Ultrastable glasses from in silico vapour deposition Sadanand Singh, M. D. Ediger and Juan J. de Pablo. Nature Materials 12, (2013).

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1 Ultrastable glasses from in silico vapour deposition Sadanand Singh, M. D. Ediger and Juan J. de Pablo Nature Materials 12, (2013). In the version of the Supplementary Information originally published, in Table T1, the values of the energies E IS and <U>, and the Q 6 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.

2 Ultrastable glasses from in silico vapor deposition Sadanand Singh, 1 M. D. Ediger, 2 and Juan J. de Pablo 1,3 1 Department of Chemical and Biological Engineering 2 Department of Chemistry University of Wisconsin-Madison, Madison, WI 53706, USA 3 Institute of Molecular Engineering University of Chicago, Chicago, IL 60637, USA To whom correspondence should be addressed; depablo@uchicago.edu. Thermodynamic Properties Vapor-deposited glass are grown for substrate temperatures in the range between T s = 0.15 and 0.8. To compare the energy (potential and inherent structure) of stable and ordinary glasses, vapor-deposited glass samples are cooled from T s to T = 0.05 at a cooling rate of The particles in the vapor-deposited glass sample are divided into two groups, namely the core (defined as particles which are at least 8 σ AA away from both the substrate and the free surface) and the surface (defined as particles which are within 5 σ AA of the free surface). Unless otherwise specified, the reported properties of vapor-deposited glasses refer to those of core particles. Table T1 gives the values of energy per particle (both potential and inherent structure) for ordinary glasses prepared at different cooling rates, and for vapordeposited glasses prepared at different substrate temperatures T s. For the ordinary glasses, the potential energies reported in the Table are calculated at a temperature T = 0.3. For the vapor-deposited glasses, the potential energies reported correspond to the as-deposited sample. All inherent energies correspond to the inherent structures at a temperature T = For ordinary glasses, the instantaneous configurations for which the inherent structure was calculated were selected at the end of the temperature scan, i.e. at T = All vapor-deposited glass samples were cooled to a temperature of 0.05 at a cooling rate of to get the inherent structure at T = NATURE MATERIALS 1

3 Figure S1: Representative configuration of a binary stable glass sample prepared by vapor deposition. The density of glasses prepared by vapor deposition is higher ( ) than that of the ordinary glasses prepared at constant volume at a density of ρ = 1.2. To study whether the enhanced stability of vapor-deposited glasses is a result of their higher density, we also simulated ordinary glasses with densities of ρ = 1.22, 1.24 and 1.26 using a range of cooling rates. The average inherent structure energies of those glasses at a temperature of T = 0.05, along with their corresponding potential energies at a temperature of T = 0.3, are given in Table T2. As the results on that Table show, the energies of denser ordinary glass samples are only marginally different than those of the ordinary glass prepared at ρ = 1.2, indicating that density alone 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 central atom. The structure factor, s(q) is related to the rdf by s(q)=1 + 4πρ [g(r) 1] sin(qr)dr, q 0 (E1) 2 NATURE MATERIALS

4 SUPPLEMENTARY INFORMATION Table T1: Thermodynamic properties of stable and ordinary glasses. Error bars in Density, E IS, U and Q 6 are less than 0.005, 0.01, and , respectively. Process Cooling rate or Density E IS U Q 6 substrate temperature (1/σAA 3 ) (ε AA) (ε AA ) Parameter Ordinary Cooling T s = T s = T s = Vapor T s = Deposition T s = T s = T s = T s = T s = where q represents the wave vector. Figure S2 shows the structure factor for an ordinary glass (prepared at a cooling rate of ) and for a stable glass (prepared at a substrate temperature of T s = 0.30). The two glass samples exhibit identical structure factors. It is of interest to consider the structural details of the stable and ordinary glasses in different directions, in order to assess whether the substrate introduces any anisotropy in the samples. A two-dimensional radial distribution function can be defined as: n(r) g(r)= (E2) 4π ρ Nr r where n(r) is the histogram generated by counting the number of times the distance between two particles lies between r and r + r, N is the total number of particles in the simulation box, and ρ is the bulk density of the system. Results for the corresponding radial distribution functions are shown in Fig S3. Given the heterogenous nature of the binary systems considered here, it is instructive to examine the relative distributions of A-A, A-B and B-B particles. Figure S4 shows the corresponding three-dimensional radial distribution functions for the stable and the ordinary glasses at T = 0.45 and 0.3. Again, we find little evi- NATURE MATERIALS 3

