Frustration on the way to crystallization in glass · problem of the liquid–glass transition....

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Frustration on the way to crystallization inglass

HIROSHI SHINTANI AND HAJIME TANAKA*Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan*e-mail: [email protected]

Published online: 19 February 2006; doi:10.1038/nphys235

Some liquids do not crystallize below the melting point,

but instead enter into a supercooled state and on

cooling eventually become a glass at the glass-transition

temperature. During this process, the liquid dynamics not

only drastically slow down, but also become progressively

more heterogeneous. The relationship between the kinetic

slowing down and growing dynamic heterogeneity is a key

problem of the liquid–glass transition. Here, we study this

problem by using a liquid model, with a crystalline ground

state, for which we can systematically control frustration

against crystallization. We found that slow regions having

a high degree of crystalline order emerge below the

melting point, and their characteristic size and lifetime

increase steeply on cooling. These crystalline regions lead

to dynamic heterogeneity, suggesting a connection to the

complex free-energy landscape and the resulting slow

dynamics. These findings point towards an intrinsic link

between the glass transition and crystallization.

The mechanism of the drastic slowing down of the structuralrelaxation of a liquid on cooling is one of the central issues ofthe physics of the liquid–glass transition. In relation to this,

experimental1–4 and numerical5,6 evidence has been accumulatedfor the existence of dynamic heterogeneity in a supercooled stateof liquid, whose length scale increases on cooling. The link betweenthis growing length scale and the slowing down of the structuralrelaxation has been actively discussed on the basis of the Adam–Gibbs theory7, the spin–glass model8,9, the mode-coupling theory10

and the kinetically constrained lattice model11, as it may providea clue to our understanding of the liquid–glass transition. Despiteintensive efforts, however, the physical factors that control dynamicheterogeneity remain elusive.

Many different ideas have been proposed on the origin of theliquid–glass transition1–12. Among them, frustration has often beenconsidered to play a key role in the glass transition in conjunctionwith other glassy systems such as spin glass and dipolar glass12.The most popular scenario is geometrical frustration associatedwith the fact that locally favoured structures such as icosahedralorder cannot fill up the space12–16. In other words, liquids tendtowards global icosahedral order, but they cannot achieve it,which is the origin of frustration in this scenario. There is also adifferent possibility17,18 (see ref. 19 on the difference in the physicalmechanism between the two scenarios): liquids tend to order intothe equilibrium crystal, but frustration effects of locally favouredshort-range ordering on long-range crystalline ordering preventcrystallization and help vitrification. Even a simple one-componentliquid may suffer from such frustration if the local symmetry of theinteraction potential does not perfectly match the symmetry of theequilibrium crystal.

Motivated by this frustration scenario, we have developed adifferent type of simulation model, where we can systematicallychange the degree of frustration against crystallization. Moleculardynamics simulation is a useful means for investigating thedynamics of liquids microscopically6. We can also directly controlthe intermolecular potential in molecular dynamics, whereas itis difficult to do so in experiments. We modify a sphericallysymmetric interaction potential by including an anisotropic part;more specifically, we introduce a spin director ui into particle isuch that the interaction between particle i and j depends not onlyon the interparticle distance |rij|, but also on the angles among

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= 0.2 = 0.6

= 0.5 = 0.6

= 0.65= 0.4

= 0.5

= 0.6

Liquid

AFMcrystal AFM

crystal

Plasticcrystal

1.1

1.2

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– 2.5

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.2 0.3 0.1 0.2 0.3

⟨V⟩

⟨U⟩

⟨U⟩

⟨U⟩

T T

T T

a b

c d

ΔΔ

Δ

Δ Δ

Δ

Δ

Δ

Figure 1 Phase change on cooling and heating. a,b, T-dependence of 〈V 〉 (a)and 〈U 〉 (b) for various Δ on cooling. The arrow indicates the location of Tg. ForΔ≤ 0.5, we see a step in the volume change, reflecting crystallization into a plasticcrystal. For Δ= 0.6, on the other hand, there is no step and only the slope ofd〈V 〉/dT changes at Tg. c,d, T-dependence of 〈U 〉 on heating are shown forΔ= 0.5 (c) and Δ= 0.6 (d). For Δ= 0.5, we see two steps, reflecting theAFM-to-plastic crystal transformation and the melting of the plastic crystal. ForΔ= 0.6, on the other hand, we see only one step, which reflects the melting of theAFM crystal.

