Royall, C. P., Dyre, J., Ingebrigsten, T., & Schroder, T. (2019).Crystallization Instability in Glass-Forming Mixtures. Physical ReviewX, 9(3), [031016]. https://doi.org/10.1103/PhysRevX.9.031016

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Crystallisation Instability in Glassforming Mixtures

Trond S. Ingebrigtsen,1, 2 Jeppe C. Dyre,2 Thomas B. Schrøder,2 and C. Patrick Royall3, 4, 5

1Department of Fundamental Engineering, Institute of Industrial Science,University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan.2Glass and Time, IMFUFA, Department of Science and Environment,

Roskilde University, P.O. Box 260, DK-4000 Roskilde, Denmark3H.H. Wills Physics Laboratory, Tyndall Avenue, Bristol, BS8 1TL, UK.

4School of Chemistry, University of Bristol, Cantock’s Close, Bristol, BS8 1TS, UK.5Centre for Nanoscience and Quantum Information, Tyndall Avenue, Bristol, BS8 1FD, UK.

Understanding the mechanisms by which crystal nuclei form is crucial for many phenomena suchas gaining control over crystallisation in glassforming materials or accurately modelling rheologicalbehaviour of magma flows. The microscopic nature of such nuclei, however, makes their under-standing extremely hard in experiments, while computer simulations have hitherto been hamperedby short timescales and small system sizes. Here we use highly-efficient GPU simulation techniquesto address these challenges. The larger systems we access reveal a general nucleation mechanism inmixtures. In particular, we find that the supercooled liquid of a prized atomistic model glassformer(Kob-Andersen model) is inherently unstable to crystallisation, i.e. that nucleation is unavoidableon the structural relaxation timescale, for system sizes of 10, 000 particles and larger. This is dueto compositional fluctuations leading to regions comprised of one species that are larger than thecritical nucleus of that species, which rapidly crystallise. We argue that this mechanism provides aminimum rate of nucleation in mixtures in general, and show that the same mechanism pertains tothe metallic glassformer copper-zirconium (CuZr).

I. INTRODUCTION

Crystallisation in supercooled liquids has profound im-plications in fields as diverse as the development of amor-phous materials [1], magma flows in volcanoes [2] andaqueous solutions of ions [3]. Materials in question in-clude metallic, inorganic and chalcogenide glassformers,where mixtures of a number of different constituents havethe effect of suppressing or controlling crystallisation [4].Alas, this tendency to crystallise places stringent limitson the size of the pieces of amorphous material that canbe formed: large pieces are more likely to undergo crystalnucleation [5, 6]. This “Achilles heel” of glass formationthus limits the exploitation of metallic glasses for exam-ple, whose superior mechanical properties otherwise holdgreat promise [7].

It is clear that any liquid cooled below its freezing pointmust, for a sufficiently large system, nucleate [8, 9]. How-ever, the practical limits of cooling rate versus system sizerequired for vitrification are not known in general. In ad-ditional to these practical considerations, crystallisationis one solution to the Kauzmann paradox of vanishingconfigurational entropy upon which a number theories ofthe glass transition rest [4, 10]: crystallisation avoids theneed to invoke any particular theoretical description ofdivergent viscosity in amorphous materials [11, 12].

It is known empirically that increasing the numberof constituent species and introducing a size disparar-ity among these components, together with a negativeheat of mixing, tends to suppress nucleation – this hasbeen the guiding principle in the development of bulkmetallic glasses [5]. However despite recent innovativeapproaches using model systems [13, 14] and novel sam-pling techniques [15, 16], there is still a lack of funda-

mental understanding of the mechanisms by which glass-forming mixtures crystallise.

Here we consider a crystallization mechanism that isalways present when a glass-former is produced by mixingconstituents which by themselves are poor glass-formers,as is often the case. We therefore expect this mecha-nism to be remarkably widespread. In particular, com-positional fluctuations in the supercooled liquid lead toregions containing just a single constituent, and even-tually such a region will occur that is large enough –and long-lived enough – that it will nucleate a crystal ofthat one species. Of course, depending on the specificmixture, there may be other, faster, nucleation mecha-nisms. Nucleation by compositional fluctuations never-theless provides a lower bound for the nucleation rate inmixtures. We emphasize that compositional fluctuationsoccur even in the absence of any underlying demixing be-haviour driven by a thermodynamic transition. In fact,the first mixture we investigate below is specifically de-signed not to demix, using a non-additive attractive crossinteraction between the two species. Thus the composi-tional fluctuations we consider are distinct from enhancedcrystal nucleation rates due for example to density fluctu-ations related to a nearby critical point [17, 18]. Clearly,our analysis falls within the concept of the Ostwald rule ofstages, suitably generalised to mixtures [19]. In this con-text, we emphasise that the crystals formed through suchcompositional fluctuations will in general not be thermo-dynamically stable.