5 4 3 Ordinary T = 0.3 Stable T = 0.3 s (q) q (in σ AA -1 units) Figure S2: Structure factor for all particles for the ordinary glass prepared at a cooling rate of and for the stable glass prepared at a substrate temperature of T s = 0.3. Both results correspond to T = Ordinary Stable (in XY) Stable (in YZ) Stable (in ZX) g 2D (r) r 2D (in σ AA units) Figure S3: Two-dimensional radial distribution functions of ordinary and stable glass samples. All results correspond to T = 0.3. The ordinary glass sample (shown in black) was prepared at a cooling rate of The stable glass was grown at a substrate temperature of T s = 0.3; Z represents the direction normal to the substrate. 4 NATURE MATERIALS

6 SUPPLEMENTARY INFORMATION Table T2: Potential energy of ordinary glass prepared at different densities. The error bars in the energy are of smaller than U (ε AA ) Cooling Rate ρ = 1.2 ρ = 1.22 ρ = 1.24 ρ = dence of structural differences between the two systems beyond statistical noise of the simulation. Structural Properties Bond Order Parameters Spherical harmonic order parameters, such as Q 4, W 4, Q 6 and W 6, characterize the orientational correlation of the vectors joining the centers of mass of neighboring molecules. These parameters are near 0 for the liquid phase [34]. To define these order parameters first, the complex vector q lm (i) of particle i is defined as q lm (i)= 1 N b (i) N b (i) j=1 Y lm (r ij ). (E3) Here, N b (i) is the number of nearest neighbors of particle i - defined as particles within a spherical volume with radius r = 1.8 (corresponding to second maximum in the radial distribution function), l is a free integer parameter, and m is an integer that runs from m = l to m =+l. The functions Y lm (r ij ) are the spherical harmonics and r ij 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 q l (i)= 4π 2l + 1 l m= l q lm (i) 2. (E4) NATURE MATERIALS 5

7 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-deposited stable glasses. Results at T = 0.45 have been shifted above the curves at T = 0.3 for clarity. (B) A-B rdf for the ordinary glasses and the vapor-deposited stable glasses. Results at T = 0.45 have been shifted above the curves at T = 0.3 for clarity. (C) B-B rdf for the ordinary glasses and the vapor-deposited stable glasses. 6 NATURE MATERIALS

8 SUPPLEMENTARY INFORMATION Depending on the choice of l, these parameters are sensitive to different crystal symmetries. Another class of local bond order parameters can be defined as ( ) l l l q m w l (i)= 1 +m 2 +m 3 =0 m 1 m 2 m lm1 (i)q lm2 (i)q lm3 (i) 3 ( ) l 3/2. (E5) q lm (i) 2 m= l Here, the integers m 1, m 2 and m 3 run from l to l, but only combinations with m 1 + m 2 + m 3 = 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 correspond to higher crystalline character. Liquids and crystals can be more easily differentiated in terms of averaged values of these order parameters. The locally averaged parameters q l (i) and w l (i) can be defined as w l (i)= q l (i)= 4π l 2l + 1 q lm (i) 2 (E6) m= l ( ) l l l q m 1 m 2 m lm1 (i) q lm2 (i) q lm3 (i) 3 ( ) l 3/2 (E7) q lm (i) 2 m= l m 1 +m 2 +m 3 =0 where, q lm (i)= 1 Ñ b (i) Ñ b (i) k=0 q lm (k). (E8) Here, the sum from k = 0 to Ñ b (i) runs over all neighbors of particle i plus the particle i itself. The overall Q l and W l values of a system are defined as the values of q l (i) and w l (i) averaged over all particles. Commonly, to differentiate typical crystal structures and liquids, a value of l = 4 and l = 6 have been used in literature [36]. We have calculated these order parameters for different glass samples by averaging over more than independent structures at any given condition. The values of Q 6 for different binary LJ glass samples have been listed in Table T1. Other order parameters like Q 4, W 4 and W 6 also have values in the range of 0.01 to These near zero values confirm the absence of any significant crystalline order in all glass samples. To analyze distributions of these order parameters in more detail, the distribution of q 6 (i) was studied in more detail. Figure 3 of the main text shows the results NATURE MATERIALS 7