ui, uj , and rij (see the Methods section). The modulation strengthis given by a parameter Δ. This potential locally favours five-fold symmetry (see Supplementary Information, Fig. S1), but hasa crystalline state with long-range antiferromagnetic (AFM) spinorder as the ground state. As five-fold symmetry is not consistentwith any crystallographic symmetry13,14, this Δ can be regarded asthe strength of frustration against crystallization. To the best of ourknowledge, this is the first two-dimensional glass-forming modelliquid system that consists of single-component particles and has acrystalline state as the ground state. By using this lattice-free spin-glass model, we investigate the roles of frustration in vitrification aswell as the relationship between crystallization and vitrification.

We investigated the phase behaviour of this system as a functionof Δ by studying the temperature (T) dependence of the averagevolume per particle 〈V 〉 and the average potential energy perparticle 〈U〉 at pressure P = 0.5 for a fixed cooling rate Q = 10−4

(see Fig. 1a,b). All results are given in reduced units (see Methods).For weak frustration (that is, for Δ < 0.6), the system crystallizes(into a plastic crystal) on cooling with a distinct jump in thevolume. On the other hand, for strong frustration (that is, forΔ ≥ 0.6), there is no jump in the volume itself, but there arerather sharp changes in both d〈V 〉/dT and d〈U 〉/dT aroundthe glass-transition temperature (Tg). This indicates that thesystem is vitrified without crystallization under strong frustrationeven for the same cooling rate Q: energetic frustration against

Plastic crystal

AFMcrystal

Glass

Liquid

T Supercooledliquid

0.00.0 0.2 0.4 0.6 0.8 1.0

0.1

0.2

0.3

0.4

0.5

Δ

Figure 2 The phase diagram of our model system. The horizontal axis representsΔ, which is a measure of the strength of frustration against crystallization. The filledcircles represent the melting point of a plastic crystal. The open squares representthe temperature where an AFM crystal transforms into a plastic crystal or melts onheating. The filled squares represent the Tg, which is defined as τα (Tg) = 106. Thefilled triangles represents the Vogel–Fulcher temperature T0. T0 = 0.099, 0.076 and0.026 for Δ= 0.6, 0.7 and 0.8, respectively. The increase in Tg − T0 with Δ

means that a liquid becomes stronger (that is, less fragile) with an increase in Δ

(see the text). We also show a simulated structure corresponding to each state ofour system. The arrows indicate the spin direction.

crystallization prevents crystal nucleation. To understand whatkinds of crystalline ordering take place in this system, we studiedcrystalline structures that can be formed at P = 0.5 for variousΔ. We found that the lowest-energy ground state of our systemat Δ > 0 is a crystal with AFM spin order (see Fig. 1c). Thiscrystal has translational symmetry concerning both position andorientation of particles. Note that this crystalline lattice is notperfectly hexagonal, but slightly distorted uniaxially because ofstrong anisotropic interactions. We also studied the change ofthe potential energy on heating to work out the phase behaviourfor various values of Δ (see Fig. 1c,d). For weak frustration (forΔ < 0.6), the crystal with AFM spin order first transforms into aplastic crystal to gain the rotational entropy, and then the plasticcrystal melts into a liquid at Tm(Δ), where Tm is the meltingpoint. This plastic crystal has hexagonal positional order but noorientational order. For strong frustration (for Δ ≥ 0.6), on theother hand, no plastic crystal exists and the crystal with AFM spinorder melts directly into a liquid. Interestingly, in the region where,on heating, the AFM crystal directly melts into a liquid withoutpassing through the plastic crystal (namely, for Δ ≥ 0.6), a liquidvitrifies without crystallization on cooling. Figure 2 summarizes thephase behaviour of this system including the non-equilibrium stateas a function of the degree of frustration Δ.

