Supplementary Informations

First-order phase transition in a model glass former:coupling of local structure and dynamics

Thomas Speck,1 Alex Malins,2, 3 and C. Patrick Royall2, 4, 5

1Institut fur Theoretische Physik II, Heinrich-Heine-Universitat, D-40225 Dusseldorf, Germany2School of Chemistry, University of Bristol, Bristol BS8 1TS, UK

3Bristol Centre for Complexity Sciences, University of Bristol, Bristol BS8 1TS, UK4HH Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, UK5Centre for Nanoscience and Quantum Information, Bristol BS8 1FD, UK

I. SIMULATION DETAILS

A. Model

We have performed extensive molecular dynamics simulations on the Kob-Andersen binary Lennard-Jones mix-ture [1]. It is composed of 80% large (A) and 20% small (B) particles possessing the same mass m. Here we study asmall system of N = 216 particles with NA = 173 and NB = 43. Particles interact through the continuous truncatedand shifted Lennard-Jones potentials

uαβ(r) =

{uLJ(r; εαβ , σαβ)− uLJ(2.5σαβ ; εαβ , σαβ), (r 6 2.5σαβ)

0, (r > 2.5σαβ)

where uLJ(r; ε, σ) = 4ε[(σ/r)12 − (σ/r)6]. The parameters read σAA = σ, σAB = 0.8σ, σBB = 0.88σ, εAA = ε,εAB = 1.5ε, and εBB = 0.5ε. As usual, we employ reduced Lennard-Jones units with respect to the A particles, i.e.,we measure length in units of σ, energy in units of ε, time in units of

√mσ2/ε, and we set Boltzmann’s constant to

unity.

B. Molecular dynamics simulations

Newton’s equations of motion are integrated through the velocity Verlet algorithm with time step 0.005 usingLAMMPS [2]. We use the massive stochastic collision thermostat to control temperature, where every 300 time steps(corresponding to ∆t = 1.5) we reassign all velocities drawn from a Maxwell-Boltzmann distribution. Care has to betaken to subtract the center-of-mass velocity. For energy minimization (in order to obtain the inherent state positions)we employ the FIRE algorithm [3] with a maximal number of 100 iterations.

C. Importance sampling

Just as standard Monte Carlo simulations perform importance sampling of configurations, here we carry out im-portance sampling of trajectories. Specifically, we harvest new trajectories using the half-moves from transition pathsampling (TPS) [4]. Accepting or rejecting a move X → X ′ according to the usual Metropolis criterion

min

{1,e−w(C[X′])

e−w(C[X])

}generates an ensemble of trajectories with weight

P [X] ∼ P0[X]e−w(O[X])

(the TPS moves preserve the equilibrium weight P0). Here, O[X] is a dynamical order parameter that calculates areal number from trajectory X (in this work either C or N ); and w(x) is the weight function. We use two weightfunctions: (i) The quadratic function

wα(x) =k

2(x− xα)2

2

to sample distributions. To enhance sampling we exchange trajectories between Nrep replicas with different xα. As themost simple approach we use equidistant umbrellas and a single strength k such that the barrier between umbrellasis 1. For every TPS step we attempt Nrep × 84 swaps between all of the replicas, not only neighbors. The curves inthe two ensembles presented in Fig. 2 in the main text are obtained through reweighting with factors e−sx and eµx,respectively. (ii) For the data of Fig. 3 in the main text (and Fig. 2 below) we use wα(x) = −µx to sample trajectoriesat constant dynamical chemical potential µ. We use a chain of Nrep = 15 replicas from µ = 0 to µ = 0.014, wherereplicas are clustered around µ∗ ' 0.0056.

II. STRUCTURAL ANALYSIS

A. TCC Clusters

Our structural analysis of the dynamical s-ensemble includes the following Topological Cluster Classification (TCC)clusters: 9X, 9A, 11A, 13A, and the number of mobile particles as defined in the main text. The clusters are depictedon the right side in Fig. 1. In particular, 9A is comparable in size to 11A but has a somewhat lower population.The figure shows the mean population of the different cluster types across the dynamic transition for the s and theµ-ensemble. It includes the data presented in Fig. 2 of the main text. Note that 13A, representing local icosahedralordering, is virtually absent over the whole range. The population of 11A increases upon crossing the transition. Both9A and 9X are negatively correlated with 11A, i.e., their population decreases. The response is much stronger in theµ-ensemble (right panel), most likely due to competition with 11A, which increases rather strongly to more than 30%.

