Ultrastable glasses from in silico vapour deposition · Supplementary Information 6 Figure S4: Individual three-dimensional radial distribution functions (rdf) at T = 0.45 and T =

Ultrastable glasses from in silico vapour deposition

Sadanand Singh, M. D. Ediger and Juan J. de Pablo

Nature Materials 12, 139–144 (2013).

In the version of the Supplementary Information originally published, in Table T1, the values of the energies EIS and <U>, and the Q6 parameter corresponding to ordinary glasses, and in Table T2, the <U> values, have been corrected to reflect the reported densities. These errors have been corrected in this file 13 May 2014.

© 2014 Macmillan Publishers Limited. All rights reserved

Ultrastable glasses from in silico vapor deposition

Sadanand Singh,1 M. D. Ediger,2 and Juan J. de Pablo1,3∗

1Department of Chemical and Biological Engineering2Department of Chemistry

University of Wisconsin-Madison, Madison, WI 53706, USA3Institute of Molecular Engineering

University of Chicago, Chicago, IL 60637, USA

∗To whom correspondence should be addressed;E-mail: [email protected].

Thermodynamic Properties

Vapor-deposited glass are grown for substrate temperatures in the range betweenTs = 0.15 and 0.8. To compare the energy (potential and inherent structure) ofstable and ordinary glasses, vapor-deposited glass samples are cooled from Ts toT = 0.05 at a cooling rate of 3.33× 10−6. The particles in the vapor-depositedglass sample are divided into two groups, namely the “core” (defined as particleswhich are at least 8 σAA away from both the substrate and the free surface) andthe “surface” (defined as particles which are within 5 σAA of the free surface).Unless otherwise specified, the reported properties of vapor-deposited glasses referto those of “core” particles.

Table T1 gives the values of energy per particle (both potential and inherentstructure) for ordinary glasses prepared at different cooling rates, and for vapor-deposited glasses prepared at different substrate temperatures Ts. For the ordinaryglasses, the potential energies reported in the Table are calculated at a temperatureT = 0.3. For the vapor-deposited glasses, the potential energies reported corre-spond to the as-deposited sample. All inherent energies correspond to the inherentstructures at a temperature T = 0.05. For ordinary glasses, the instantaneous con-figurations for which the inherent structure was calculated were selected at the endof the temperature scan, i.e. at T = 0.05. All vapor-deposited glass samples werecooled to a temperature of 0.05 at a cooling rate of 3.33×10−6 to get the inherentstructure at T = 0.05.

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Supplementary Information 2

Figure S1: Representative configuration of a binary stable glass sample preparedby vapor deposition.

The density of glasses prepared by vapor deposition is higher (1.22−1.26) thanthat of the ordinary glasses prepared at constant volume at a density of ρ = 1.2. Tostudy whether the enhanced stability of vapor-deposited glasses is a result of theirhigher density, we also simulated ordinary glasses with densities of ρ = 1.22, 1.24and 1.26 using a range of cooling rates. The average inherent structure energies ofthose glasses at a temperature of T = 0.05, along with their corresponding potentialenergies at a temperature of T = 0.3, are given in Table T2. As the results onthat Table show, the energies of denser ordinary glass samples are only marginallydifferent than those of the ordinary glass prepared at ρ = 1.2, indicating that densityalone cannot explain the low energy of stable glasses.

Radial Distribution Functions and Structure Factor

A common measure of local structure is provided by the radial distribution function(rdf), which measures the particle density as a function of distance from a centralatom. The structure factor, s(q) is related to the rdf by

s(q) = 1+4πρ

q

∫ ∞

0[g(r)−1]sin(qr)dr, (E1)

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Table T1: Thermodynamic properties of stable and ordinary glasses. Error bars inDensity, EIS, �U� and Q6 are less than 0.005, 0.01, 0.003 and 0.0001, respectively.

Process Cooling rate or Density EIS �U� Q6substrate temperature (1/σ 3

AA) (εAA) (εAA) Parameter3.33×10−3 1.2 -8.207 -7.775 0.02573.33×10−4 1.2 -8.231 -7.789 0.0257

Ordinary 3.33×10−5 1.2 -8.248 -7.804 0.0261Cooling 3.33×10−6 1.2 -8.266 -7.823 0.0264

3.33×10−7 1.2 -8.284 -7.841 0.02663.33×10−8 1.2 -8.306 -7.857 0.0268

Ts = 0.18 1.18 -8.20 -7.76 0.021Ts = 0.2 1.21 -8.32 -7.81 0.021Ts = 0.25 1.23 -8.53 -7.89 0.020

