Nucleation rate (m -3 s -1 ) Radius of water nano droplet (Å) 1e+00 1e-64 1e-128 1e-192 1e-256

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1 Supplementary Figures Nucleation rate (m -3 s -1 ) 1e+00 1e-64 1e-128 1e-192 1e-256 Calculated R in bulk water Calculated R in droplet Modified CNT Radius of water nano droplet (Å) Supplementary Figure S1: Comparison of the computed ice nucleation rate in the mw water droplet at 230 K with the prediction from CNT modified using the melting temperature of nano droplet. The calculated ice nucleation rates in droplets, in bulk water, and the prediction by incorporating Gibbs-Thomson effect on the melting temperature (Eqn. (S3)) are represented by red solid line, black squares, and blue solid line, respectively. 1

2 a 1.00 Density (g cm -3 ) r=2.4 nm r=3.1 nm r=3.9 nm r=4.9 nm r=6.1 nm b Radius (Å) r=6.1 nm Density (g cm -3 ) Å Radius (Å) Supplementary Figure S2: Density profile of droplets. a: Calculated density profile of water in the mw nano droplets at 230 K. Inset is the zoom-in of the density profile that shows the increase of the average density of the water with the decrease of the radius of droplet. b: Density oscillates near the boundary of the droplet. The red solid and dashed lines indicate the immediate surface of the droplet and position where the density oscillation vanishes, respectively. The region in between is of the thickness of about 9 Å, and is defined as the surface region of the droplet. All droplets studied show the similar oscillations. 2

3 Density (g cm -3 ) Liquid water Ice Ic Ice Ih r=2.4 nm r=3.1 nm r=3.9 nm r=4.9 nm r=6.1 nm Temperature (K) Supplementary Figure S3: Variation of density of mw water with temperature. Calculated density of bulk liquid water, water nano droplet, and ice, as a function of temperature for the mw water model. 3

4 Density (g cm -3 ) mw water mw ice Real water (Ref. [29]&[30]) Real ice Temperature (K) Supplementary Figure S4: Variation of the density of water and ice with temperature. The density of supercooled water was measured down to 238 K 29, below which the density was extrapolated to 227 K 30 (represented by the blue dotted line). 4

5 Nucleation rate (m -3 s -1 ) 1e+40 1e+35 1e+30 1e+25 1e+20 1e+15 1e+10 1e+05 1e+00 1e-05 Bulk Slab Temperature (K) Supplementary Figure S5: Comparison of the calculated ice nucleation rates in supercooled bulk water and water slab based on the mw water model at 230 K. 5

6 1 Committor p B r=2.4 nm r=3.1 nm r=3.9 nm r=4.9 nm r=6.1 nm Number of water molecules in ice nucleus Supplementary Figure S6: Determination of critical size. Calculated committor p B as a function of number of water molecules contained in ice nuclei at 230 K in the mw water droplets. According to the definition of p B, the size of the critical nucleus is determined by the intersect of the p Bc = 0.5 (dashed line) with the computed p B. Thus the critical nucleus contains about 570, 510, 500, 490, and 465 water molecules when it nucleates from the droplet with the radius from 2.4 nm to 6.1 nm, respectively. The critical size of ice nucleus in bulk supercooled water at the same temperature was estimated to be around 435 molecules 17. 6

7 40 30 NS= p(ns) NS=R G /R GS Supplementary Figure S7: Calculated distribution of the nonsphericity (NS) of the critical ice nuclei in the mw droplet with the radius of 2.4 nm at 230 K. An NS of unit corresponds to the perfect sphere. The distribution is obtained by an ensemble of 119 critical nuclei. The average of NS is calculated to be ±

8 r=2.4 nm a Distribution r=3.1 nm Surface region b r=4.9 nm c Radius (Å) Supplementary Figure S8: Spatial distribution of ice critical nuclei. Radial distribution of the geometric center (blue) and the outer boundary (black), i.e., the boundary furtherest to the center of the droplet, of the critical nuclei in droplets. The red solid and dashed lines indicate the outer (i.e., immediate surface), and inner boundary of the surface region in droplets, respectively. The two arrow heads indicate the average positions of the geometric center and outer boundary of the critical nuclei, respectively. For the smallest droplet r = 2.4 nm, almost all the critical ice nuclei have their outer boundary located within the surface region. The average radii of gyration R G for the critical nucleus in (a), (b), and (c) are calculated to be 12.9 Å, 12.6 Å, and 12.4 Å, respectively. 8

9 1e+00 1e-04 Growth probability P(λ λ 0 ) 1e-08 1e-12 1e-16 1e-20 1e-24 1e-28 r=2.4 nm r=3.1 nm r=3.9 nm r=4.9 nm r=6.1 nm 1e Nucleus size (λ) Supplementary Figure S9: Calculated growth probability P (λ λ 0 ) of ice nucleus as a function of nucleus size λ in the mw water droplets with various radii at 230 K. For each curve, the abscissa of each data point corresponds to the defined interface λ i. 9