B. Crystalline structures

Besides the cluster analysis we have measured other structural order parameters that are routinely employed todescribe crystalline states. First, we have quantified bond-orientational order following Steinhardt et al. [5]. To thisend a complex vector

qlm(k) ≡ 1

Nb

Nb∑k′=1

Ylm(θkk′ , φkk′)

is defined for every particle k, where Ylm are spherical harmonics and the angles θkk′ and φkk′ describe the orientationof the displacement vector between particles k and k′ with respect to a fixed reference frame. The sum is over allNb neighbors in the first coordination shell of A particles with radius 1.42. The normalized scalar product of the

4 8 12

11A

〈n〉 s

,〈c〉 s

s × 103

exc

0.00

0.20

9A0.40

0.60

13A

0.80

1.00

9X

0 4

11A

8 12

exc

〈n〉 µ

,〈c〉 µ

µ× 103

9A

0.00

0.20

13A0.40

0.60

0.80

1.00

0

9X

9X

11A

9A

13A

FIG. 1: Population of different clusters across the dynamical transition from the active to the inactive phase for K = 200.Black line is the concentration of mobile particles. The vertical dashed lines indicate the values of s∗ and µ∗, respectively.

3

CNA-155

0

0.5

biased

1

1.5

2

2.5

3

-0.2 0 0.2 0.4 0.6 0.8 1pr

obab

ility

ψ6

(a)

0

500

1000

1500

2000

2500

0 0.1 0.2 0.3

freq

uenc

y

fraction of bonds

(b)CNA-142

unbiased

FIG. 2: (a) Distribution of the bond orientational order parameter ψ6 in the unbiased (µ = 0, solid red line) and biasedensemble (µ = 0.014, dashed black line). (b) Common neighbor analysis (CNA) for two motifs representing crystalline fcc order(CNA-142) and icosahedral ordering (CNA-155).

q-vectors is

S(k, k′) ≡∑6m=−6 q6m(k)q∗6m(k′)√∑6

m=−6 |q6m(k)|2√∑6

m=−6 |q6m(k′)|2.

The average over neighbors

ψ6(k) ≡ 1

Nb

Nb∑k′=1

S(k, k′) (1)

is the bond order parameter. It is ψ6 = 1 for a particle in a perfect crystal and acquires a broad distribution withmean 0.2 − 0.3 for particles in a disordered environment. Results are shown in Fig. 2(a) for the unbiased ensembleand the biased ensemble at fixed µ = 0.014. Both curves are typical for a liquid and demonstrate that no long-rangeorder has developed. There is a minute shift in the inactive phase towards smaller values, i.e., a more unorderedenvironment.

Second, we have performed a common neighbor analysis (CNA) [6, 7] for two structural motifs: CNA-142 andCNA-155. Following the naming scheme CNA-1mb of Ref. 6, for a bond of nearest neighbors m is the number ofcommon neighbors and b is the number of bonds between common neighbors. CNA-142 is present in an fcc lattice(which forms part of the zero-temperature crystal of the KA mixture [8]), whereas CNA-155 quantifies icosahedralordering. Fig. 2(b) confirms that no crystallization occurs in the inactive phase. Even though the relevant structure11A is not an icosahedra, the formation of clusters in the inactive phase shows up in the distribution of CNA-155,which is shifted to higher bond fractions.

III. SCALING

In Fig. 3(a) the fluctuations of the density c of mobile particles is plotted in both ensembles. The peak in theµ-ensemble is both higher and more narrow, indicating a stronger response to structural changes.

The reason why we have to study rather small systems is two-fold: While it of course reduces the computationaleffort to harvest many thousands of trajectories, the principal reason is that the probability of the inactive trajectorieswe are after decreases exponentially with system size and trajectory length. One is of course confronted with the sameissue in more traditional phase transitions, where one can resort to finite size scaling [9]. While such a powerful toolis not yet available for dynamical phase transitions, we can nevertheless demonstrate that the scaling with space-timevolume NK [Fig. 3(b)] is compatible with a first-order transition, i.e., the peak susceptibility χ∗ diverges, indicatinga jump of the corresponding order parameter. The two susceptibilities are defined as

χ(s) = 〈c2〉s − 〈c〉2s, χ(µ) = 〈n2〉µ − 〈n〉2µwith the peak value χ∗ being χ(s∗) or χ(µ∗), respectively. The data for the longer trajectories for the s-ensemble istaken from Ref. 10. Note that the peak height is somewhat less susceptible to a change of the system size comparedto the trajectory length.