Vapor Ts = 0.27 1.25 -8.57 -7.91 0.019Deposition Ts = 0.3 1.26 -8.61 -7.93 0.019

Ts = 0.32 1.25 -8.55 -7.88 0.019Ts = 0.35 1.24 -8.41 -7.83 0.019Ts = 0.4 1.22 -8.37 -7.70 0.020Ts = 0.45 1.2 -8.27 -7.57 0.020

where q represents the wave vector. Figure S2 shows the structure factor for anordinary glass (prepared at a cooling rate of 3.33 × 10−8) and for a stable glass(prepared at a substrate temperature of Ts = 0.30). The two glass samples exhibitidentical structure factors.

It is of interest to consider the structural details of the stable and ordinaryglasses in different directions, in order to assess whether the substrate introducesany anisotropy in the samples. A two-dimensional radial distribution function canbe defined as:

g(r) =n(r)

4π�ρ�Nr∆r(E2)

where n(r) is the histogram generated by counting the number of times the distancebetween two particles lies between r and r+∆r, N is the total number of particlesin the simulation box, and �ρ� is the bulk density of the system. Results for thecorresponding radial distribution functions are shown in Fig S3.

Given the heterogenous nature of the binary systems considered here, it is in-structive to examine the relative distributions of A-A, A-B and B-B particles. Fig-ure S4 shows the corresponding three-dimensional radial distribution functions forthe stable and the ordinary glasses at T = 0.45 and 0.3. Again, we find little evi-






6 9 12 15 18 21q (in σAA

-1 units)

0

1

2

3

4

s (q)

Ordinary T = 0.3Stable T = 0.3

Figure S2: Structure factor for all particles for the ordinary glass prepared at a cool-ing rate of 3.33×10−8 and for the stable glass prepared at a substrate temperatureof Ts = 0.3. Both results correspond to T = 0.30.

1 2 3 4r2D (in σAA units)

0.9

0.95

1

1.05

1.1

g 2D (r

)

OrdinaryStable (in XY)Stable (in YZ)Stable (in ZX)

Figure S3: Two-dimensional radial distribution functions of ordinary and stableglass samples. All results correspond to T = 0.3. The ordinary glass sample(shown in black) was prepared at a cooling rate of 3.33× 10−8. The stable glasswas grown at a substrate temperature of Ts = 0.3; Z represents the direction normalto the substrate.






Table T2: Potential energy of ordinary glass prepared at different densities. Theerror bars in the energy are of smaller than 0.005.

�U� (εAA)

Cooling Rate ρ = 1.2 ρ = 1.22 ρ = 1.24 ρ = 1.263.33×10−3 -7.775 -7.792 -7.798 -7.7833.33×10−4 -7.789 -7.810 -7.816 -7.8063.33×10−5 -7.804 -7.830 -7.839 -7.8293.33×10−6 -7.823 -7.849 -7.857 -7.8503.33×10−7 -7.840 -7.867 -7.877 -7.8733.33×10−8 -7.857 -7.888 -7.900 -7.895

dence of structural differences between the two systems beyond statistical noise ofthe simulation.

Structural Properties

Bond Order Parameters

Spherical harmonic order parameters, such as Q4, W4, Q6 and W6, characterize theorientational correlation of the vectors joining the centers of mass of neighboringmolecules. These parameters are near 0 for the liquid phase [34]. To define theseorder parameters first, the complex vector qlm(i) of particle i is defined as

qlm(i) =1

Nb(i)

Nb(i)

∑j=1

Ylm(ri j). (E3)

Here, Nb(i) is the number of nearest neighbors of particle i - defined as particleswithin a spherical volume with radius r = 1.8 (corresponding to second maximumin the radial distribution function), l is a free integer parameter, and m is an in-teger that runs from m = −l to m = +l. The functions Ylm(ri j) are the sphericalharmonics and ri j is the vector from particle i to particle j. A set of parameters,which hold the information of the local structure, are local bond order parameters,or Steinhardt order parameters [34], defined as

ql(i) =

√

√

√

√

4π2l +1

l

∑m=−l

|qlm(i)|2. (E4)






Figure S4: Individual three-dimensional radial distribution functions (rdf) at T =0.45 and T = 0.3. (A) A-A rdf for the ordinary glasses and the vapor-depositedstable glasses. Results at T = 0.45 have been shifted above the curves at T = 0.3for clarity. (B) A-B rdf for the ordinary glasses and the vapor-deposited stableglasses. Results at T = 0.45 have been shifted above the curves at T = 0.3 forclarity. (C) B-B rdf for the ordinary glasses and the vapor-deposited stable glasses.