10 Supplementary Note 1: Influence of nucleation rate due to the suppression of melting temperature in water droplet The equilibrium melting temperature of water droplet decreases with size, an effect well explained by the Gibbs-Thomson equation. Here we show that the suppression of ice nucleation rates can not be explained in terms of a shift of the melting temperature. To a first order approximation, the chemical potential difference between bulk liquid and solid of the mw model was estimated by µ b = H(Tm T )/Tm, (S1) where H = 4878 J mol 1 is a constant 17, and Tm = K is the melting temperature of the mw water model. It was also shown that 17 the homogeneous ice nucleation rate can be expressed as: ( ) 1 R = A exp C (T Tm) 2 T, (S2) where A is the kinetic pre-factor, and C = 16πγ 3 ls T 2 m/(3k B ρ 2 H 2 ). The kinetic factor A plays a significant role near and below the temperature of maximum crystallization rate, T f max, where the nucleation rate is dominated by kinetics, while at higher temperature, the nucleation rate is mainly controlled by free energy barrier. Since the temperature range of this study is well above T max f of the mw water model 33, we approximate the kinetic factor A as a constant. This would not be the case for real water, for which the diffusion coefficient is more sensitive to the temperature, and the T max f is 225 K 33. We also note that even if the kinetic factor should be treated explicitly when considering the temperature dependence of ice nucleation rate near T max f for real water, the difference in diffusivity between bulk and droplets is small, therefore the ratio Rp/R b is much less 10

11 sensitive to the kinetic effect. By fitting the Eqn. (S2) with respect to the calculated homogeneous ice nucleation rate, one obtains A = m 3 s 1 and C = K 3. If the suppression of the ice nucleation rate originated from the change of melting temperature due to the Gibbs-Thomson effect, one would be able to estimate the nucleation rate based on Eqn. (S2), by replacing Tm by Tm(r), where Tm(r) is the equilibrium melting temperature of the water nano droplet with the radius r. We show in the following that this is not the case. In particular, we write C = DTm, 2 where D = 16πγ ls 3 /(3k B ρ2 H 2 ) = K is a constant, and Eqn. (S2) becomes: ( T 2 ) R = A exp D m(r) (T Tm(r)) 2 2 T. (S3) Using the reported melting temperature for the mw water droplet in Ref. 22, Eqn. (S3) predicts the ice nucleation rate in droplet at 230 K, shown as the blue curve along with the simulation results in Supplementary Figure S1. It can be seen that while incorporating the Gibbs-Thomson effect in CNT through Eqn. (S3) yields the correct trend, it does not allow one to reproduce the computed ice nucleation rate. Supplementary Note 2: Effect of the p v term on nucleation rate According to classical nucleation theory, the nucleation rate can be expressed by the following equation, by assuming spherical critical nucleus: [ 16πγ 3 ] R b = Aexp ls 3k B T (ρ µ b ) 2, (S4) where µ b denotes the chemical potential difference between liquid and solid phase in bulk phase under zero pressure at temperture T, i.e., µ b = µ l µs > 0. When a small pressure p is applied 11

12 to the liquid, the chemical potential difference µ p has an additional contribution p v, µ p = µ b + p v = µ b + p(v l vs) (S5) Assuming that γ ls (liquid-solid interface tension) and µ b do not vary under small pressure, the ratio of nucleation rate R b /Rp can be obtained as follow: Rp = exp 16πγ3 ls 1 R b 3k B T ρ 2 ( µ b + p v) 1 2 µ 2 b (S6) When p v µ b, Rp = exp 32π ( ) 3 γls p v R b 3k B T ρ 2 µ b (S7) Estimate of Rp/R b in the vicinity of flat surface of supercooled liquid silicon The previous numerical study 16 reported the enhancement of nucleation rate of crystalline silicon in liquid silicon slab relative to that in the bulk liquid by three orders of magnitude at 0.95 Tm. This rate change can also be predicted by Eqn. (S7). In the vicinity of a flat liquid surface with normal direction is along z, a small lateral pressure p T (= pxx = pyy) is induced by surface tension, and the normal pressure p N = pzz = 0. The thermodynamic pressure p is thus one third of the trace of the pressure tensor 34, i.e., p = 2/3p T. In Tersoff silicon, the chemical potential difference between liquid and solid was computed by thermodynamic integration µ b = ev per atom 32 =2409 J mol 1 at 2400 K (0.95 Tm). The solid-liquid surface tension γ ls = 0.29 J m 2 was also estimated by the calculated homogeneous nucleation rate based on classical nucleation 32. The density of the liquid and solid silicon are calculated to be ρ l = mol m 3 and ρs = mol m 3, respectively. By calculating the lateral pressure p T = 0.18 GPa in the liquid slab containing 12