We calculated the T-dependence of the radial distributionfunction g(r). Figure 3a shows such an example for Δ = 0.6.There is no long-range correlation even at low temperatures, whichindicates that the system forms a glassy state for this value ofΔ. Thus, we confirm that a liquid vitrifies on cooling withoutcrystallization for Δ ≥ 0.6. As explained in the Methods section,our system tends to form locally favoured structures of five-foldsymmetry. From the T-dependence of g(r), we can clearly see thatboth clusters with crystallographic symmetry and locally favouredstructures progressively develop with a decrease in T . We confirmedthat the T-dependence of the height of a peak corresponding to the

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1 1.5 2 2.5

T*m T*m

1/T 1/T

Log

g (r

)

g (r

)0

2

4

6

8

5

0

1

2

3

4

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0.6

0.7

0.8

0.9

1 2 3 4 5 61 2 3 4 5 6

r

β

01234

0 5 10 15r

a

b c

τ α

Figure 3 Change in structures and dynamics on cooling. a, T-dependence of theradial distribution function g(r ) for Δ= 0.6. T is 0.18, 0.22, 0.25 and 0.4 from thetop to the bottom. g(r ) in a wider r range for T = 0.18 is shown in the inset. Thearrows at r ∼ 1.13 and r ∼ 1.63 correspond to a peak of the AFM crystal and apeak of the locally favoured structure of five-fold symmetry, respectively (see theblack lines in the diagrams of the two structures). This can also be confirmed in realspace in Fig. 4. b, T-dependence of τα for Δ= 0.6. The dashed line is theArrhenius T-dependence fitted to the data above T ∗

m, whereas the solid line is theVogel–Fulcher behaviour fitted to all of the data. c, T-dependence of β for Δ= 0.6.Owing to strong anisotropic interactions, the relaxation function is not a singleexponential (β ∼ 0.95) even at high temperatures. Such behaviour is known forhydrogen-bonding liquids such as glycerol.

locally favoured structure is consistent with the prediction of ourtwo-order-parameter model of liquid (see equation (2) in ref. 20)that assumes that a liquid consists of only two types of structure;namely, locally favoured structures (blue particles in Fig. 2) andnormal-liquid structures (green particles in Fig. 2).

We also calculated the rotational autocorrelation functionCR(t) = (1/N )〈∑i ui(t) ·ui(0)〉, where t is time and N is thenumber of particles. Then we fitted the stretched exponentialfunction to CR(t) as CR(t) ∝ exp(−(t/τα)

β). Figure 3b and cshows the T-dependence of the characteristic rotational relaxationtime τα and the Kohlausch exponent β obtained by the fitting,respectively, for Δ = 0.6. τα can be well-fitted by the Vogel–Fulcher law τα = τ0 exp(DT0/(T − T0)) with τ0 = 0.61, D = 7.4and the Vogel–Fulcher temperature T0 = 0.099. The increase of τα

on cooling is often classified between strong and fragile extremesusing Tg as a scaling parameter3,5. Liquids whose τα obeys theArrhenius law are called ‘strong’, whereas ‘fragile’ liquids havesuper-Arrhenius behaviour. D is an indicator of the strength ofa liquid: the larger D is, the stronger the liquid is. Here D = 7.4corresponds to a fragile liquid. The T-dependence of τα starts todeviate from the Arrhenius behaviour below T∗

m (=0.46), which wedefine as Tm of a crystal free from frustration (that is, at Δ = 0)17.β is almost constant above T∗

m, whereas it monotonically decreases

with a decrease in T below T∗m. Both results indicate that a liquid

shows complex cooperative behaviour only below T∗m. We also

calculated the mean square displacement �r2(t) (detailed later)and the intermediate scattering function and found that there is atwo-step relaxation (fast β and α relaxation) at temperatures lowerthan T∗

m. We confirmed dynamic heterogeneity of a supercooledliquid by means of a time-dependent four-point density correlationfunction26 (see the Supplementary Information) and found thatdynamic heterogeneity becomes more pronounced with a decreasein T (we provide further details later). Thus, we may say that thissystem reproduces the most essential features of the liquid–glasstransition well, which are known from previous studies1–6.

Here we focus on the relationship between the dynamics andthe structure; specifically, the medium-range crystalline order.It was pointed out16,21 that revealing the relationship betweendynamics and structure is key to the understanding of the slowdynamics associated with the liquid–glass transition. This problemis also related to the problem of the complex energy landscape5,22–24.The most popular method is to focus on an inherent structure25

or an ideal glass structure, which is believed to form local minima(basins) in the energy landscape. In contrast to this popular model,hereafter we propose a possibility that a crystalline structure is a keystructure of a supercooled state.