Having argued for compositional fluctuations as a rele-vant mechanism for crystal nucleation, we turn our atten-tion to situations in which this mechanism may dominate.We begin with the Kob-Andersen (KA) binary Lennard-Jones mixture. Since its inception in 1994, this model,

2

b ca

BCCHCP

LFS

N=10,000

FCCN=100,000

HCP

FCC

LFS

FIG. 1: Structural analysis of crystallisation in the Kob-Andersen glassformer. a Particle snapshot at time t ≈ 36ταat T = 0.40 and N = 10, 000. We observe an FCC dominated crystallite of A particles. Light green and yellow particles areFCC for A and B particles respectively; light blue and orange are HCP; purple and dark pink are bicapped antiprism liquidlocally favoured structure (LFS) and dark blue and light pink particles are liquid. b Population of local structures as a functionof time reveals rapid crystallisation of FCC and HCP for N = 10, 000. BCC is found in very small quantities and the bicappedsquare antiprism liquid LFS (which is incompatible with FCC and HCP) is also shown. Inset shows that at short times (. 5τα),we cannot infer any crystal growth within the fluctuations. c Even more rapid crystallisation occurs when N = 100, 000, hereirreversible growth in the FCC population is found within one relaxation time. Inset: Liquid LFS and HCP populations arecompared with FCC population as shown in the main figure.

based on the metallic glassformer Nickel-Phosphorous,has been a mainstay of model systems with which totackle the glass transition [20]. Prized for its simplicity,speed of computation and its stability against crystallisa-tion, the KA model is among the most widely used atom-istic glassformers in computer simulations. It is only re-cently, with the advance of high performance GPU com-puting, that the KA model has been crystallized by directsimulation [14], where an estimate was made of the nucle-ation rate at a single temperature and system size. Hereinstead we carry out large-scale simulations and focus onthe mechanism for crystallisation.

Our results reveal that nucleation in the KA model isinduced by composition fluctuations as discussed above.The KA model is thus representative of systems crystal-lizing via this mechanism, and we expect our results tohave profound consequences for the glassforming abilityof mixtures, such as metallic glasses [7] and oxides [21].We illustrate the generality of our results by presentingresults for the model metallic glassformer CuxZr1−x us-ing a range of compositions from x = 0.15 to 64.5.

II. FREEZING IN THE KOB-ANDERSENMODEL GLASSFORMER

We begin the presentation of our results by studyingcrystallisation in the KA mixture using a global struc-tural analysis for ρ = 1.204 and T = 0.40 in Figs. 1band c. We use the NVT ensemble with an Nose-Hooverthermostat [22]. Figures 1b and c show, respectively,the time evolution of the population of liquid local struc-tures (bicapped square antiprisms) and crystalline struc-tures for system sizes of N = 10, 000 and N = 100, 000.

Here and henceforth we scale time by the structural re-laxation time τα. For T = 0.40, we have that the struc-tural relaxation time τα = 2.91 × 105 simulation timeunits. A snapshot of a crystal nucleus, comprised pre-dominantly of the majority A species, is shown in Fig.1a. We identify particles in liquid local structures andFCC, HCP crystalline with the topological cluster clas-sification (TCC) algorithm [23] and BCC crystalline re-gions with bond-orientational order (BOO) parameters[24, 25]. Our choice of order parameter is motivated bythe ability of the TCC to identify the liquid local struc-ture (and HCP and FCC), and we have in any case con-firmed that our results for identification of the crystalstructures are very similar between the two methods (seeMethods for simulation details and order parameters).

We see from Fig. 1b that the liquid begins to freezeon a timescale of a few structural relaxation times τα.Thus, for these parameters of T = 0.40 and N = 10, 000,it is hard to regard the KA mixture as anything but aremarkably poor glassformer. We further see that the lo-cally favoured structure (LFS) in the liquid, the bicappedsquare antiprism (pictured in Fig. 1b), transforms intothe crystal in much the same way as in one-componenthard spheres where the liquid LFS competes with thecrystal symmetry [26].

Here of course we have a binary system, but the pre-dominant crystal structures we find are FCC and hexag-onal close-packed (HCP) of the large A species only, andvery little mixed AB BCC. The lack of BCC is consis-tent with predictions that the crystal nucleation barrieris much higher relative to FCC [27] and with the equi-librium KA phase diagram [28]. For the KA model, wetherefore neglect the BCC structure and focus on theHCP and FCC crystals in the following.

3

In Fig. 1b, we see that there seems to be very lit-tle incubation time. However close inspection (Fig. 1binset) reveals that for timescales of a few τα, the fluctu-ations in crystal population are larger than the increase,so the liquid may in fact be regarded as metastable onshort timescales. In Fig. 1c, we show that upon a fur-ther increase of system size, to N = 100, 000, this shorttime metastability vanishes, and the crystal nuclei growimmediately.

