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@example.com
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 liquidglass 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 ndings 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 liquidglass transition. In relation to this,experimental14 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 AdamGibbs theory7, the spinglass model8,9, the mode-coupling theory10
and the kinetically constrained lattice model11, as it may providea clue to our understanding of the liquidglass transition. Despiteintensive eorts, however, the physical factors that control dynamicheterogeneity remain elusive.
Many dierent ideas have been proposed on the origin of theliquidglass transition112. 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 ll up the space1216. 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 adierent possibility17,18 (see ref. 19 on the dierence in the physicalmechanism between the two scenarios): liquids tend to order intothe equilibrium crystal, but frustration eects of locally favouredshort-range ordering on long-range crystalline ordering preventcrystallization and help vitrication. Even a simple one-componentliquid may suer 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 adierent 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 dicult to do so in experiments. We modify a sphericallysymmetric interaction potential by including an anisotropic part;more specically, 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
0.0 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
Figure 1 Phase change on cooling and heating. a,b, T-dependence of V (a)and U (b) for various on cooling. The arrow indicates the location of Tg. For 0.5, we see a step in the volume change, reecting crystallization into a plasticcrystal. For= 0.6, on the other hand, there is no step and only the slope ofdV /dT changes at Tg. c,d, T-dependence of U on heating are shown for= 0.5 (c) and= 0.6 (d). For= 0.5, we see two steps, reecting theAFM-to-plastic crystal transformation and the melting of the plastic crystal. For= 0.6, on the other hand, we see only one step, which reects the melting of theAFM crystal.
ui, uj , and rij (see the Methods section). The modulation strengthis given by a parameter . This potential locally favours ve-fold symmetry (see Supplementary Information, Fig. S1), but hasa crystalline state with long-range antiferromagnetic (AFM) spinorder as the ground state. As ve-fold symmetry is not consistentwith any crystallographic symmetry13,14, this can be regarded asthe strength of frustration against crystallization. To the best of ourknowledge, this is the rst 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 vitrication aswell as the relationship between crystallization and vitrication.
We investigated the phase behaviour of this system as a functionof 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 xed cooling rate Q = 104(see Fig. 1a,b). All results are given in reduced units (see Methods).For weak frustration (that is, for < 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, for 0.6), there is no jump in the volume itself, but there arerather sharp changes in both dV /dT and dU /dT aroundthe glass-transition temperature (Tg). This indicates that thesystem is vitried without crystallization under strong frustrationeven for the same cooling rate Q: energetic frustration against
0.00.0 0.2 0.4 0.6 0.8 1.0
Figure 2 The phase diagram of our model system. The horizontal axis represents, which is a measure of the strength of frustration against crystallization. The lledcircles 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 lled squares represent the Tg, which is dened as (Tg) = 106. Thelled triangles represents the VogelFulcher temperature T0. T0 = 0.099, 0.076 and0.026 for= 0.6, 0.7 and 0.8, respectively. The increase in Tg T0 withmeans that a liquid becomes stronger (that is, less fragile) with an increase in(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 various. We found that the lowest-energy ground state of our systemat > 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 (see Fig. 1c,d). For weak frustration (for< 0.6), the crystal with AFM spin order rst transforms into aplastic crystal to gain the rotational entropy, and then the plasticcrystal melts into a liquid at Tm(), where Tm is the meltingpoint. This plastic crystal has hexagonal positional order but noorientational order. For strong frustration (for 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 0.6), a liquidvitries without crystallization on cooling. Figure 2 summarizes thephase behaviour of this system including the non-equilibrium stateas a function of the degree of frustration.
We calculated the T-dependence of the radial distributionfunction g(r). Figure 3a shows such an example for = 0.6.There is no long-range correlation even at low temperatures, whichindicates that the system forms a glassy state for this value of. Thus, we conrm that a liquid vitries on cooling withoutcrystallization for 0.6. As explained in the Methods section,our system tends to form locally favoured structures of ve-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 conrmedthat the T-dependence of the height of a peak corresponding to the
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1 1.5 2 2.5
1 2 3 4 5 61 2 3 4 5 6
0 5 10 15r
Figure 3 Change in structures and dynamics on cooling. a, T-dependence of theradial distribution function g(r ) for= 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 ve-fold symmetry, respectively (see theblack lines in the diagrams of the two structures). This can also be conrmed in realspace in Fig. 4. b, T-dependence of for= 0.6. The dashed line is theArrhenius T-dependence tted to the data above T m, whereas the solid line is theVogelFulcher behaviour tted to all of the data. c, T-dependence of for= 0.6.Owing to strong anisotropic interactions, the relaxation function is not a singleexponential ( 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 tted the stretched exponentialfunction to CR(t) as CR(t) exp((t/)). Figure 3b and cshows the T-dependence of the characteristic rotational relaxationtime and the Kohlausch exponent obtained by the tting,respectively, for = 0.6. can be well-tted by the VogelFulcher law = 0 exp(DT0/(T T0)) with 0 = 0.61, D = 7.4and the VogelFulcher temperature T0 = 0.099. The increase of on cooling is often classied between strong and fragile extremesusing Tg as a scaling parameter3,5. Liquids whose 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 starts todeviate from the Arrhenius behaviour below Tm (=0.46), which wedene as Tm of a crystal free from frustration (that is, at = 0)17. is almost constant above Tm, whereas it monotonically decreases
with a decrease in T below Tm. Both results indicate that a liquidshows complex cooperative behaviour only below Tm. We alsocalculated the mean square displacement r2(t) (detailed later)and the intermediate scattering function and found that there is atwo-step relaxation (fast and relaxation) at temperatures lowerthan Tm. We conrmed 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 liquidglasstransition well, which are known from previous studies16.
Here we focus on the relationship between the dynamics andthe structure; specically, 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 liquidglass transition. This problemis also related to the problem of the complex energy landscape5,2224.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 ( 0.6). The orderparameter i characterizing it can then be dened as i(t) =(1/6)
i,j |ui(t) uj(t)|, where i, j represents the summation
over the nearest neighbours of particle i. When i 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; specically,if |ui uj| 0.9. To avoid the eects of thermal noise and detectregions with slow dynamics, we averaged the order parameter overthe duration of as
i(t) = (1/) t +/2
t /2dt i(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. 4ac, 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. 4df and translational dynamics (particle...