9 T = 0.8 Ordinary Glass Vapor-Deposited Glass T = 0.5 # of Clusters T = 0.4 T = 0.35 T = 0.3 T = Size of Clusters Figure S5: Number of ordered clusters of particles (on a logarithmic scale) with local q 6 (i) > 0.12, as a function of cluster size at different temperatures. Data have been collected over 1000 frames across 10 independent trajectories. At the optimal substrate temperature maximum difference can be observed between the vapor-deposited (red) and ordinary (black) glass samples. The curves at different temperatures have seen shifted for clarity. 8 NATURE MATERIALS

10 SUPPLEMENTARY INFORMATION for the local distribution of q 6 (i). The local distribution was further analyzed by performing a cluster analysis using the DBSCAN [37] algorithm. For the cluster analysis, inherent structures of a glass sample at a given temperature were considered. For a given temperature, we have selected 1000 inherent structures from 10 independent simulations, hence encompassing statistics from over 10,000 inherent structures. The clusters were designated based on their local q 6 (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 temperature T s = 0.3 shows significant differences from ordinary glass samples. The effect 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 local q 6 (i). As temperature in lowered, the vapor-deposited glasses show significantly different clusters as compared to ordinary glasses. The difference increases until an optimal value of deposition temperature T s is reached, below which, once again the 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 structural features like Voronoi-Delaunay analysis [38]. By definition, the Voronoi polyhedron (VP) of an atom is that region of space that is closer to the given atom than to any other atom of the system. A dual system spanning space is formed by the Delaunay Simplices (DS). These are tetrahedra formed by four atoms that lie on the surface of a sphere that does not contain any other atom. Both VP and DS fill the space of the system without gaps and overlaps. In our calculations we do a Voronoi-Delaunay tessellation of the inherent structures of different glasses by the algorithm described in Reference [39]. Two main types of DS are predominant in monoatomic glasses [40, 41], namely DS similar to ideal tetrahedron (tetrahedricity, Γ) and DS resembling a quarter of a regular octahedron (octahedricity, O). Similar to the definition in Reference [38], the two quantities are defined as follows. { (l i l j ) Γ = 2 6 } 1 i< j 15 l 2, and O = g m O 1 m, (E9) m=1 NATURE MATERIALS 9

11 where l = m=1 l m, g m = and O m = e3δ m/σ 6 e 3δ i/σ i=1 i< j; i, j m, δ m = l [ ] m l 1, σ = l 6 1/2 δm 2, (l i l j ) 2 10 l 2 + i m ( l i l m / ) l 2. Here l i denotes the length of the i th edge of the DS. A value of Γ = 0 represents a perfect 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 q 6 (i), the local Γ(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, the cluster analysis was performed. Cluster analysis was performed over glass samples with Γ(i) < and O(i) < As seen in Figure S6 (a) and (b), results show a similar trend as the local q 6 (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 coordination can be resolved through a Voronoi tessellation of space [45, 46, 47, 48] that allows one to characterize the local, particle-level environment. The Voronoi polyhedron associated with any given particle is characterized in terms of two quantities: the number of faces (equal to the coordination number), and the number of edges on each face. Following the notation used by Sheng et al. [42, 44], we use the Voronoi index n 3,n 4,n 5,n 6,..., where n i denotes the number of i-edged faces of the Voronoi polyhedron of a given particle and i n i is the total coordination number of that particle. For example, the designation 0,2,8,0 indicates a coordination number of 10 with two 4-edged faces and eight 5-edged faces. The designation 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 a coordination number of 12 with twelve 5-edged faces. The results for the ordinary glasses produced in this work exhibit the same trends reported by Sheng et al. [42] for Ni 80 P 20 metallic glasses. Most of the polyhedra centered around B -type particles have coordination numbers between 9 and 12. Figure 3(B) of the main text shows a distribution of the major types of B-particle polyhedra having coordination numbers 10, 11 and 12, labeled using the 10 NATURE MATERIALS