We now consider the formation of critical crystal nucleiand estimate their size. In Figs. 2a and b, we showthe number of particles Nxtal in the largest connectedregion of crystal particles (HCP or FCC) for differentsystem sizes. Here we select a run with a relatively longincubation period (Fig. 2a). We see that the crystalregions are smaller than 100 particles for around 40ταbefore growing. These data enable us to infer a criticalnucleus size of approximately 50–100 particles for T =0.40. Figure 2b shows the run at N = 100,000 wherecrystal growth is immediate and thus it is difficult toinfer a critical nucleus size in this case.

Next, we consider the statistics of nucleation in theKA glassformer. From the ten runs we performed forN = 10, 000 and T = 0.40 (all of which crystallised),we determine the mean nucleation time from τX =∑ni=1 tX(i)/n, where n is the total number of simulations,

to be τX = 38.4± 26.8τα (the error is the standard devi-ation). Here tX(i) is the time when the size of the largestcrystal region reaches, and does not subsequently dropbelow, 100 particles. At higher temperatures, the driv-ing force for crystallisation is of course reduced, but thedynamics is much faster. We find that the system doescrystallise at higher temperatures (we probed up to T= 0.45) but that not all the runs do so. In this case,we determine the mean nucleation time for each statepoint following the method of Ref. [29]. In particular,we presume that nucleation is exponentially distributedin time, such that the probability of a nucleation eventhappening at time t is p(t) = 1/τX exp(−t/τX). Theprobability that a given run of length trun crystallises is

then∫ trun

0p(t)dt = 1 − exp(−trun/τX). The fraction of

runs which crystallised then gives us τX . Errors are es-timated by considering the case that one more, or onefewer, simulation runs underwent crystallisation. Whilemore sophisticated analyses have been developed, whichenable accurate determination of the critical nucleus size[30], even with the considerable computational resourceswe have used, it has only been possible to carry out tenruns per state point. This limits the extent to which wecan implement such methods.

We see from Fig. 3a that, when scaled by the re-laxation time, the time to nucleate drops rapidly withtemperature at N = 10, 000 and that well before the dy-namical divergence temperature predicted from a Vogel-Fulcher-Tamman fit to the temperature dependence ofthe relaxation time (T0 ≈ 0.30, see Supplementary Ma-terial), the nucleation time τX is expected to fall belowτα at T ≈ 0.38. Moreover in the range T . 0.43, we find

an exponential scaling with temperature, τX/τα ∼ eAT

with A ≈ 97. Of course this observation rests on onlythe four data points which we fit, but given the signif-icant magnitude of the fall in τX/τα with temperature,we are confident that, were this trend to continue, theobservation that for some T > T0, τX < τα would hold.

When we simply plot the nucleation time in simulationtime units (Fig. 3a inset), we make two observations.Firstly, the absolute nucleation time does not changehugely (around one order of magnitude) throughout thetemperature range in question, while the relaxation timechanges by three orders of magnitude. Secondly, there isan upturn at the lowest temperature that we consider,T = 0.395. The reason for the minimum in τX(T ) pre-sented in the inset of Fig. 3a is then competition betweenthe decrease in the average nucleation time (for a givensystem size) and the increase in relaxation time uponcooling though we emphasise that this is only one datapoint and more statistics would be helpful to confirm thisobservation. In any case, this is dwarfed by the increasein relaxation time, so the scaled quantity τX/τα contin-ues to drop. Turning to the system size dependence ofnucleation in Fig. 3b we find, as expected, a system sizescaling consistent with τX ∼ 1/N . Note that in Fig. 3b,we consider a single temperature, T = 0.40, so that ταdoes not enter into the scaling.

III. COMPOSITION FLUCTUATIONS

Next, we proceed to investigate the role of composi-tional fluctuations in crystallization. To quantify these,we use the order parameter illustrated in Fig. 4b. Weseek to find the largest region of liquid A particles whichis devoid of any B particles. We presume that such alarge compositional fluctuation would be most likely todrive crystallisation.

Therefore, we use the following procedure for a givensnapshot.

1. We find the A particle which is furthest from thenearest B particle.

2. We define a sphere, centred on the A particle,whose radius is its distance to the nearest B parti-cle.

3. The number of particles in the sphere, ns, is takenas the current largest compositional fluctuation.

4. We iterate to smaller AB separations and hencesmaller spheres, avoiding particles already con-tained in a previous sphere, and updating ns if alarger region is encountered.