First, we show evidence for the existence of long-lived clustersof medium-range crystalline order in a supercooled liquid state.In our system, we know the equilibrium crystalline structures (seeFig. 2). Note that only one type of crystal, which has AFM spinorder, exists in the glass-forming region (Δ ≥ 0.6). The orderparameter Ψi characterizing it can then be defined as Ψi(t) =(1/6)

∑〈i,j〉 |ui(t) ·uj(t)|, where 〈i, j〉 represents the summation

over the nearest neighbours of particle i. When Ψi ≥ 0.8, we judgethat particle i is involved in clusters of high crystalline order. Wejudge that particle j is also involved in the same cluster if the spin ofparticle j is parallel or antiparallel to that of particle i; specifically,if |ui ·uj| ≥ 0.9. To avoid the effects of thermal noise and detectregions with slow dynamics, we averaged the order parameter overthe duration of τα as

Ψi(t) = (1/τα)

∫ t ′+τα/2

t ′−τα/2

dt ′Ψi(t ′).

We emphasize that this orientational order parameter is muchmore sensitive to the crystalline order than the positional orderparameter, which is one of the merits of our model system.In Fig. 4a–c, we present snapshots of the spatial distributionof the orientational order parameter for three temperatures.At T = 0.35, there are few clusters of high crystalline order.With a decrease in T , clusters of high crystalline order appearand grow in size and lifetime. We also present rotationaldynamics in Fig. 4d–f and translational dynamics (particletrajectories) in Fig. 4g–i corresponding to Fig. 4a–c. Despitethe fact that Fig. 4a–c are snapshots, Fig. 4d–f show the time-averaged information and Fig. 4g–i show trajectories, theyare well-correlated: particles are less mobile both rotationallyand translationally in ordered regions than in disorderedregions. To demonstrate this correlation more quantitatively, wealso calculated the rotational correlation function 〈CR(Ψ , t)〉and the mean square displacement 〈�r2(Ψ , t)〉 for regionshaving a certain range of the orientational order parameter(Ψi ∈ Ψ ) at t = 0. Here CR(Ψ , t) = (1/NΨ )

∑Ψi∈Ψ

ui(t) ·ui(0), where NΨ is the number of particles belonging toΨi ∈Ψ and �r2(Ψ , t) = (1/NΨ )

∑Ψi∈Ψ [ri(t)− ri(0)]2. As shown

in Fig. 5a, we reveal that the higher the degree of crystalline order,the slower the local rotational relaxation. The correlation betweentranslational motion and the degree of the crystalline order can also


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a b c

d e f

g h i

Figure 4 Medium-range crystalline order and dynamic heterogeneity. a–c, Snapshots of liquid structures (at a certain time ts) for three different temperatures:a, T = 0.35, b, T = 0.18 and c, T = 0.17 (see also Supplementary Information, Video S1, which corresponds to b). Red particles haveΨi greater than 0.75. Blue particles,on the other hand, represent locally favoured structures with five-fold symmetry. The arrows indicate the spin direction or the direction of ui. d–f, The slowness of therotational dynamics. The red and blue particles denote those slowly rotating during ts < t < ts + τα/2: the particles belonging to the locally favoured structures are colouredblue. g–i, The trajectories of corresponding to a–c. The trajectories in g–i are drawn from ts to ts + τα , ts +2τα and ts +4τα , respectively. Note that a–c are snapshots at ts.

be seen in Fig. 5b. In Fig. 5c and d we also show the distributions ofCR(Ψ , t1) (t1 = 16,400) and �r2(Ψ , t2) (t2 = 7,900), respectively,where t1 roughly corresponds to the rotational relaxation time,τα (more precisely, t1 = 0.87τα), and t2 is the time when theheterogeneity of the translational motion, which is characterizedby χ4 (see the Supplementary Information), becomes maximum.These results clearly indicate a positive correlation between thedegree of crystalline order and the slowness of the structuralrelaxation, that is, a link between crystalline order and dynamicheterogeneity.

