-size systemexpectedly demonstrates upon compression a strong metasta-bility.

Infinite compressibility in the spinodal domain manifestsin diverging long-wavelength limit of S�Q�, a measure of thedensity fluctuations on the system-size scale. In a macro-scopic system, S�0� is related to the isothermal compressibil-ity �T by the compressibility equation: S�0�=�kBT�T.29 Fig-

ure 2�d� shows the density-dependence of the small-Qbehavior of S�Q� along the T=0.78 isotherm. A trend fordivergence of the small-Q limit of S�Q� is clearly visible atthe densities where the spinodal instability was detectedfrom the respective isotherm in Fig. 2�c�. Note that there isno divergence in the small-Q limit of S�Q� for the LTL inFig. 2�b�, indicating that at T=0.68, �=0.85 it is below thespinodal domain.

As we mentioned above, the presence of a liquid-liquidspinodal in the supercooled metastable liquid domain greatlyenhances the crystalline nucleation18,19 by reducing the re-spective free-energy barrier. Therefore, our observation thatthe two liquid phases coexist in equilibrium may be regardedas an indication that, within the liquid-liquid coexistence do-main, the system remains thermodynamically stable with re-spect to a possible crystallization.

Figure 3 presents a real-space picture of the spinodaldecomposition of a system of 128 000 particles along the T=0.78 isotherm. The plots depict cross-sections of thecoarse-grained spatial distribution of energy at different den-sities. At �=0.84 representing the HTL phase, the energy isdistributed uniformly. At �=0.87, precipitation of the LTLphase appears as distinct large-scale low-energy domains.Upon further isothermal compression, the LTL domains growand eventually percolate as the system leaves the spinodaldomain. This is indicated by the reduction of the isothermalcompressibility that can be detected from the increase in the

0.7 0.72 0.74 0.76 0.78 0.8

5.6

5.8

6

6.2

6.4

6.6

6.8

7

7.2

T

H

HeatingCooling

0 5 10 15 200

1

2

3

4

5

6

7

8

Q

S(Q)

0.8 0.82 0.84 0.86 0.88 0.93.8

4

4.2

4.4

4.6

4.8

5

ρ

P

T=0.78

T=0.75

T=0.72

0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Q

S(Q)

0.8450.860.870.880.89d

ba

c

FIG. 2. Liquid-liquid phase transition. The data represent a system of N=128 000 particles, unless indicated otherwise. �a� Isochoric temperature variation ofthe enthalpy H at density �=0.85. Circles, cooling; squares, heating. �b� The structure factors S�Q� of the HTL at T=0.78 �dashed line� and LTL at T=0.68 �solid line�, both at the density �=0.85. �c� Isotherms crossing the region of the liquid-liquid transition. Open squares: HTL. Filled circles represent the�=0.85 isochore shown in �a�. Open circles and crosses: LTL points obtained by isothermal compression and expansion, respectively. Small dots: LTL statessimulated by the system of N=3456 particles. The lines are included as a guide to the eye. �d� Density variation of the low-Q behavior of the structure factoralong the T=0.78 isotherm for the indicated densities.

174502-3 A simple monatomic ideal glass former J. Chem. Phys. 133, 174502 �2010�

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slope of the isotherms shown in Fig. 2�c�. Respectively, theHTL domains shrink and become disconnected. The de-scribed transformation of the pattern of phase distributioncan arguably account for the development of a low-Q pre-peak arising in the respective S�Q� in Fig. 2�d�. This obser-vation is consistent with the conjecture31 that a low tempera-ture phase produced by a polyamorphous transition undercooling must be intrinsically heterogeneous.

In order to get further insight into the nature and themicroscopic mechanism of the observed phase transition, weperformed a detailed analysis of the underlying structuraltransformation. To remove thermally induced structural fluc-tuations, the investigated liquid configurations were sub-jected to the steepest descent energy minimization. Becauseof the generally tetrahedral structure of both phases, statisti-cal geometry of the local order has been analyzed in terms oftetrahedral and icosahedral configurations of the nearestneighbors. These configurations were identified using an ear-lier developed algorithm.7 All the particles comprising thefirst peak of the radial distribution function g�r� were as-sumed to be the nearest neighbors. Because of the narrowand sharp shape of the peak, this definition of neighborsinvolved no significant ambiguity with respect to the cutoffdistance.