4

6 8 10

χ∗

NK × 10−4

(b)

(µ)

(s)

0

10

20

30

40

50

60

70

80

0 4 8 12

〈δc2〉 s,µ

s, µ× 103

K = 100(a)

0

50

100

K = 200

150

200

250

300

0

N = 216

2 4

N = 343

FIG. 3: (a) Fluctuations 〈δc2〉 = 〈c2〉s−〈c〉2s of the mobile particle density in the s-ensembles (dashed lines) and the µ-ensemble(solid lines). (b) Scaling of the peak height χ∗.

IV. DYNAMICAL TRANSITION

In Fig. 4(a) the data from Fig. 2(d) of the main text for different trajectory lengths K is replotted with the x-axisshifted and rescaled. Close to the peak all three curves collapse as expected from the central limit theorem. Thetails of the distributions are non-Gaussian, but only for the longer trajectories K = 100 and K = 200 a non-convexshape is observed. In longer trajectories (higher K) it thus becomes relatively easier to form 11A clusters, whichshows that dynamical correlations play a role. This indicates that not only the absolute number of particles withinclusters is relevant, but also dynamical properties like the cluster’s lifetime. There is a minimal trajectory length Kmin

below which we do not observe a phase transition between active and inactive phases. The system size dependenceof Kmin remains to be studied in more detail; leaving open the possibility that if Kmin → 0 there is a conventionalorder-disorder transition with the population of 11A clusters serving as an order parameter.

V. BIASING WITH 9A CLUSTERS

Is the formation of 11A clusters necessarily related to slow dynamics and the dynamic phase transition, or is itsufficient to induce any kind of local structure? As a “control experiment” we have performed simulations in whichthe field µ couples to particles bound in 9A clusters, the size of which are comparable to 11A clusters (see the inset ofFig. 4(b) and Fig. 1). As shown in Fig. 4(b), the distribution for the population n is Gaussian to a very good degreeover a wide range [note the scale of the y-axis, which is more than three times that of Fig. 4(a)]. In particular, no

0.4

(c)

K = 100

-25

-20

-15

-10

-5

0

9X

-0.5 0

11A

0.5 1

exc

1.5 2

9A

2.5

lnp(n)

13A

√K(n − nmax)

(a)

-80

-60

-40

-20

0

0 0.02 0.04

K = 200

0.06

0.6

0.8

1

-20 0 20

K = 20

40

〈n〉 µ

,〈c〉 µ

0.08 0.1

lnp(n)

n

(b)

0

0.2

µ9A × 103

FIG. 4: (a) As Fig. 2(d) of the main text but shifted by nmax ' 0.092 and scaled by a factor√K. The collapse of the

curves close to the peak confirms the scaling predicted from the central limit theorem. (b) Probability distribution p(n) forthe population n of 9A clusters (see inset) for K = 100. The dashed line is a parabola, demonstrating that the distribution isGaussian to a very good degree. (c) Mean density of mobile particles and population of the different clusters in the dynamicalensemble where the biasing field µ9A couples to 9A clusters.

5

phase transition can be associated with such a uni-modal distribution. This is emphasized in Fig. 4(c), which clearlydemonstrates the absence of a phase transition. Note the strength of the biasing field, which is about four times thatof Fig. 1. It is thus rather hard to induce the formation of this local motive. Moreover, the 9A population and thedensity of mobile particles c are weakly positively correlated, quite in contrast to the behavior observed in Fig. 1.

[1] W. Kob and H. C. Andersen, Phys. Rev. Lett. 73, 1376 (1994).[2] S. Plimpton, J. Comp. Phys. 117, 1 (1995), available at http://lammps.sandia.gov.[3] E. Bitzek, P. Koskinen, F. Gahler, M. Moseler, and P. Gumbsch, Phys. Rev. Lett. 97, 170201 (2006).[4] C. Dellago, P. G. Bolhuis, and P. L. Geissler, Adv. Chem. Phys. 123, 1 (2002).[5] P. J. Steinhardt, D. R. Nelson, and M. Ronchetti, Phys. Rev. B 28, 784 (1983).[6] J. D. Honeycutt and H. C. Andersen, J. Phys. Chem. 91, 4950 (1987).[7] H. Tsuzuki, P. S. Branicio, and J. P. Rino, Comp. Phys. Comm. 177, 518 (2007).[8] J. R. Fernandez and P. Harrowell, Phys. Rev. E 67, 011403 (2003).[9] K. Binder, Rep. Prog. Phys. 50, 783 (1987).

[10] T. Speck and D. Chandler, J. Chem. Phys. 136, 184509 (2012).