12 SUPPLEMENTARY INFORMATION Figure S6: (A) Number of ordered clusters of particles (on a logarithmic scale) with local Γ(i) < 0.025, as a function of cluster size at temperature T = 0.3. Data have been 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. NATURE MATERIALS 11

13 Voronoi index format defined above. The inset of the figure shows the distribution of coordination number in the system. Stable glasses exhibit primarily a single type of polyhedron for each coordination number. Ordinary glasses exhibit several types of polyhedra. Stable glasses have a more homogenous local structure than ordinary glasses. Coslovich [44] found that, above T g, as the liquid is cooled the abundance of polyhedra centered around B -type particles of types 0,2,8,0, 0,2,8,1, and 0,0,12,0 gradually increases. At T g that growth slows down considerably, and the number of such regular polyhedra remains almost constant. At T g, other types of irregular polyhedra such as 0,2,6,2, 1,2,7,1, and 0,2,8,2 are also relatively frequent in the ordinary glass. In contrast, for stable glasses, the abundance of 0,2,8,0, 0,2,8,1, and 0,0,12,0 polyhedra continues to grow below T g, down to the temperature corresponding to the optimal substrate temperature T s = 0.3. In such stable glasses, there are essentially no irregular polyhedra of types 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 a function of temperature for both ordinary vapor-deposited glasses as a function of temperature. For ordinary glasses, at higher temperatures (TK B /ε AA > 0.4), consistent with the results of Coslovich [44], the percentage content of such polyhedra increases rapidly. At lower temperatures, in the glassy regime of the material, the growth 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 increases monotonically until, for the most stable glass formed at a substrate temperature T s = 0.3, it reaches a maximum. Vapor-deposited glasses at a lower substrate temperature are not as stable and have much lower content of these polyhedra. A direct 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 of local q 6 (i), tetrahedricity Γ(i) and octahedricity O(i), described in previous sections. In particular, the Voronoi indices for polyhedra on particles belonging to the larger clusters that only arise in ordinary glasses, are precisely of the type that is absent in stable glasses. Debye-Waller Factors The molecular mobility of vapor-deposited glasses is determined by measuring the Debye-Waller factor, denoted by u 2 (z), as a function of the distance (z) from the substrate layer. Debye-Waller factors are common in the glass literature [49, 50, 51] because they can be extracted from both simulations and neutron 12 NATURE MATERIALS

14 SUPPLEMENTARY INFORMATION % content of <0,2,8,0> polyhedra Ordinary Vapor-deposited T k B /ε AA Figure S7: Percentage content of 0, 2, 8, 0 polyhedra centered around B -type particles as a function of temperature for ordinary glasses (black circles) and as a function of the substrate temperature for vapor-deposited glasses (red squares). As the temperature is reduced, the percentage of 0, 2, 8, 0 polyhedra increases gradually until the simulated glass transition temperature is reached (T 0.4). Below that temperature, in ordinary glasses the percentage of these polyhedra is nearly 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 T s = 0.3. At substrate temperatures below T s < 0.3, the percentage of such polyhedra is again comparable to that observed in ordinary glasses, suggesting that a correlation exists between the stability of the vapor-deposited glasses and molecular structure as measured by the number of 0, 2, 8, 0 polyhedra. NATURE MATERIALS 13