Depending on the choice of l, these parameters are sensitive to different crystalsymmetries. Another class of local bond order parameters can be defined as

wl(i) =∑

m1+m2+m3=0

(

l l lm1 m2 m3

)

qlm1(i)qlm2(i)qlm3 (i)

(

l∑

m=−l|qlm(i)|2

)3/2 . (E5)

Here, the integers m1, m2 and m3 run from −l to l, but only combinations with m1+m2+m3 = 0 are allowed. The term in parentheses is the Wigner 3− j symbol [35].For liquids all such order parameters have a value near 0. A larger value correspondto higher crystalline character.

Liquids and crystals can be more easily differentiated in terms of averagedvalues of these order parameters. The locally averaged parameters ql(i) and wl(i)can be defined as

ql(i) =

√

√

√

√

4π2l +1

l

∑m=−l

|qlm(i)|2 (E6)

wl(i) =∑

m1+m2+m3=0

(

l l lm1 m2 m3

)

qlm1(i)qlm2(i)qlm3(i)

(

l∑

m=−l|qlm(i)|2

)3/2 (E7)

where, qlm(i) =1

Nb(i)

Nb(i)

∑k=0

qlm(k). (E8)

Here, the sum from k = 0 to Nb(i) runs over all neighbors of particle i plus theparticle i itself. The overall Ql and Wl values of a system are defined as the valuesof ql(i) and wl(i) averaged over all particles. Commonly, to differentiate typicalcrystal structures and liquids, a value of l = 4 and l = 6 have been used in litera-ture [36].

We have calculated these order parameters for different glass samples by av-eraging over more than 10000 independent structures at any given condition. Thevalues of Q6 for different binary LJ glass samples have been listed in Table T1.Other order parameters like Q4, W4 and W6 also have values in the range of 0.01to 0.03. These near zero values confirm the absence of any significant crystallineorder in all glass samples.

To analyze distributions of these order parameters in more detail, the distribu-tion of q6(i) was studied in more detail. Figure 3 of the main text shows the results






2 4 6 8 10Size of Clusters

# of

Clu

ster

s

Ordinary GlassVapor-Deposited GlassT = 0.8

T = 0.5

T = 0.4

T = 0.35

T = 0.3

T = 0.2

Figure S5: Number of ordered clusters of particles (on a logarithmic scale) withlocal q6(i) > 0.12, as a function of cluster size at different temperatures. Datahave been collected over 1000 frames across 10 independent trajectories. At theoptimal substrate temperature maximum difference can be observed between thevapor-deposited (red) and ordinary (black) glass samples. The curves at differenttemperatures have seen shifted for clarity.






for the local distribution of q6(i). The local distribution was further analyzed byperforming a cluster analysis using the DBSCAN [37] algorithm. For the clusteranalysis, inherent structures of a glass sample at a given temperature were consid-ered. For a given temperature, we have selected 1000 inherent structures from 10independent simulations, hence encompassing statistics from over 10,000 inherentstructures. The clusters were designated based on their local q6(i) values (> 0.12)and their position in the simulation box. As, shown in Figure 3 of the main text,the most stable glass formed by the vapor deposition process at a substrate temper-ature Ts = 0.3 shows significant differences from “ordinary” glass samples. Theeffect of temperature on clusters can be seen in Figure S5. At high temperatures(T > 0.6), the vapor-deposited and ordinary glasses show identical clusters of localq6(i). As temperature in lowered, the vapor-deposited glasses show significantlydifferent clusters as compared to ordinary glasses. The difference increases untilan optimal value of deposition temperature Ts is reached, below which, once againthe vapor-deposited and ordinary glasses show a similar behavior.

Tetrahedricity and Octahedricity

Local structures of glasses have also been studied in terms of higher order structuralfeatures like Voronoi-Delaunay analysis [38]. By definition, the Voronoi polyhe-dron (VP) of an atom is that region of space that is closer to the given atom thanto any other atom of the system. A dual system spanning space is formed by theDelaunay Simplices (DS). These are tetrahedra formed by four atoms that lie onthe surface of a sphere that does not contain any other atom. Both VP and DSfill the space of the system without gaps and overlaps. In our calculations we doa Voronoi-Delaunay tessellation of the inherent structures of different glasses bythe algorithm described in Reference [39]. Two main types of DS are predominantin monoatomic glasses [40, 41], namely DS similar to ideal tetrahedron (tetra-hedricity, Γ) and DS resembling a quarter of a regular octahedron (octahedricity,O). Similar to the definition in Reference [38], the two quantities are defined asfollows.