Since there will be small noncritical fluctuations ofcrystal particles in the liquid, and we are looking for thelargest region of liquid A particles, we seek to avoid theeffects of such A particles in crystalline environments.Therefore, we accept a maximum of 10% of the particles

4

a bN=10,000 N=100,000

FIG. 2: Time-evolution of the largest crystalline region. All data are for temperature T = 0.40 and shading is to guidethe eye. a Run with incubation period of around 40τα prior to growth of crystalline region (N = 10, 000). b Immediate crystalgrowth at N = 100, 000.

a b

~N-1

FIG. 3: Nucleation times τX with respect to temperature and system size. a Nucleation time scaled by the relaxationtime τα as a function of temperature T at N = 10, 000. Line is a fit to τX/τα ∼ eAT with A ≈ 97. Inset: Non-scaled nucleationtimes τX with a curve to guide the eye (dashed line). b Nucleation times as a function of system size at T = 0.40. Linerepresents expected 1/N scaling for τX in the case of a constant nucleation rate. We determine τX as described in the text.

in the sphere to be in a crystalline environment. The timeevolution of the largest compositional fluctuation in eachsnapshot, ns, is shown in Fig. 4c. We only sample wherethe system has yet to crystallise, under our criterion ofa nucleus size of less than 100 particles. Simply becausethe system has not yet crystallised does not mean thatits properties are stationary, as shown in Figs. 1 and 2.However, it is still instructive to apply the same metricfor the larger systems as for the smaller systems (whoseproperties are stationary for timescales beyond the struc-tural relaxation time), and this we do, with the caveatthat the distributions are sampled from a non-stationarysystem.

In Fig. 4a, we see that for the KA system at T = 0.40and N = 5000, 10, 000 and 100, 000, the distribution oflargest composition fluctuations ns of liquid A particleshas a significant dependence on the system size N . Twoeffects are apparent. Firstly, the typical size of compo-sitional fluctuations increases with N . Second, the dis-tribution has a “fat tail” indicating more fluctuations of

larger ns than a symmetric distribution such as a Gaus-sian would predict. We note 50–100 particles was a roughestimate of the critical nucleus size and that fluctuationscomparable to this are seen in the tails of the distribu-tions.

IV. STATISTICS OF COMPOSITIONALFLUCTUATIONS

What can we say about the origin of the distributionof the largest compositional fluctuations in Fig. 4a? Letus suppose that the distribution of all fluctuations of Aparticles is exponential P (nA) ∝ exp(−nAλ) where nA isthe number of A particles around a given A particle thatare closer than the nearest B particle calculated for everyA particle. Here λ is the decay constant. The extremevalues of such a distribution, i.e. those fluctuations largeenough to initiate nucleation, should then follow a Gum-

bel distribution given by P [z(ns)] ∝ e−(z+e−z) in which

5

a d

N=5000

N=10,000

N=100,000

b

c

e

n s(t)

t/

Cu64.5Zr35.5

KA2:1

KA4:1

Cu25Zr75

Cu15Zr85

N=100,000N=10,000N=500

FIG. 4: Compositional fluctuations of majority A-particles. a Distributions of largest compositional fluctuations ofA-particles ns for several system sizes N at T = 0.40. Each system size is fitted with a Gumbel distribution (see text). bSchematic indicating the order parameter ns for compositional fluctuations. Central pink particle is the A particle underconsideration and dark pink particle the nearest B particle. Blue particles are A particles lying within the sphere as shown.The compositional fluctuation shown has 15 A particles. c Time-evolution of largest compositional fluctuations ns in the liquid.d Scaling of the median 〈ns〉 calculated from fitted Gumbel distributions with system size; lines are fits (see text). Shown isdata for the KA mixture at various compositions along with CuxZr1−x metallic glassformers (see Methods section for details).Circles are constant pressure data (P = 0) and squares are constant density data (ρ = 1.204). For the KA mixtures, constantpressure data is taken in the NVT ensemble fixing the mean pressure at 〈P 〉 = 0 and T = 0.80 and constant density data istaken at T = 0.40. CuxZr1−x is simulated in the NPT ensemble at T = 1270K or T = 1500K and P = 0. e Distribution for allcompositional fluctuations for several system sizes (4:1 KA mixture) at T = 0.40. The distribution is independent of systemsize above nA > 25. Dashed line denotes exponential decay with decay constant λ = 0.22 (see text).

z ≡ (ns − µ)/β where µ is the mode of the probabilitydistribution, i.e. the highest probability point and β isthe scale of the function. Furthermore, the median of theextremes of an exponentially distributed process followsm = 1/λ [lnn − ln(ln(2))] in which n is the number ofsamples [31].

In Fig. 4e, for several system sizes we plot the distribu-tion of all compositional A-particle fluctuations P (nA).The dashed line indicates that, for large nA, P (nA) ex-hibits an exponential decay virtually independent of thesystem size as expected. This motivates us to fit theGumbel distribution to P (ns) in Fig. 4a (full lines). ForN & 10, 000, the agreement is remarkable. Moreover,the median of the fitted Gumbel distributions 〈ns〉 ex-hibits a logarithmic dependence on the system size N asshown in Fig. 4d. We conclude that because the scal-ing and distribution follows the Gumbel distribution, thelargest compositional fluctuation is consistent with expo-nentially distributed fluctuations.

In fact, from Fig. 4e we find that the decay constant ofthe exponential is λ ≈ 0.22. This value of λ correspondsto a completely random distribution of A and B particlesindicating that the large regions of one species are mainlyentropic and thus present irrespective of the particularsystem; the probability to find n A-particles in a clusterof n particles, assuming indistinguishable A and B parti-cles, is P (n) = (xA)n = exp(n · ln(0.8)) ≈ exp(−0.22 ·n).