The structure analysis revealed a remarkable peculiarityof this phase transformation. The total number of icosahedrain the LTL phase was found to be by about 30% smaller thanin the HTL phase, whereas the statistics of tetrahedral orderremained generally unchanged. The transformation of the lo-cal order resulting in this counterintuitive effect can be un-derstood in detail by inspecting a characteristic LTL clustershown in Fig. 4�a�. The cluster was identified within a lowtemperature domain observed in Fig. 3 that we concluded tobe a LTL precipitation. It can be described as a fivefold tet-rahelical configuration composed of axially stacked pentago-nal bipyramides. However, the bipyramides’ pentagonal sym-metry appears to be broken, as shown in Fig. 4�b�. Thisresults in a linear strain of nontetrahedral defects that in-volves two adjacent helical lines of atoms, marked as blueand green in Fig. 4�a�. Such a defect reduces the energy ofthe cluster by removing the geometric frustration inherent tothe fivefold packing of tetrahedra. Moreover, we found that

these linear defects facilitate aggregation of the tetrahelicalclusters into an extended tetrahedral network. Figure 4�c�shows two clusters which share a line of helically stackedfourfold defects as depicted in Fig. 4�d�. In this way, theobserved linearly shaped tetrahedral aggregations, apparentlyincompatible with periodic order, can form under cooling anextended stable long-lived network, which is expected to re-sult in a dramatic increase in the viscosity. The apparentinability of these aggregations to fill the space uniformlyrenders the LTL phase intrinsically heterogeneous.

We next investigate the evolution of the system’s dy-namics upon isochoric cooling at �=0.85. Figure 5 shows anArrhenius plot of diffusivity within the entire range of tem-peratures explored in this simulation. Within the HTL do-main, the temperature variation of system’s diffusion is of aperfectly Arrhenius kind above the crossover temperatureTA=1. Below it, the diffusion clearly demonstrates a fragilebehavior, which can be described by the Vogel–Tamman–Fulcher equation. The temperature variation of the LTL dif-fusivity below the spinodal transition domain appears to beperfectly exponential. This enables us to conclude that theobserved polyamorphous phase transition upon cooling is ac-companied by a fragile to strong crossover similar to thatobserved in other glass formers.16,32 We note that the diffu-sivity does not significantly change across the phase transi-tion, which is in contrast to the polyamorphous glass transi-tion in supercooled silicon16 where it was found to drop bymore than two orders of magnitude. This difference in be-havior between the two systems can presumably be attrib-uted to the structural heterogeneity of the LTL phase that wasobserved in Fig. 3, with the faster diffusing part of the sys-tem being confined to the high-energy fluid domains. Thedata presented in Fig. 5 also demonstrate that the LTL diffu-sion rate is independent of the system-size. This made itpossible to use the small system to explore the low-T diffu-sion within affordable computer time.

FIG. 3. The coarse-grained energy distribution in a system of 128 000 par-ticles at the thermodynamic states indicated in the plots. The distribution isshown in a plane crossing the simulation box parallel to the XZ axes. Eachdistribution has been averaged over 103 time steps.

ba

dc

FIG. 4. Structure of the LTL. �a� A representative tetrahelical configurationdiscerned in a low-energy region observable in Fig. 2 for �=0.87. Thecolors are a guide to the eye distinguishing the six constituent lines ofatoms. �b� A fragment of the configuration shown in a demonstrating adefect in pentagonal packing of tetrahedra. The colors are the same as in �a�.�c� Aggregation of the tetrahelical cluster shown in a �green� with a similarcluster �blue�. �d� The linear strain of nontetrahedral defects interfacing thetwo tetrahelical configurations is shown in �c� using the same colors. Atomsfulfilling the neighbor condition as described in the text are connected bybonds.

174502-4 Elenius, Oppelstrup, and Dzugutov J. Chem. Phys. 133, 174502 �2010�

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The structural relaxation is commonly discussed in termsof dissipating density-density correlations as measuredby the intermediate scattering function F�Q , t�= ��Q , t���−Q ,0�. The system’s approach to the ergodic equilibriumis thus controlled by the slowest dissipating component ofF�Q , t�, within the structurally relevant range of Q. The ini-tial part �small t� of the F�Q , t� decay can be approximated29

as 1−F�Q , t� /S�Q��TQ2 /S�Q�t2. We, therefore, expect thesmall-Q prepeak of S�Q� that can be observed in Fig. 2�b� tobe the slowest dissipating structural feature of the LTL phase.This conclusion is confirmed by the results presented in theinset of Fig. 6. The main panel of Fig. 6 shows the timevariation of F�Qpp , t�, Qpp being the position of the prepeak,for T=0.65 and T=0.6. For both temperatures, the relaxationtimes clearly exceed by several orders of magnitude the timeinterval shown in the plot which, in terms of argon interpre-tation of the reduced units of time, correspond to 6.5·10−5 s.This makes it possible to conclude that the LTL is a structur-ally arrested glassy state.