Γ = ∑i< j

(li − l j)2

15l2 , and O =

{

6

∑m=1

gmO−1m

}−1

, (E9)






where

l =16

6

∑m=1

lm, gm =e3δm/σ

6∑

i=1e3δi/σ

, δm =lm − l

l, σ =

[

16 ∑δ 2

m

]1/2,

and Om = ∑i< j; i, j �=m

(li − l j)2

10l2 + ∑i�=m

(

li − lm/√

2)2

5l2 .

Here li denotes the length of the ith edge of the DS. A value of Γ = 0 represents aperfect tetrahedron and a value of O = 0 represents a perfect quarter-octahedron.Higher values represent more imperfect/distorted local structural features.

As described in the above section on the calculation of local q6(i), the localΓ(i) and O(i) (defined as average Γ and O of all DS to which a particle i belongs)were calculated for 10,000 inherent structures. Similar to the previous analysis, thecluster analysis was performed. Cluster analysis was performed over glass sampleswith Γ(i)< 0.025 and O(i)< 0.06. As seen in Figure S6 (a) and (b), results showa similar trend as the local q6(i).

Voronoi Polyhedra

The local structure of glasses can be also quantified in terms of the distribution of“Voronoi volume” [42, 43, 44]. Specifically, the local nearest-neighbor coordina-tion can be resolved through a Voronoi tessellation of space [45, 46, 47, 48] thatallows one to characterize the local, particle-level environment. The Voronoi poly-hedron associated with any given particle is characterized in terms of two quanti-ties: the number of faces (equal to the coordination number), and the number ofedges on each face. Following the notation used by Sheng et al. [42, 44], we usethe Voronoi index �n3,n4,n5,n6, . . .�, where ni denotes the number of i-edged facesof the Voronoi polyhedron of a given particle and ∑i ni is the total coordinationnumber of that particle. For example, the designation �0,2,8,0� indicates a coor-dination number of 10 with two 4-edged faces and eight 5-edged faces. The des-ignation �0,2,8,1� indicates a coordination number of 11 with two 4-edged faces,eight 5-edged faces and one 6-edged face. The designation �0,0,12,0� indicates acoordination number of 12 with twelve 5-edged faces.

The results for the ordinary glasses produced in this work exhibit the sametrends reported by Sheng et al. [42] for Ni80P20 metallic glasses. Most of thepolyhedra centered around ‘B’-type particles have coordination numbers between9 and 12. Figure 3(B) of the main text shows a distribution of the major types ofB-particle polyhedra having coordination numbers 10, 11 and 12, labeled using the






Figure S6: (A) Number of ordered clusters of particles (on a logarithmic scale) withlocal Γ(i)< 0.025, as a function of cluster size at temperature T = 0.3. Data havebeen collected over 1000 frames across 10 independent trajectories. (B) Similarly,number of ordered clusters of particles (on a logarithmic scale) with local O(i) <0.06, as a function of cluster size at temperature T = 0.3.






Voronoi index format defined above. The inset of the figure shows the distributionof coordination number in the system. Stable glasses exhibit primarily a singletype of polyhedron for each coordination number. Ordinary glasses exhibit severaltypes of polyhedra. Stable glasses have a more homogenous local structure thanordinary glasses.

Coslovich [44] found that, above Tg, as the liquid is cooled the abundance ofpolyhedra centered around ‘B’-type particles of types 〈0,2,8,0〉, 〈0,2,8,1〉, and〈0,0,12,0〉 gradually increases. At Tg that growth slows down considerably, andthe number of such regular polyhedra remains almost constant. At Tg, other typesof irregular polyhedra such as 〈0,2,6,2〉, 〈1,2,7,1〉, and 〈0,2,8,2〉 are also rel-atively frequent in the ordinary glass. In contrast, for stable glasses, the abun-dance of 〈0,2,8,0〉, 〈0,2,8,1〉, and 〈0,0,12,0〉 polyhedra continues to grow belowTg, down to the temperature corresponding to the optimal substrate temperatureTs = 0.3. In such stable glasses, there are essentially no irregular polyhedra oftypes 〈0,2,6,2〉, 〈1,2,7,1〉, and 〈0,2,8,2〉.