Merely demonstrating the existence and size of thesefluctuations is of course not sufficient. We need to showalso that they are sufficiently long-lived to initiate thecrystallisation as well. In order to address this ques-tion, we now consider dynamics. At T = 0.40, as notedabove, the structural relaxation time τα = 2.91 × 105.

This is wildly in excess of the nucleation time in the one-component system at these temperatures which is τX ≈41 for a system size of N = 13,500 [32]. Thus, since thelifetime of the compositional fluctuations must be on theorder of τα at least and we do not see any signs of phaseseparation, i.e. other mechanisms of crystallisation (seeFig. 4c and composition-composition correlation func-tions in the SM), we conclude that the compositionalfluctuations we identify lead to crystallisation.

Before we explore the compositional fluctuations forother systems, we provide some considerations as to thecrystallisation mechanism. One alternative possibilityis enhancement of nucleation related to density fluctu-ations. Now the liquid-gas binodal has been measuredas lying at a temperature not much less than T ∼ 0.40to which we simulate [33]. It is conceivable that somedensity fluctuations related to the proximity of liquid-gas phase separation might act to enhance nucleation, asis known for protein-like systems [17, 18]. However thesystem is not in or near the two-phase region: the den-sity of ρ = 1.204 we consider is much higher than thecritical isochore (around 0.3). In the SM we investigatebut see little evidence for density fluctuations [33]. Inany case, any such nucleation enhancement would stillneed to invoke a mechanism for A-B demixing, which isabsent. Indeed to observe demixing in similar binary sys-tems, one needs to weaken the interaction between thespecies so that it is again non-additive but weaker thanthe additive case, i.e. a positive enthalpy of mixing [34].In fact, we see very little evidence for A–B demixing (seeSM). In short, we provide evidence that the composi-tional fluctuations we identify here are unrelated to thedensity fluctuations known to enhance nucleation in (ef-

6

fective one-component) protein-like systems [17, 18].We also consider the consequences of our choice of an

instantaneous quench protocol (see Methods). In Supple-mentary Fig. S3, we see that the median of the largestregion of liquid A particles, 〈ns(T )〉, shows very little de-pendence upon temperature. Thus, as the system sam-ples from a nearly temperature independent distributionand due to the long mean nucleation time for T > 0.40(more than 100τα), we argue that our quenching protocoldoes not affect our conclusions to any significant extent.The independence with respect to temperature is intrigu-ing: we interpret this in the context that the structureof the liquid is dominated by the hard core [35], in whichcase a weak temperature dependence is expected.

V. DEPENDENCE OF FLUCTUATIONS ONSYSTEM COMPOSITION

We now consider other compositions of the KA mix-ture. The temperature independence of the composi-tional fluctuations suggests that the scaling leading tolarge compositional fluctuations may be identified at hightemperature where timescales are amenable to computersimulation, without recourse to simulations of the deeplysupercooled liquid. This suggests that it may be possibleto use our approach to predict the glassforming ability ofmixtures in the liquid state.

Usually, as above, the 4:1 KA mixture is simulated, butupon changing the composition to be more equimolar, weexpect smaller regions of pure A particles. We focus onthe 2:1 KA mixture at zero pressure and at the highertemperature of T = 0.80, where the relaxation times for4:1 and 2:1 KA are comparable [36]. We also consideredthe 3:1 composition, which turns out to lie close to the2:1 system. In Fig. 4d we see that at zero pressureand T = 0.80, the 4:1 mixture has a value of 〈ns〉 verysimilar to that at which we see crystallisation (T = 0.40).We infer that the change in pressure also has little effecton the compositional fluctuations, which is reasonable asthey are largely random, according to the exponentialdistribution (Fig. 4e).

As expected, the 2:1 KA mixture in Fig. 4d has verymuch smaller values of 〈ns〉, as its composition is closer toequimolar. To predict where crystallisation might occurwe fit each composition to a logarithmic increase as indi-cated by the dashed lines in Fig. 4d. From this we findthat the 2:1 KA system reaches the value of 〈ns(N)〉 = 31(corresponding to the 4:1 system with N ≈ 10, 000) ata system size of N = 1.2 × 109. Thus we expect that,for the mechanism of crystallisation we consider here,the 2:1 composition should be a very much better glass-former than the usual 4:1 system. We confirm this byvery lengthy simulations of the 2:1 (and 3:1) KA systemsat comparable supercoolings (i.e. T = 0.40 for KA 4:1)and N = 10,000, 100,000 and 1,000,000 where no crys-

tallisation was observed. We simulated around 9 billiontime steps for N = 10,000 and 100,000 and 2 billion timesteps for N = 1,000,000.