We have earlier concluded that the density fluctuationsgiving rise to the prepeak featured by the structure factor ofthe LTL phase correspond to the spatial distribution of thelow-energy and high-energy domains like those shown inFig. 3. The observed nondissipation of S�Qpp� implies thatthe domain pattern gets frozen as the system is cooled belowthe spinodal. To support this conclusion, we present in Fig. 7the evolution of the coarse-grained distribution of the poten-tial energy in the LTL phase at T=0.65 and T=0.60. For eachtemperature, the time interval separating the two depicteddistributions corresponds to that spanned by the respectiveF�Qpp , t� shown in Fig. 6. The apparent time invariance ofthe general patterns of energy distribution indicates that thetetrahedrally ordered low-energy clusters like those shown inFig. 4 form a percolating network which remains topologi-cally unchanged on the explored time-scale. Figure 8 showsa cluster discerned within a low-energy domain at T=0.65.The cluster, percolating through the simulation box, repre-sents a configuration composed of face-sharing tetrahedra.Due to this structure, it is expected to possess a great degreeof mechanical stability. We note that the low-dimensionalmanner of the tetrahedra aggregation forming the cluster en-ables its unlimited growth, avoiding accumulation of thegeometric frustration that limits bulk packing oftetrahedra.30,33

0.5 1 1.5 2 2.510−6

10−5

10−4

10−3

10−2

10−1

1/T

D

1.3 1.4 1.5

10−3

0.4 0.6 0.8 1 1.2 1.4 1.6 1.810−4

10−3

10−2

10−1

1 / T

D

Simulation dataVFT fit

FIG. 5. �top� Arrhenius plot of the diffusivity at the density �=0.85.Crosses, HTL �Ref. 24�. Open circles: LTL for the system of N=128 000particles. Dots and open squares: LTL for the system of N=3456 particles,under cooling and heating, respectively. The inset shows an enlargement ofthe transition area. The straight lines indicate Arrhenius fits to the data.�bottom� HTL diffusion. Circles: simulation results; solid line: the Vogel–Fultcher–Tamman equation D=D0 exp�−BT0 / �T−T0��, with D0=0.12, B=3.7, and T0=0.386.

104 105 106 1070

0.2

0.4

0.6

0.8

1

t

F(Qpp,t)/S(Qpp)

T = 0.60T = 0.65

0 0.5 1 1.50

0.1

0.2

0.3

0.4

Q

F(Q,t)

t = 0t = 104

t = 5⋅106

FIG. 6. Main panel: the intermediate scattering function F�Qpp , t�, Qpp beingthe position of the low-Q prepeak of S�Q�, for the LTL at �=0.85 and thetwo indicated temperatures. Inset: the time evolution of F�Q , t� in thesmall-Q domain.

−0.3

−0.2

−0.1

−0.4

0.0

−0.3

−0.2

−0.1

−0.4

0.0

FIG. 7. �Top� Two cross-sections of the coarse-grained energy distributionat �=0.85 and T=0.65 separated by the time interval of 0.5·107. �Bottom�The same as in top panel but for T=0.6 and the separating time interval of3 ·107.

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IV. DISCUSSION

The general picture of the thermodynamic liquid-glasstransition we report here can be summarized as follows.Upon cooling the system at �=0.85 from the equilibriumliquid state, a domain of spinodal instability is observed be-low T=0.72 where two structurally and thermodynamicallydistinct liquid phases, HTL and LTL, coexist in equilibrium;this is shown in Figs. 2 and 3. Upon further cooling, the LTLclusters grow and percolate around T=0.68, and, below thattemperature, the estimated rate of structural relaxation for thepercolating LTL network is found to be smaller than thevalue that defines the glass transition. Thus, we observe athermodynamic transition transforming an equilibrium liquidthrough an equilibrium coexistence domain into the glassystate. The system thus remains in equilibrium within the en-tire temperature range above the glass transition. Moreover,based on earlier results,18,19 the apparent equilibrium coex-istence of the two phases may be viewed as an indication thatthe system remains thermodynamically stable within the co-existence domain. This makes it possible to classify thepresent model as a candidate for an ideal glass former.

The structure analysis demonstrates that the observedpolyamorphous transformation is driven by the pronouncedpredisposition of this model for tetrahedral local ordering.Based on the earlier results,24 this behavior can be accountedfor by the narrow and deep first minimum of the pair poten-tial restricting the variation in the first-neighbor distance.Breaking the pentagonal symmetry of the tetrahelical clustersby creating a linear strain of octahedral defects as shown inFig. 4 introduces an elegant novel way of avoiding the geo-metric frustration inherent to the tetrahedral packing.30,33 Itresults in the formation of a network of stable low-energyconfigurations of perfect tetrahedral order, dramatically ex-tending the range of structural coherence and thereby lower-ing the entropy. We note that the extended range of structuralordering within the tetrahedrally ordered clusters of the LTLphase gives rise to the anomalously high main peak of S�Q�,shown in Fig. 2�b�. This structural peculiarity, unusual for

simple liquids, can be compared with the structure of me-sophases, e.g., those formed by proteins.34 It is of interest tomention that, as a possible resolution of his entropy paradox,Kauzmann conjectured13 the existence of some kind of stateof high order for the liquid at low temperature which differsfrom the normal crystalline state.