13 5832 silicon atoms, we obtained p v = J mol 1, i.e., p v µ b. Therefore Eqn. (S7) applies and yields Rp/R b = 101, in agreement with the simulation result. Estimate of Rp/R b in the nucleation of ice in the vicinity of flat surface For the mw water model, our simulation (Supplementary Figure S5) shows no appreciable difference in the nucleation rate between bulk supercooled water and water slab, at variance with the case in silicon shown above. Here we apply Eqn. (S7) to further elucidate the absence of surface induced nucleation in the mw water, and predict its existence in real water. In the mw water, the chemical potential difference µ b at 230 K is estimated to be J mol 1, through Eqn. (S1). In the same work the ice-water surface tension was also obtained based on calculated nucleation rates: γ ls = J m 2, in excellent agreement with the direct estimate of γ ls J m 2 based on free energy method 18. We also calculated the density of liquid and solid for the mw water model at 230 K: ρ l = mol m 3 and ρs = mol m 3. The surface tension of water surface exerts a lateral pressure p T = 0.02 GPa within the slab containing 4096 molecules, yielding p v = 4.56 J mol 1 ( µ b ). The Eqn. (S7) thus yields Rp/R b = 3, i.e., the difference in the rates is well within the error bar of the calculated nucleation rate. Therefore we conclude no appreciable change on the computed ice nucleation rate is expected in water slab with respect to that in bulk water based on the mw water model. For real water, the density difference between liquid and water is larger than that predicted by the mw model, thus an enhancement on the rate of ice nucleation is expected. At 240 K, the density difference between supercooled water 29 and ice is ρ = g cm 3, which consequently yields 13

14 p v = J mol 1 (0.025 µ b ). Applying Eqn. (S7), we obtain Rp/R b Therefore for real water, our model predicts the strong tendency for ice to nucleate in the vicinity of the flat liquid-vacuum surface of supercooled water. Supplementary Note 3: Radial density profile and the density variation with temperature in the mw nano droplet The radial density profiles are shown for the mw water nano droplets with different radii in Supplementary Figure S2. The density profile for each droplet was computed based on the NVT simulation longer than 4 ns, and the width of the bin along the radial direction is chosen to be 1.6 Å, centralized at each r, for computing the density average. For all the droplets studied, the density of water oscillates within about 9 Å from the immediate liquid surface (Supplementary Figure S2(b)). Below this surface region, the density is uniform. The effective volume for nucleation is defined as the total volume of the droplet minus the surface volume. Supplementary Figure S3 shows the calculated temperature dependence of the density of liquid water and ice based on the mw water model. The density of mw water shows its maximum at 250 K, while the density of the mw ice monotonically increases as the temperature is lowered. The densities of hexagonal ice I h and cubic ice I c were found to be virtually identical. 14

15 Supplementary Note 4: Shape of the critical nuclei We compute the radius of gyration R G, R 2 G = 1 N 2 N rij 2, i>j (S8) for the critical ice nuclei to evaluate their nonsphericity (NS). Following the definition used in Ref 35, the NS of an ice cluster is defined as the ratio between R G of the ice cluster and that of a perfect sphere containing the same number of molecules. For perfect sphere, the radius of gyration R GS = 3/5R, where R is the radius of the sphere. Therefore, the perfect sphere yields an NS of 1, while a less spherical shape yields higher NS. For example, a cylindrical rod with an aspect ratio of 5 yields an NS of For the droplets with radii of 2.4 nm, 3.1 nm, and 4.9 nm at 230 K, the calculated average NS for the critical nuclei are ± , ± , and ± respectively. The calculated distribution of NS for the critical nuclei in the droplet with radius of 2.4 nm is shown in Supplementary Figure S7. Therefore the critical nuclei are found to be nearly spherical, which is consistent with recent molecular simulations 36 using the mw water model, where the ice clusters were identified as nearly spherical at 220 K. 15

16 Supplementary References 33. Moore, E. B. & Molinero, V. Structural transformation in supercooled water controls the crystallization rate of ice. Nature 479, (2104). 34. Jiménez-Serratos, G., Vega, C. & Gil-Villegas, A. Evaluation of the pressure tensor and surface tension for molecular fluids with discontinuous potentials using the volume perturbation method. J Chem Phys 137, (2012). 35. Moore, E. B. & Molinero, V. Growing correlation length in supercooled water. J Chem Phys 130, (2009). 36. Reinhardt, A. & Doye, J. P. K. Free energy landscapes for homogeneous nucleation of ice for a monatomic water model. J Chem Phys 136, (2012). 16

hydrate systems Gránásy Research Institute for Solid State Physics & Optics H-1525 Budapest, POB 49, Hungary László

hydrate systems Gránásy Research Institute for Solid State Physics & Optics H-1525 Budapest, POB 49, Hungary László Phase field theory of crystal nucleation: Application to the hard-sphere and CO 2 Bjørn Kvamme Department of Physics, University of Bergen Allégaten 55, N-5007 N Bergen, Norway László Gránásy Research