Note that we are only considering the crystallisationmechanism based on compositional fluctuations. Whilewe expect the mechanism to be present in all mixtures,crystallisation may be dominated by other, faster, mech-anisms. For example the 1:1 KA mixture forms a mixedBCC crystal quite rapidly [37].

VI. CRYSTALLISATION INCOPPER-ZIRCONIUM

To address whether the mechanism described abovepertains to other systems, we consider the metallic glass-former copper-zirconium. Here we use Embedded AtomModel (EAM) simulations (see Methods for more de-tails). In Fig. 5a, we show that, like the KA mixture, theextreme values of the composition fluctuations in CuZralso follow a Gumbel distribution. To determine the mag-nitude of the composition fluctuations in the liquid, weuse the higher temperatures of T = 1500K or T = 1270Krespectively, so that we can run the simulations quickly.Here we use the NPT ensemble with a Nose-Hoover ther-mostat [22], see Methods for further details.

The system size dependence of 〈ns(N)〉, where we con-sider fluctuations of the majority species is shown in Fig.4d. Again we see the logarithmic scaling, moreover com-positions such as Cu64.5Zr35.5 exhibit weaker fluctuationscompared to the KA model. Following our analysis ofthe 2:1 KA mixture, here we estimate that the systemmay be susceptible to crystallisation at a system size ofN = 3.5 × 1016 (when ns ≈ 31). However, upon chang-ing to the more asymmetric compositions Cu25Zr75 andCu15Zr85, we see a marked increase in the fluctuations.

Given these larger fluctuations, the logarithmic scal-ing in Fig. 4d would indicate that the metallic glass-former should crystallise for those more asymmetric com-positions on simulation timescales already around N =10000, assuming similar behavior to KA. To investigatecrystallisation, we run simulations at lower temperaturesfor two compositions, Cu25Zr75 and Cu15Zr85. Here thesystem was first equilibrated at T = 2000K or 1500K de-pending on the composition and then rapidly cooled tothe temperature of interest. For Cu25Zr75 and Cu15Zr85,and the temperatures at which we see crystallization T =900K and 1100K, the cooling rates are ∆T/∆t = 3.0×105

and 2.0× 105 K/ps, respectively.We find that CuZr indeed crystallises with representa-

tive runs freezing after 930τα and 95τα for Cu25Zr75 andCu15Zr85 respectively. Here we consider the BCC crystal,as the FCC and HCP are found only in trace quantities.In the snapshot in Fig. 5b, we find that the nucleus forCu25Zr75 is dominated by the majority species Zr, in amanner similar to that in Fig. 1a, although the growth is

7

a

N=1000 N=10,000

b

FIG. 5: Crystal nucleation in CuZr. a Compositional fluctuations of majority Zr atoms in Cu25Zr75 at T = 1500K. Linesare fits to a Gumbel distribution. b Nucleus in the early stage for Cu25Zr75 at T = 900K. Large grey particles are Zr (BCC),large pink are Cu (BCC), smaller grey and pink are liquid Zr and Cu respectively. Black are Zr in a local FCC environment asdetermined with bond-orientational order parameter [24]. We see that the nucleus is dominated by Zr.

more rapid in the case of this binary metallic glassfomer(See SM). Note that this higher rate of growth contrastswith slow growth previously observed in CuZr with re-spect to other metallic glassformers, for a binary crystal[38].

We thus infer that the mechanism for nucleation, atleast for these compositions is the same as that for crys-tallisation in KA. Again, like KA, other mechanisms arealso possible in which the crystal may be mixed [39, 40].However we argue that we have presented a general crys-tallisation mechanism in mixtures, which occurs in theabsence of faster, specific, crystallisation pathways.

VII. OUTLOOK

Before concluding, we consider the consequences forthe long-term stability of supercooled mixtures. We haveshown that crystal nuclei are expected in mixtures ingeneral. But by how much should they grow? By con-sidering the KA mixture, the growth of FCC nuclei ofA particles will deplete the remaining liquid of A par-ticles. This depletion will tend to slow and may evenarrest the growth of the one-component A crystals. Inthe case of the KA system, we note that if the liquidapproaches a 1:1 composition then crystallisation, not ofthe one-component FCC, but of the 1:1 composition BCCcrystal may be expected. We noted in the introductionthat the Ostwald rule of stages, generalised to mixtures,would provide pathways by which the nuclei may grow[19].

Given the small dimensions of the nuclei we find,and despite the developments we present here, our sim-

ulations are still small compared to experimental sys-tem sizes, and thus it seems reasonable to suppose thatthe final material may be comprised of nano-crystals.Nanocrystals are known to have important consequencesfor the mechanical properties of glassforming materials[41]. While this behaviour has been seen in experiments[42], our work suggests that such nanocrystals may berather prevalent in metallic glasses. Because identifyingtiny crystalline regions is hard with x-ray scattering, re-quiring techniques such as fluctuation TEM or 3D atomprobe tomography [42], it is possible that such nanocrys-tals may go undetected. The detection of such nanocrys-tals, for example with techniques such as nanobeam elec-tron diffraction [43], or fluctuation electron microscopy[44] is an exciting avenue for future research.