Figure S7 shows the percentage content of 〈0,2,8,0〉 B-particle polyhedra as afunction of temperature for both ordinary vapor-deposited glasses as a function oftemperature. For ordinary glasses, at higher temperatures (T KB/εAA > 0.4), con-sistent with the results of Coslovich [44], the percentage content of such polyhedraincreases rapidly. At lower temperatures, in the glassy regime of the material, thegrowth of such polyhedra slows down as the temperature is lowered. In contrast,for the vapor-deposited glasses, the percentage content of 〈0,2,8,0〉 polyhedra in-creases monotonically until, for the most stable glass formed at a substrate temper-ature Ts = 0.3, it reaches a maximum. Vapor-deposited glasses at a lower substratetemperature are not as stable and have much lower content of these polyhedra. Adirect correlation appears to exist between the stability of the glass and its structure.

The homogeneity of stable glasses is also evident in our results for clusters oflocal q6(i), tetrahedricity Γ(i) and octahedricity O(i), described in previous sec-tions. In particular, the Voronoi indices for polyhedra on particles belonging to thelarger clusters that only arise in ordinary glasses, are precisely of the type that isabsent in stable glasses.

Debye-Waller Factors

The molecular mobility of vapor-deposited glasses is determined by measuringthe Debye-Waller factor, denoted by 〈u2(z)〉, as a function of the distance (z)from the substrate layer. Debye-Waller factors are common in the glass litera-ture [49, 50, 51] because they can be extracted from both simulations and neutron






0.2 0.4 0.6 0.8 1T kB/εAA

15

20

25

30

% c

onte

nt o

f <0,

2,8,

0> p

olyh

edra

OrdinaryVapor-deposited

Figure S7: Percentage content of �0,2,8,0� polyhedra centered around ‘B’-typeparticles as a function of temperature for ordinary glasses (black circles) and asa function of the substrate temperature for vapor-deposited glasses (red squares).As the temperature is reduced, the percentage of �0,2,8,0� polyhedra increasesgradually until the simulated glass transition temperature is reached (T ≈ 0.4).Below that temperature, in ordinary glasses the percentage of these polyhedra isnearly constant. In contrast, for the vapor-deposited glasses, the percentage of�0,2,8,0� polyhedra increases steadily until the most stable glass is formed at Ts =0.3. At substrate temperatures below Ts < 0.3, the percentage of such polyhedra isagain comparable to that observed in ordinary glasses, suggesting that a correlationexists between the stability of the vapor-deposited glasses and molecular structureas measured by the number of �0,2,8,0� polyhedra.






scattering experiments. Furthermore, they are often interpreted as measures ofthe free volume surrounding a given particle [51] or, within the framework of theharmonic model, as measures of the stiffness of a glassy matrix [49, 50]. Vari-ous approaches have been adopted to calculate this parameter [49, 51, 52], one ofwhich is to calculate the mean-squared atomic displacement after the beginningof the caging regime [51] as a measure of free volume in the system. That is theapproached followed here.

0 10 20 30 40Z (in σAA units)

1

10

100

1000

<u2 >

/ <u2 > bu

lk

Figure S8: The Debye-Waller factors normalized with respect to the core valueof the stable glass sample as a function of distance from the substrate. Differentcurves show values at transient times during the preparation of stable glasses.

Figure S8 shows the corresponding Debye-Waller factor profiles, as a func-tion of position along the vapor-deposited glass, at different times during the glasspreparation process. The results represent an average over the plane parallel to thesubstrate, and over slabs of thickness 2σAA at a substrate temperature Ts of 0.3.One can see that the Debye-Waller factors at the surface are approximately 2 to 3orders of magnitude larger than in the core of the material.

On comparing the absolute Debye-Waller factors of the ordinary glass withthat of the most stable glass prepared at a substrate temperature of Ts = 0.3, it isfound that the Debye-Waller factors in the stable glasses have reduced significantly(

〈u2〉ordinary/〈u2〉stable = 7.64)

. A universal scaling between structural relaxationτα and Debye-Waller factors has been proposed by Larini et al. [53],

τα ∝ exp

(

a2

2〈u2〉 +σ 2

a2

8〈u2〉2

)

, (E10)






where a2 and σ 2a2 are average and variance in the square of a molecular displace-

ment required for local structural relaxation. Using such a scaling, we estimatethe alpha relaxation time for our most stable glass at 1024 times the value for theordinary glass. This is consistent with our other, independent estimates.