The above results show that clusters of high crystalline orderemerge below T∗

m and their average size and lifetime increaseon cooling. The lifetime is longer than τα (see SupplementaryInformation, Fig. S2) and reaches the order of 10τα near Tg. Thisis consistent with experimental findings2. Particles do not rotate fora long time in these clusters, as can be seen from SupplementaryInformation, Video S1. As this heterogeneity has a finite lifetime,it can be regarded as ‘dynamic heterogeneity’. The clusters are

temporally created and annihilated, and their average size remainsconstant with time. Thus, we speculate that the size of the clustersis still smaller than the size of the critical crystal nucleus, but thisneeds to be checked carefully in the future.

Figure 6a shows the T-dependence of the characteristic sizeof clusters, ξ. We define this size ξ(T) as ξ(T) =

√N , where

N is the average number of particles contained in a clusterwith Ψ ≥ 0.75. We also plot the dynamic coherence lengthobtained by the analysis using the four-point time-correlationfunction26, ξ4 (see the Supplementary Information), in Fig. 6a.Note that ξ4 characterizes the heterogeneity of translational motion(see also Fig. 4g–i). We found that ξ4 is proportional to ξ,and both ξ and ξ4 can be fitted by a function of ξ(4) (T) =A(4) (T − T0)

−2/d (d is the spatial dimensionality; d = 2 in ourcase) with T0 = 0.099, which is obtained by the fitting of theVogel–Fulcher relation to the T-dependence of τα (see Fig. 3b).The behaviour of ξ and ξ4 above is consistent with the scalingargument8,9, which connects the increase of ξ and ξ4 to that of τα.


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Log t Log t

SlowSlow

Freq

uenc

y

Freq

uenc

y

1.0101

100

10 – 1

10 – 2

0.8

0.6

0.4

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0

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Total

= 0.65

= 0.70

= 0.75

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Total

= 0.40 (LFS)

= 0.65

= 0.70

= 0.75

= 0.85

= 0.80

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= 0.40 (LFS)

= 0.65

= 0.70

= 0.75

= 0.85

= 0.80

Total

= 0.65

= 0.70

= 0.75

= 0.80

= 0.85⟨C

R(,t

)⟩

CR( ,t1)

⟨Δr2 (

,t)⟩

r 2( ,t2)

a b

c d

Ψ

Ψ

Ψ

Ψ

Ψ

Ψ

Ψ

Ψ

Ψ

Ψ

Ψ

ΨΨΨΨ

ΨΨ

ΨΨΨΨ

ΨΨ

ΔΨ Ψ

Ψ

Figure 5 Relation of rotational and translational dynamics to the degree of local crystalline order. a, 〈CR(Ψ , t )〉 for T = 0.17 and Δ= 0.6. Note that the dynamicheterogeneity smears out for large t. b, 〈�r2(Ψ , t )〉 for T = 0.17 and Δ= 0.6. c, The distribution of CR(Ψ , t1) (t1 = 16,400). HereΨ values are ±0.02. The particles withlargerΨ are slower in rotation, but the particles belonging to the locally favoured structures (LFS,Ψ ∼ 0.4) are also slow. d, The distribution of �r2(Ψ , t2) (t2 = 7,900). Theparticles with largerΨ are slower in translation. Unlike rotational motion, the particles belonging to the locally favoured structures are rather fast in translational motion. Notethat these distribution functions are normalized such that the integral becomes one.

In addition to the growth of clusters, the increase in the fractionof ordered clusters and the resulting percolation and/or theincrease in the frictional contacts between clusters may also causeslow dynamics. Although we cannot specify which is the mostrelevant mechanism for slow dynamics, we may say that dynamicheterogeneity and the slow dynamics are intrinsically related tomedium-range crystalline ordering in a supercooled liquid. Nextwe focus on another important structure, that is, locally favouredstructure, which induces frustration against crystallization. Wefound that the rotational dynamics of particles in locally favouredstructures is slow (see Figs 4d–f and 5c), although their translationaldynamics are not so slow (see Fig. 5d). Thus, both medium-rangecrystalline ordering and short-range bond ordering contributeto slow rotational dynamics near Tg. However, as the spatialdistribution of locally favoured structures is rather random, theirconnection to dynamic heterogeneity (or the growing length scale)remains the future problem.