8

VIII. DISCUSSION AND CONCLUSIONS

We have demonstrated a general mechanism of crys-tallisation in multicomponent systems. Our large scalesimulations of the widely used Kob-Andersen model su-percooled liquid reveal that it has a fatal flaw as a glass-former which is general to mixtures. Local composi-tional fluctuations lead to regions populated only by onespecies. These regions can be larger than the criticalcrystal nucleus size of the one-component system undersimilar conditions. Nucleation in these regions is fast onthe timescale of this deeply supercooled liquid, appar-ently requiring little re-arrangement of the particles, asis known to be the case for hard spheres at deep super-cooling [45, 46]. Our findings are important as the resultswe reveal here pose a fundamental challenge for the devel-opment of glassforming materials: mixtures whose com-ponents crystallise easily are themselves inherently un-stable to crystallisation and thus ultimately compromisedas glassformers. Our findings rationalise the empiricalrule of thumb that increasing the number of componentstends to increase glassforming ablility, as the chances thata critical nucleus of one particular species are formed arereduced in that case.

We find a scaling with system size which, once param-eterised, may be used to predict the largest system whichis stable against crystallisation, and therefore the largestpieces of amorphous material which can be prepared froma given mixture. That the compositional fluctuationsare rather random and insensitive to temperature, sug-gests that simulations in the liquid at higher tempera-ture where the dynamics are much faster may be usedto predict the system size at which crystallisation maybe expected. We have demonstrated this principle usingthe 2:1 (and 3:1) KA mixtures and have predicted thatboth can reach system sizes, for comparable simulationtimes and supercoolings, very much larger than the usual4:1 mixture before crystallisation occurs. These compo-sitions may thus be used when a better glassformer isneeded in simulations than the standard 4:1 model.

The binary model we use demonstrates the use of amixture to suppress crystallisation, as is typically em-ployed in metallic and inorganic glassformers and is en-countered in vitreous magmas. Although prevalent andaccessible to computer simulation for the model systemswe consider, we expect the same mechanism will operatefor more general binary mixtures, and indeed for mul-ticomponent systems frequently employed in the questfor ever-better glassforming alloys [7]. We demonstratethis by considering the well-studied CuZr metallic glass-former, which exhibits the same behaviour. Experimen-tal evidence in support of the mechanism we find has beenseen in some metallic glasses [42] and we suggest that thepresence of such nanocrystals as we identify here wouldbe worth investigating further in metallic glasses.

Crystallisation via compositonal fluctuations thusforms a lower bound to nucleation: other mechanismsinvolving more complex crystal structures may prove

faster, as indeed seems to be the case for some models[39, 47] and for certain compositions of the Kob-Andersen[37] and CuZr [39] models considered here. Neverthelesswe have shown that liquids which rely on mixing for theirstability against crystallisation are fundamentally com-promised and provide a principle by which their glass-forming ability may be optimised.

In addition to the number of components, crystallisa-tion may be suppressed in alloys by the use of systemswith a negative heat of mixing. Here, the Kob-Andersenmixture is engineered in that way, precisely to inhibitcrystallisation. However, our analysis in section IV sug-gests that such negative heat of mixing has little impact:the Gumbel distribution assumes that the two speciesare randomly distributed in space, so given its success indescribing the statistics we infer that our analysis is ro-bust to the case where there is a negative heat of mixing.Noting that small size disparities will permit rapid crys-tallisation, and that for certain size ratios, binary crys-tals form an addition route to crystallisation as noted insection VII [13, 40], we expect that increasing the sizeratio will inhibit crystallisation. We leave the prospectof a detailed analysis of the role of size disparity for thefuture.

In addition to the metallic glasses we consider here,an intriguing case is aqueous ionic solutions. Here crys-tallisation of water occurs through segregation to ion-richand ion-poor regions, the latter being where the ice nu-cleates, which appears similar to that we observe here,for the A particles [3]. However, the various anomalies inthe thermodynamic behaviour of water, not least increas-ing fluctuations, which may be related to an (avoided)liquid-liquid transition [48, 49], mean that further studyof that system would be needed to ascertain whether themechanism we have identified here dominates water crys-tallisation in some aqueous solutions.

Acknowledgements

It is a pleasure to thank Nick Bailey, Daniele Coslovich,Daan Frenkel, Felix Hofling, Peter Harrowell, TakeshiKawasaki, Rob Jack, Ken Kelton, Heidy Mader, Kuni-masa Miyazaki, John Russo, Hajime Tanaka, and Jian-guo Wang for stimulating discussions. CPR acknowl-edges the Royal Society, and Kyoto University SPIR-ITS fund. European Research Council (ERC consolida-tor grant NANOPRS, project number 617266) for finan-cial support. TSI acknowledges support from a JSPSPostdoctoral Fellowship under which parts of this workwas carried out. TSI, TBS, and JCD are supported bythe VILLUM Foundations Matter Grant (No. 16515).The authors are grateful to Ioatzin Rios de Anda forinspirational help with the figures. LAMMPS simula-tions was carried out using the computational facilitiesof the Advanced Computing Research Centre, Universityof Bristol. RUMD simulations were out carried using theHPC cluster at Department of Science and Environment,

9

Roskilde University.