Swallen et al. [54] have proposed that the enhanced stability of vapor-depositedglasses is due to the enhanced mobility of surface molecules, which can undergosignificant relaxations as new molecules continue to arrive at the glass-vacuuminterface during the growth process [55]. The deposition occurs sequentially andevery molecule being deposited is part of the surface at some point during thatprocess. In this view, the enhanced surface dynamics enable sampling of lower-energy inherent structures. To examine this hypothesis, a local mobility profilecan be generated in terms of local Debye-Waller factors calculated at small timescales [51]; as discussed above, the mobility of surface particles is indeed muchhigher than that of the core of the stable glass throughout the entire depositionprocess, consistent with Swallen et al.’s [54] proposed mechanism. This finding isalso consistent with observations by Shi et al. [56], who have proposed that surfaceparticles can sample the energy landscape more effectively than those in the core.

Mechanical Properties

The mechanical properties of both ordinary glasses (prepared by slow cooling ofthe liquid at a cooling rate of 3.33 × 10−8) and stable glasses (vapor-depositedglass prepared at a substrate temperature Ts = 0.3) were calculated using an uniax-ial strain deformation. For the ordinary glasses, we first calculated the pressure ofthe NV T glass at ρ = 1.2 and at T = 0.3. Using the calculated values of pressureas a set-point, an additional NσT run was performed. An additional stress was ap-plied along one of the directions to deform the material. Similar calculations wereperformed in each of the three Cartesian directions. The strains were sufficientlysmall for the systems to remain in the linear response regime. For stable glasses,the pressure was controlled only in the plane of the substrate. In the direction nor-mal to the substrate, a repulsive wall potential was introduced on top of the film inorder to apply a stress. The stress-strain response in the plane of the substrate wasmeasured in a similar manner to that employed for the ordinary glasses.

Figure S9 shows the average normal uniaxial stress-strain response for ordi-nary and stable glasses. From the slope of these curves we can extract an elasticmodulus. We find that stable glass samples have a modulus C = 54.9 that is about18% higher than that of the ordinary glass, which is C = 46.2.






0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008strain

0

0.05

0.1

0.15

0.2

0.25

0.3

stre

ss

Ordinary GlassStable Glass

Figure S9: Stress as a function of strain in the linear regime of the stress-straincurve at T = 0.30. The ordinary glass refers to the glass prepared at by cooling liq-uid at cooling rate of 3.33×10−8 and the stable glass refers to the vapor-depositedglass with a substrate temperature Ts = 0.3. The slope of the curve determines theelastic constant of the material. The modulus C of ordinary and the stable glassesis found to be 46.2 and 54.9, respectively.

van Hove Self-Correlation Function

To evaluate the difference in the dynamics of particles in an ordinary glass andthose at the surface of a vapor-deposited glass, we calculated the self intermediatescattering function Fs(q, t) [57], the Fourier transform at wave vector q of the vanHove self-correlation function

Gs(r, t) =1N

N

∑i=1

�δ (|ri(t)− ri(0)|− r)� . (E11)

This function describes the probability of finding a particle at time t at a dis-tance r from where it was at t = 0. We evaluate Fs(q∗, t) at a wave vector ofq∗ = 7.21σ−1

AA , (see Figure S2) which corresponds to the first peak of the structurefactor. A characteristic alpha relaxation time τα is defined as the time at whichFs(q∗, t) = 1/e [58]. Figure S10 shows plots of Fs(q∗, t) as a function of time forthe ordinary glass and for the surface region of the vapor-deposited glass at differ-ent temperatures. At any given temperature, the surface of the stable glass exhibitsfaster relaxation times than the ordinary glass. In contrast, the particles in the coreof the stable glass exhibit much slower dynamics than those in the ordinary glass.






1 10 100 1000 10000t - t0

0

0.2

0.4

0.6

0.8

1

F s(q*,

t-t0) /

Fs(q

*,t 0)

Ordinary T = 0.6Stable T = 0.6Ordinary T = 0.45Stable T = 0.45Ordinary T = 0.3Stable T = 0.3

Figure S10: van Hove correlation function Fs(q, t) of the ordinary glass, shown bysolid lines, and the surface region of a stable glass (vapor-deposited glass preparedat a substrate temperature Ts = 0.3), shown by dotted lines, measured at differenttemperatures. The function is displayed with a shift in the time origin to t0 = 1.2,and has been normalized to the value at t0. This transformation has been used as aconvenient procedure to eliminate the Gaussian time dependence at short t [57].