We note that medium-range crystalline ordering can also beconfirmed in the radial distribution function g(r) shown in Fig. 3a.We can clearly see that the height of the peak correspondingto the crystalline order increases with decreasing T . We do notfind any significant change in the structure factor S(q) (q is thewavenumber) in the low-q region (q ∼ 1/ξ). This is consistent withthe well-known fact that dynamic heterogeneity cannot be detected

in small-angle scattering experiments. The reason may be due tothe fact that clusters with high crystalline order appear randomly inspace without spatial correlation. Our simulation results indicatethat there is a possibility to detect ‘dynamic heterogeneity’, or‘medium-range ordering’, in g(r) and S(q) ‘not’ in a low-q regionwhere q ∼ 1/ξ, but rather in a high-q region where q ∼ 1/σ (σ isthe particle diameter) through medium-range crystalline ordering.In relation to this, it is worth mentioning the experimentalstudy27 on the T-dependence of S(q) in propylene carbonateand phenyl salicilate (salol). There it was found that shouldersin S(q), which were identified as peaks from clusters of highcrystalline order, progressively develop in the supercooled liquidsbelow T∗

m on cooling. Our simulation results are consistent withthis experimental finding, which may imply that this behaviour isgeneric to any glass-forming liquids. Further experimental studiesalong this line are highly desirable.

Our study indicates that glassy slow dynamics can beinduced by frustration between long-range density orderingtowards crystallization and short-range bond-ordering towards theformation of locally favoured structures. According to this17–19, aliquid suffering from stronger frustration should be stronger. In oursimulations, we found that D increases with an increase in Δ, asshown in Fig. 6b: D = 7.4, 10.9, 17.2, 29.7 and 84.3 for Δ = 0.6,0.65, 0.7, 0.75 and 0.8, respectively. We stress that this range of D


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8

6

4

2

6

5

4

3

2

1

00.2 0.3 0.4 0.2 0.4 0.6 0.8 1.0

Log ( T

)

T Tg/T

4

= 0.60= 0.65= 0.70= 0.75= 0.80

a b

ξ

ξ

ξ

ΔΔΔΔΔτ α

Figure 6 Growth of dynamic heterogeneity on cooling and the relationshipbetweenΔ and the fragility. a, T-dependence of the characteristic size ofcrystal-like clusters ξ and the correlation length of translational motion ξ4 obtainedfrom the analysis of the four-point time-correlation function (see the SupplementaryInformation) for Δ= 0.6. The solid and dashed lines are the fittings of A/(T− T0)with T0 = 0.099 for ξ and ξ4, respectively. We found that ξ ∝ ξ4. The error barindicate the range of ξ observed during 10τα . b, The Angell plot of τα against Tg/T.Tg = 0.150, 0.158, 0.163, 0.167 and 0.166 for Δ= 0.6, 0.65, 0.7, 0.75 and0.8, respectively.

almost covers from the fragile (for example, o-terphenyl, D ∼ 5) tothe strong extremes (for example, silica, D ∼ 60). This is anotherindication that the frustration against crystallization controls thenature of the liquid–glass transition including the fragility of liquid.Network-forming tendency seems not to be a prerequisite formaking liquids strong, in contrast to common beliefs. Our studysuggests that the model where the slow dynamics are caused byjamming owing to dense packing may not be enough to understandthe glass transition. It may be important to consider why a systemcan remain in a disordered state and be jammed without ordering;namely, frustration adversely affects crystallization.

METHODS

MODEL

Here we use molecular dynamics simulation to investigate the roles ofcrystallization in glass transition in the spirit of our two-order-parametermodel of the liquid–glass transition17–19. For this purpose, we introduce apotential that can directly and systematically control the strength of frustrationagainst crystallization.