The arrest of the structural relaxation in the LTL phase isapparently caused by the formation of a percolating networkof linearly shaped low-energy domains of tetrahedral orderwhich is presented in Figs. 7 and 8. This can be comparedwith the glass transition in polymeric ideal glass-formerswhich too was found to occur as a result of a percolationtransition.12 There are reasons to consider this mechanism ofclassification as generic. It was argued35 that the develop-ment of the spatial heterogeneity commonly observed in afragile liquid attaining the glass transition7,36 resembles theprocess of equilibrium polymerization, with the respectivepercolation point hidden below Tg. The percolation transitionwe observe in this simulation is also reminiscent of gelationin colloids forming a percolating network of linear clusterscomposed of tetrahedra.37 We mention that a gelationlikepercolation transition was observed in the present system atlow densities.38

The structural heterogeneity of the LTL phase and itsrelatively high diffusion rate distinguish it from the typicalinorganic glasses forming continuous tetrahedral networks. Afeasible conjecture is that these distinctions of the presentglass may possibly render it ductile and capable of self-repair, in contrast to the well-known brittleness of the inor-ganic glasses. This possibility is of significant technologicalinterest and deserves further investigation.

The metallic nature of the interaction potential used inthis model, incorporating characteristic Friedel oscillations,suggests a possible new approach to the design of bulk me-tallic glass formers. The efforts have so far been focused onthe search for eutectic mixtures of atomic species with astrong mismatch in size. This is assumed to increase thepacking density and create a strong chemical ordering,thereby reducing the free energy of the melt relative to therespective crystal and inducing strong liquid behavior.5,10

The present result demonstrates an alternative way of glassformation in metallic systems where the entropy of a fragileliquid is reduced discontinuously by a polyamorphous phasetransition from the equilibrium stable liquid state driven by apurely topological structural ordering. As a relevant observa-tion, we mention a strong-fragile transition with an indica-tion for hysteresis found in a bulk metallic glass-former.39

Although the pair potential used in this model is quiterealistic in describing interionic interaction in a �hypotheti-cal� single-component metallic liquid, no elemental metal re-produces it with a sufficient accuracy. Nevertheless, it is pos-sible to suggest a feasible strategy in the search for a realmetallic system with the pattern of phase behavior as wereported here. The Friedel oscillations26 like the one incor-porated in the present pair potential are ubiquitous in effec-tive interionic potentials conjectured for liquid metals.25

These oscillations are controlled by the electron densitywhich is a function of the composition of a metallic alloy.Therefore, it is conceivable that by appropriately manipulat-

FIG. 8. A cluster composed of face-sharing tetrahedra detected within alow-energy domain of the configuration shown in the top panel of Fig. 7.The variation of color indicates the variation of Z-coordinate. The clusterincludes 10 456 atoms.

174502-6 Elenius, Oppelstrup, and Dzugutov J. Chem. Phys. 133, 174502 �2010�

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ing the composition, the effective interionic potentials in realmetallic systems can be tuned to approximate the presentmodel.

V. CONCLUSIONS

In summary, these results introduce a novel type of theliquid-glass transition, whereby a first-order polyamorphousphase transition precludes crystallization upon cooling by di-rectly transforming an equilibrium thermodynamically stableliquid phase into a glass. The transition has been observed ina molecular dynamics simulation of a one-component systemwith a metal-like pair potential. This is the first observationof a thermodynamic glass transition from an equilibriumstable liquid phase. In this way, the present model may beregarded a reasonable candidate for the first nonpolymericsingle-component ideal glass-former observed so far. Its pro-totypically metallic nature suggests a possible new strategyfor designing bulk metallic glass-forming alloys in search fora noncrystallizing metallic liquid. The finding of a simpleone-component low temperature amorphous phase that suc-cessfully competes in thermodynamic stability with crystal-line phases may possibly be insightful for the continuingefforts to resolve the long-standing Kauzmann paradox.

ACKNOWLEDGMENTS

We are grateful to Professor C. A. Angell for reading themanuscript, valuable comments, and several illuminatingdiscussions. We thank Professor F. Sciortino for a discussion.We also thank the Centre for Parallel Computers �PDC� atthe Royal Institute of Technology, Stockholm and the EkmanConsortium for providing computer resources.

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174502-7 A simple monatomic ideal glass former J. Chem. Phys. 133, 174502 �2010�

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