Alpha-relaxation time for liquids and stable glasses

For liquid configurations at temperatures above 0.6, τα is extracted from Fs(q∗, t).As mentioned above, we define τα according to Fs(q∗,τα ) = 1/e. We use a Vogel-Tammann-Fulcher (VTF) equation to estimate α-relaxation times (τα ) of super-cooled liquid samples [59]. A VTF equation can be written as

log(τα/τα ,0) =B

T −T0, (E12)

In Equation E12 τα ,0, B and T0 are adjustable parameters. Using the τα calculatedfrom the van Hove self-correlation function at temperatures in the range of T = 0.6to T = 1.5 (see triangles in Figure S11), and also using results from Kob et al. [58](see squares in Figure S11), we get log(τα ,0) = 0, B = 0.607 and T0 = 0.301 (seeFigure S11).

Using these parameters, the values of τα can be estimated at lower tempera-tures. To estimate the alpha-relaxation in stable glasses, as compared to that of theordinary glasses at their glass transition temperature (Tg = 0.432), we calculate ταat T = 0.432 and T = 0.322. We use T = 0.322 because a stable glass prepared at






0.4 0.6 0.8 1 1.2 1.4T kB / εAA

0

5

10

15

20

25

30

35

40

log

(τα )

VTF equation fitExtrapolated valuesfrom Kob et al. [58]Values from Fs(q*,t)

Figure S11: Alpha-relaxation times (τα ) of the supercooled liquid as a functionof temperature. The green triangles represent values obtained from the van Hoveself-correlation function. The blue squares represent values taken from Kob etal. [58]. Equation E12 is fitted to the data obtained at high temperatures (squares).The dashed line shows the fitted curve. Two red points represent the values of ταestimated by Equation E12 at T = 0.432 and T = 0.322, respectively.






this substrate temperature has similar energy to that of the most stable glass (pre-pared at a substrate temperature of 0.3), and its potential energy lies very near tothe equilibrium supercooled liquid line (see Figure 1B of the main text). We findthat τα at a temperature T = 0.322 is about 24 orders of magnitude larger than atT = 0.432.

Vibrational density of states

A characteristic feature of glasses is that g(ω) exhibits departures from the Debyeform (g(ω) ∝ ω2), whereby an excess of states - usually referred to as the “bosonpeak” [60, 61] - appears at characteristic frequencies. Various definitions of theboson peak have been introduced [62], including (1) a peak in Raman scatteringdata [63], (2) a peak in the difference between g(ω) of the glass and the correspond-ing crystal, and (3) a peak in g(ω)/ω2, for g(ω) extracted from neutron scatteringdata [64]. In simulations, g(ω) can be extracted directly through diagonalizationof the Hessian matrix of the inherent structures. The vibrational frequencies (ω)correspond to the square root of the eigenvalues of the Hessian matrix [61].

The configurational entropy is calculated as the difference between the entropyof the liquid and that of the disordered solid [65]. The entropy of the disorderedsolid can be obtained directly from vibrational frequency data:

Sdisordered−solid (T,V ) =3N−3

∑j=1

1− ln(β hω j) (E13)

The liquid entropy can be found using procedures outlined in the literature [65].

References

[34] Steinhardt, P. J., Nelson, D. R. & Ronchetti, M. Bond-orientational order inliquids and glasses. Physical Review B 28, 784–805 (1983).

[35] Landau, L. & Lifschitz, E. Quantum Mechanics (Pergamon, London, 1965).

[36] Lechner, W. & Dellago, C. Accurate determination of crystal structures basedon averaged local bond order parameters. Journal of Chemical Physics 134,114524 (2011).

[37] Ester, M., Kriegel, H.-p., Sander, J. & Xu, X. A density-based algorithmfor discovering clusters in large spatial databases with noise. In Proceedingsof the Second International Conference on Knowledge Discovery and DataMining (KDD-96), 226–231 (AAAI Press, 1996).






[38] Luchnikov, V. A., Medvedev, N. N., Naberukhin, Y. I. & Schober, H. R.Voronoi-delaunay analysis of nornal modes in a simple model glass. PhysicalReview B 62, 3181–3189 (2000).

[39] Medvedev, N. N. The algorithm for three-dimensional voronoi polyhedra.Journal of Computational Physics 67, 223 – 229 (1986).

[40] Medvedev, N. N. Aggregation of tetrahedral and quartoctahedral delaunaysimplices in liquid and amorphous rubidium. Journal of Physics: CondensedMatter 2, 9145 (1990).

[41] Voloshin, V. P., Naberukhin, Y. I. & Medvedev, N. N. Can various classes ofatomic configurations (delaunay simplices) be distinguished in random densepackings of spherical particles? Molecular Simulation 4, 209–227 (1989).

[42] Sheng, H. W., Luo, W. K., Alamgir, F. M., M., B. J. & Ma, E. Atomic packingand short-to-medium-range order in metallic glasses. Nature 439, 419–425(2006).