U (rij ,Ωij) = U (rij)+�U (rij ,Ωij),

where rij is the distance between two particles i and j, and Ωij expresses theorientation between these particles. The first term of the potential is anisotropic potential that tends to maximize the density of a liquid and encouragecrystallization. The second term is an anisotropic potential that leads to theformation of locally favoured structures. In this paper, we adopted theLennard–Jones potential as the isotropic potential U (rij):

U (rij) = 4ε

((σ

rij

)12

−(

σ

rij

)6)

,

where ε is the well depth of the potential and σ is the diameter of the particle.The anisotropic part of the potential is given by

�U (rij ,Ωij) = −4ε�

(σ

rij

)6[h

(θi − θ0

θc

)+h

(θj − θ0

θc

)− 64

35πθc

],

h(x) = 1−3x2 +3x4 −x6 (−1 < x < 1); h(x) = 0 (x ≤ −1, x ≥ 1).

θi is an angle between the relative vector rji = rj −ri and the unit vector ui ,which represents the orientation of the axis of particle i. θj is an angle betweenthe relative vector rij = ri −rj and the unit vector uj (see SupplementaryInformation, Fig. S1). The function h((θ− θ0)/θc ) (θ0 = 126◦ and θc = 53.1◦,where θc is the width of the angular part of the potential) has a maximum atθ = 126◦, and θ0 is an energetically favoured value of θi , and is the angle whereh((θ− θ0)/θc ) becomes maximum. Thus, this term stabilizes the locallyfavoured structure of five-fold symmetry, as shown in SupplementaryInformation, Fig. S1.

Δ controls the stability of the locally favoured structure as well as thestrength of frustration against crystallization. Our system has frustration that isrelated not only to the orientation of the particle axis, which may be regarded asspin, but also to the position of the particle. In this sense, our model can beconsidered as a lattice-free spin system with translational degrees of freedom(lattice-free spin glass).

SIMULATION METHOD

The number of particles in our simulations is N = 1,024. We used periodicboundary conditions. We cut off and smoothed both the potential and itsderivative at 3.0σ to save computational time. Here we show all results inreduced units. We used particle mass m, σ and ε as the basic unit of mass,length and energy, respectively. Thus, the moment of inertia I , the temperatureT , the pressure P, the distance r and the time t are scaled as I ≡ I/mσ2,T ≡ kBT/ε (kB is Boltzmann’s constant), P ≡ Pσ2/ε, r ≡ r/σ and t ≡ t/τ

(τ = √mσ2/ε).

To control T and P of a system, we used the Nose–Poincare–Andersen(NPA) thermostat28 and extended the NPA method to include the rotationaldegrees of freedom of particles. Our modified NPA hamiltonian (HNPA) isgiven by

HNPA(t) = (HNA(t)−HNA(0))s,

HNA =∑ p2

2mV s2+

∑ p2φ

2I s2+U (qV 1/2,{φi})+ gkBT lns

+ Π 2V

2MV+ Π 2

s

2Ms+PV ,

where NA is the Nose–Anderson hamiltonian, p is the momentum, pφ is theangular momentum, φi is the angle between x axis and ui , g is the total numberof the degrees of freedom of this system, s is the variable to controltemperature, V is the two-dimensional volume of the system, ΠV is themomentum conjugate to V , Πs is the momentum conjugate to s, MV is themass of V , Ms is the mass of s and q is a scaled position vector. Here we usedI = 0.04, MV = 0.0001 and Ms = 1.0. We confirmed the equipartition of theenergy into the translational and the rotational degrees of freedom of particles.According to the theoretical estimate, the variance of the internal temperaturedistribution should be T2/N and 2T2/N for translation and rotation,respectively, which is also confirmed in our simulations within an accuracy of2%. We also proved that the above NPA method extended to a system with therotational degrees of freedom constitutes the N PT ensemble. The resultsshown in this article are for P = 0.5 and a cooling rate of Q ≡ dT/dt = 10−4. Asystem with the Lennard–Jones potential U (r) melts at T∗

m = 0.46 at P = 0.5.After quenching the system into a target temperature, we annealed it for a longtime (during 30–200 τα) before making any analysis.

Received 21 September 2005; accepted 18 January 2006; published 19 February 2006.

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AcknowledgementsThe authors are grateful to C. P. Royall for a critical reading of our manuscript. This work was partiallysupported by a grand-in-aid from the Ministry of Education, Culture, Sports, Science andTechnology, Japan.Correspondence and requests for materials should be addressed to H.T.Supplementary Information accompanies this paper on www.nature.com/naturephysics.

Competing financial interestsThe authors declare that they have no competing financial interests.

Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/


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