Methods

Simulation and model details. — We simulate the KAbinary mixture in the NVT ensemble (Nose-Hoover ther-mostat [22]) at ρ = 1.204 using the Roskilde UniversityMolecular Dynamics (RUMD) package [50] optimizedfor highly-efficient GPU simulations; the longest simu-lations took more than 100 days. The interatomic inter-actions of the 4:1 binary mixture is defined by vij(r) =εαβ[(σαβr )12 − (

σαβr )6] with parameters σAB = 0.80,

σBB = 0.88 and εAB = 1.50, εBB = 0.50 (α, β = A,B). The pair potential is cut-and-shifted at rc = 2.5σαβ .We employ a unit system in which σAA = 1, εAA = 1,and mA = mB = 1. We study system sizes N = 125, 250,500, 1000, 2000, 4000, 5000, 7000, 10,000, 20,000, 30,000,50,000, 80,000, 100,000 and 200,000 at T = 0.40. Severaldifferent temperatures T = 0.395, 0.40, 0.415, 0.43, 0.45are studied at N = 10,000. The protocol for studyingcrystallisation in the KA mixture is identical for all tem-peratures and system sizes studied. We equilibrate at T= 2.00 and then perform an instantaneous quench to lowtemperatures simulating between 9 and 36 billion timesteps after the quench (∆t = 0.0025). The cooling ratesare ∆T/∆t = 642, 640, 634, 628, and 620 in reducedunits. For each temperature and system size we perform10 independent quenches. Additionally, 4:1, 3:1 and 2:1KA mixtures were also simulated in the NVT ensembleat a mean pressure 〈P 〉 = 0 and T = 0.80 with N =1000, 5000, and 10,000 at which the relaxation times ofthe systems are similar.

Simulations of CuxZr1−x mixtures were performedin the NPT ensemble using a Nose-Hoover thermostatand barostat with the LAMMPS package [22, 51] andcompositions of x = 15, 20, 25, 35.5, and 64.5%. TheEmbedded Atom Model (EAM) method of Finnis-Sinclair was applied [52] simulating at a pressure P = 0(∆t = 0.002 ps). Three system sizes were simulated N =1000, 5000, and 10,000 at high temperatures for composi-tion statistics and nucleation was studied for N = 10,000.

Relaxation Time Determination. — For the KAmodel, we determine the relaxation time of the liquidτα from the self-part of the intermediate scattering

function Fs(q, t) ≡ 〈exp[iq∆r]〉 of the A-particles usingthe criterion FsA(q, τα) = 0.2; the length of the wavevector is q = 7.25. A system size of N = 1000 is usedfor these simulations to suppress nucleation but hasa minor effect on τα. In the case where we cannotmeasure τα directly in simulations due to extremely longsimulation timescales we extrapolate using a VFT fit(see SM for more details). For CuxZr1−x we obtainedthe intermediate scattering function from the Zr atomsand used a wave vector with q ≈ 26 nm−1.

Identifying Local Structure. — To detect the FCCand HCP crystals, and bicapped square antiprism liquidlocally favoured structure, we use the topological clusterclassification (TCC), employed previously to identify lo-cal structures in the KA mixture [23]. That is to say,we carry out a standard Voronoi decomposition and seekstructures topologically identical to geometric motifs ofparticular interest.

For the BCC crystal, we employ a bond-orientationalorder (BOO) parameter analysis [24]. For each par-ticle i we define complex order parameters qilm ≡1/nb

∑nbj=1 Ylm(θij , φij), where Ylm is the spherical har-

monic function with degree l and order m, θ and φ arethe spherical coordinates for the vector rij ≡ rj − ri,and nb is the number of neighbours defined from the 12nearest neighbors. We use the complex order parametersto differentiate between solid and liquid particles usingthe criteria that for at least 7 nearest-neighbor bondsthe scalar product qi6 ·q

j6/|qi6||q

j6| should be greater than

0.70 to be classified as a solid particle. qi6 is a (2l + 1)-dimensional complex vector. The identity of each solidparticle is then determined [53] using the third-order in-variant order parameters

W il ≡

l∑m1,m2,m3=0

(l l lm1 m2 m3

)Qilm1

Qilm2Qilm3

|Qil|3

,

where the term in the parentheses is the Wigner 3 − jsymbol and Qilm ≡ 1/(nb + 1)

∑nb+1i=1 qilm is the average

bond-orientational order parameter [25]. BCC particlesare identified as all solid particles having W i

6 > 0. Wechecked that the TCC and BOO methods for the detec-tion of FCC and HCP gave similar results.

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