15 scattering experiments. Furthermore, they are often interpreted as measures of the free volume surrounding a given particle [51] or, within the framework of the harmonic model, as measures of the stiffness of a glassy matrix [49, 50]. Various approaches have been adopted to calculate this parameter [49, 51, 52], one of which is to calculate the mean-squared atomic displacement after the beginning of the caging regime [51] as a measure of free volume in the system. That is the approached followed here. <u 2 > / <u 2 > bulk Z (in σ AA units) Figure S8: The Debye-Waller factors normalized with respect to the core value of the stable glass sample as a function of distance from the substrate. Different curves show values at transient times during the preparation of stable glasses. Figure S8 shows the corresponding Debye-Waller factor profiles, as a function of position along the vapor-deposited glass, at different times during the glass preparation process. The results represent an average over the plane parallel to the substrate, and over slabs of thickness 2σ AA at a substrate temperature T s of 0.3. One can see that the Debye-Waller factors at the surface are approximately 2 to 3 orders of magnitude larger than in the core of the material. On comparing the absolute Debye-Waller factors of the ordinary glass with that of the most stable glass prepared at a substrate temperature of T s = 0.3, it is found ( that the Debye-Waller factors in the stable glasses have reduced significantly u 2 ordinary / u 2 stable = 7.64 ). A universal scaling between structural relaxation τ α and Debye-Waller factors has been proposed by Larini et al. [53], ( a2 τ α exp 2 u 2 + σ ) 2 a 2 8 u 2 2, (E10) 14 NATURE MATERIALS

16 SUPPLEMENTARY INFORMATION where a 2 and σ 2 a 2 are average and variance in the square of a molecular displacement required for local structural relaxation. Using such a scaling, we estimate the alpha relaxation time for our most stable glass at times the value for the ordinary glass. This is consistent with our other, independent estimates. Swallen et al. [54] have proposed that the enhanced stability of vapor-deposited glasses is due to the enhanced mobility of surface molecules, which can undergo significant relaxations as new molecules continue to arrive at the glass-vacuum interface during the growth process [55]. The deposition occurs sequentially and every molecule being deposited is part of the surface at some point during that process. In this view, the enhanced surface dynamics enable sampling of lowerenergy inherent structures. To examine this hypothesis, a local mobility profile can be generated in terms of local Debye-Waller factors calculated at small time scales [51]; as discussed above, the mobility of surface particles is indeed much higher than that of the core of the stable glass throughout the entire deposition process, consistent with Swallen et al. s [54] proposed mechanism. This finding is also consistent with observations by Shi et al. [56], who have proposed that surface particles 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 of the liquid at a cooling rate of ) and stable glasses (vapor-deposited glass prepared at a substrate temperature T s = 0.3) were calculated using an uniaxial strain deformation. For the ordinary glasses, we first calculated the pressure of the NVT glass at ρ = 1.2 and at T = 0.3. Using the calculated values of pressure as a set-point, an additional NσT run was performed. An additional stress was applied along one of the directions to deform the material. Similar calculations were performed in each of the three Cartesian directions. The strains were sufficiently small 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 normal to the substrate, a repulsive wall potential was introduced on top of the film in order to apply a stress. The stress-strain response in the plane of the substrate was measured in a similar manner to that employed for the ordinary glasses. Figure S9 shows the average normal uniaxial stress-strain response for ordinary and stable glasses. From the slope of these curves we can extract an elastic modulus. We find that stable glass samples have a modulus C = 54.9 that is about 18% higher than that of the ordinary glass, which is C = NATURE MATERIALS 15

17 stress Ordinary Glass Stable Glass strain Figure S9: Stress as a function of strain in the linear regime of the stress-strain curve at T = The ordinary glass refers to the glass prepared at by cooling liquid at cooling rate of and the stable glass refers to the vapor-deposited glass with a substrate temperature T s = 0.3. The slope of the curve determines the elastic constant of the material. The modulus C of ordinary and the stable glasses is 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 and those at the surface of a vapor-deposited glass, we calculated the self intermediate scattering function F s (q,t) [57], the Fourier transform at wave vector q of the van Hove self-correlation function G s (r,t)= 1 N N i=1 δ( r i (t) r i (0) r). (E11) This function describes the probability of finding a particle at time t at a distance r from where it was at t = 0. We evaluate F s (q,t) at a wave vector of q = 7.21σAA 1, (see Figure S2) which corresponds to the first peak of the structure factor. A characteristic alpha relaxation time τ α is defined as the time at which F s (q,t) =1/e [58]. Figure S10 shows plots of F s (q,t) as a function of time for the ordinary glass and for the surface region of the vapor-deposited glass at different temperatures. At any given temperature, the surface of the stable glass exhibits faster relaxation times than the ordinary glass. In contrast, the particles in the core of the stable glass exhibit much slower dynamics than those in the ordinary glass. 16 NATURE MATERIALS