[43] Cheng, Y., Cao, A., Sheng, H. & Ma, E. Local order influences initiationof plastic flow in metallic glass: Effects of alloy composition and samplecooling history. Acta Materialia 56, 5263 – 5275 (2008).

[44] Coslovich, D. Locally preferred structures and many-body static correlationsin viscous liquids. Physical Review E 83, 051505 (2011).

[45] Finney, J. L. Random packing and the structure of simple liquids. Proceed-ings of the Royal Society A 319, 479–493 (1970).

[46] Finney, J. L. Modeling structures of amorphous metals and alloys. Nature266, 309–314 (1977).

[47] Borodin, V. A. Local atomic arrangements in polytetrahedral materials.Philosophical Magazine A 79, 1887–1907 (1999).

[48] Borodin, V. A. Local atomic arrangements in polytetrahedral materials. ii.coordination polyhedra with 14 and 15 atoms. Philosophical Magazine A 81,2427–2446 (2001).

[49] Cicerone, M. T. & Soles, C. L. Fast dynamics and stabilization of proteins:Binary glasses of trehalose and glycerol. Biophysical Journal 86, 3836–3845(2004).

[50] Zaccai, G. Biochemistry - how soft is a protein? a protein dynamics forceconstant measured by neutron scattering. Science 288, 1604–1607 (2000).






[51] Starr, F. W., Sastry, S., Douglas, J. F. & Glotzer, S. C. What do we learnfrom the local geometry of glass-forming liquids? Physical Review Letters89, 125501 (2002).

[52] Dirama, T. E., Carri, G. A. & Sokolov, A. P. Role of hydrogen bonds inthe fast dynamics of binary glasses of trehalose and glycerol: A moleculardynamics simulation study. Journal of Chemical Physics 122, 114505 (2005).

[53] Larini, L., Ottochian, A., Michele, C. D. & Leporini, D. Universal scaling be-tween structural relaxation and vibrational dynamics in glass-forming liquidsand polymers. Nature Physics 4, 42–45 (2008).

[54] Swallen, S. F. et al. Organic glasses with exceptional thermodynamic andkinetic stability. Science 315, 353–356 (2007).

[55] Leonard, S. & Harrowell, P. Macroscopic facilitation of glassy relaxationkinetics: Ultrastable glass films with frontlike thermal response. Journal ofChemical Physics 133, 244502 (2010).

[56] Shi, Z., Debenedetti, P. G. & Stillinger, F. H. Properties of model atomicfree-standing thin films. Journal of Chemical Physics 134, 114524 (2011).

[57] Hansen, J.-P. & McDonald, I. R. Theory of simple liquids (Academic Press,London, 1986).

[58] Kob, W. & Andersen, H. C. Testing mode-coupling theory for a super-cooled binary lennard-jones mixture .2. intermediate scattering function anddynamic susceptibility. Physical Review E 52, 4134–4153 (1995).

[59] Angell, C. A. & Smith, D. L. Test of the entropy basis of the vogel-tammann-fulcher equation. dielectric relaxation of polyalcohols near tg. Journal ofChemical Physcics 86, 3845–3852 (1982).

[60] Sette, F., Krisch, M. H., Masciovecchio, C., Ruocco, G. & Monaco, G. Dy-namics of glasses and glass-forming liquids studied by inelastic x-ray scatter-ing. Science 280, 1550–1555 (1998).

[61] Middleton, T. F. & Wales, D. J. Energy landscapes of some model glassformers. Physical Review B 64, 024205 (2001).

[62] Grigera, T. S., Martin-Mayor, V., Parisi, G. & Verrocchio, P. Phonon inter-pretation of the ’boson peak’ in supercooled liquids. Nature 422, 289–292(2003).






[63] Sokolov, A. P., Rossler, E., Kisliuk, A. & Quitmann, D. Dynamics of strongand fragile glass formers - differences and correlation with low-temperatureproperties. Physical Review Letters 71, 2062–2065 (1993).

[64] Buchenau, U. et al. Low-frequency modes in vitreous silica. Physical ReviewB 34, 5665–5673 (1986).

[65] Sciortino, F., Kob, W. & Tartaglia, P. Thermodynamics of supercooled liq-uids in the inherent-structure formalism: a case study. Journal of Physics:Condensed Matter 12, 6525 (2000).





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Ultrastable glasses from in silico vapour deposition · Supplementary Information 6 Figure S4: Individual three-dimensional radial distribution functions (rdf) at T = 0.45 and T =