18 SUPPLEMENTARY INFORMATION 1 F s (q*,t-t 0 ) / F s (q*,t 0 ) Ordinary T = 0.6 Stable T = 0.6 Ordinary T = 0.45 Stable T = 0.45 Ordinary T = 0.3 Stable T = t - t 0 Figure S10: van Hove correlation function F s (q,t) of the ordinary glass, shown by solid lines, and the surface region of a stable glass (vapor-deposited glass prepared at a substrate temperature T s = 0.3), shown by dotted lines, measured at different temperatures. The function is displayed with a shift in the time origin to t 0 = 1.2, and has been normalized to the value at t 0. This transformation has been used as a convenient 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, τ α is extracted from F s (q,t). As mentioned above, we define τ α according to F s (q,τ α )=1/e. We use a Vogel- Tammann-Fulcher (VTF) equation to estimate α-relaxation times (τ α ) of supercooled liquid samples [59]. A VTF equation can be written as log(τ α /τ α,0 )= B T T 0, (E12) In Equation E12 τ α,0, B and T 0 are adjustable parameters. Using the τ α calculated from the van Hove self-correlation function at temperatures in the range of T = 0.6 to 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(τ α,0 )=0, B = and T 0 = (see Figure S11). Using these parameters, the values of τ α can be estimated at lower temperatures. To estimate the alpha-relaxation in stable glasses, as compared to that of the ordinary glasses at their glass transition temperature (T g = 0.432), we calculate τ α at T = and T = We use T = because a stable glass prepared at NATURE MATERIALS 17

19 40 35 log (τ α ) VTF equation fit Extrapolated values from Kob et al. [58] Values from F s (q*,t) T k B / ε AA Figure S11: Alpha-relaxation times (τ α ) of the supercooled liquid as a function of temperature. The green triangles represent values obtained from the van Hove self-correlation function. The blue squares represent values taken from Kob et al. [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 τ α estimated by Equation E12 at T = and T = 0.322, respectively. 18 NATURE MATERIALS

20 SUPPLEMENTARY INFORMATION this substrate temperature has similar energy to that of the most stable glass (prepared at a substrate temperature of 0.3), and its potential energy lies very near to the equilibrium supercooled liquid line (see Figure 1B of the main text). We find that τ α at a temperature T = is about 24 orders of magnitude larger than at T = Vibrational density of states A characteristic feature of glasses is that g(ω) exhibits departures from the Debye form (g(ω) ω 2 ), whereby an excess of states - usually referred to as the boson peak [60, 61] - appears at characteristic frequencies. Various definitions of the boson peak have been introduced [62], including (1) a peak in Raman scattering data [63], (2) a peak in the difference between g(ω) of the glass and the corresponding crystal, and (3) a peak in g(ω)/ω 2, for g(ω) extracted from neutron scattering data [64]. In simulations, g(ω) can be extracted directly through diagonalization of 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 entropy of the liquid and that of the disordered solid [65]. The entropy of the disordered solid can be obtained directly from vibrational frequency data: 3N 3 S disordered solid (T,V )= 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 in liquids and glasses. Physical Review B 28, (1983). [35] Landau, L. & Lifschitz, E. Quantum Mechanics (Pergamon, London, 1965). [36] Lechner, W. & Dellago, C. Accurate determination of crystal structures based on averaged local bond order parameters. Journal of Chemical Physics 134, (2011). [37] Ester, M., Kriegel, H.-p., Sander, J. & Xu, X. A density-based algorithm for discovering clusters in large spatial databases with noise. In Proceedings of the Second International Conference on Knowledge Discovery and Data Mining (KDD-96), (AAAI Press, 1996). NATURE MATERIALS 19

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