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New metastable form of ice and its role in thehomogeneous crystallization of water

Supplementary Information for

“New metastable form of ice and its role

in the homogeneous crystallization of water”

John Russo1, Flavio Romano1,2, and Hajime Tanaka1

1Institute of Industrial Science, University of Tokyo

4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan

2 Physical and Theoretical Chemistry Laboratory,

Department of Chemistry, University of Oxford, South Parks Road,

Oxford, OX1 3QZ, United Kingdom

This Supplementary Information is divided into five sections. In Section I we provide struc-

tural information on the Ice 0 phase for both the mW and TIP4P/2005 potentials. In

Section II we show that Ice 0 is a metastable crystalline phase for the TIP4P/2005 model

of water, and calculate its melting line. In Section III we describe in detail some of the tech-

niques that were developed for this study: (A) a new bond orientational order parameter

for tetrahedral crystal, (B) the loop analysis procedure, and (C) the CNT-US scheme for

the calculation of free energy barriers. In Section IV we look at the crystallization trajec-

tories from a microscopic perspective to elucidate the nucleation pathway. In Section V we

focus on the structural similarities between the liquid phase and the Ice 0 crystal that help

understanding the role of Ice 0 in the ice crystallization process.

1

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I. STRUCTURAL PROPERTIES OF THE ICE 0 CRYSTAL

We have shown that Ice 0 is a metastable crystalline phase for two popular models of

water, the mW and the TIP4P/2005 models. In analogy with the other high-temperature

crystal phases of water, we assume that Ice 0 is a proton-disordered phase, in which the

oxygen atoms occupy the positions of the t12 allotrope [1]. The unit cell of Ice 0 is tetrag-

onal and contains 12 molecules. The space group is P42/ncm, where the oxygens occupy

the Wyckoff positions 4b and 8i. The crystal structure is fully specified via the following

parameters (where we employ the standard crystallographic notation): the size a and c of

the tetragonal unit cell and the parameters x and z for the Wyckoff position 8i. In Table I

we report these parameters both for the mW potential at T = 187 K and P = 0 bar, and

for the TIP4P/2005 potential at T = 220 K and P = 1 bar.

For convenience, we provide in Table II the equilibrium positions of all 12 oxygen atoms

in the unit cell for the mW potential at T = 187 K and P = 0 bar. We have also added

hydrogen atoms positions for a configuration with proton ordering. Ice 0 is a proton dis-

ordered phase, and the coordinates in Table II have to be equilibrated to ensure that the

hydrogen bond network is disordered. The coordinates can be replicated in all three direc-

tions to obtain a crystal of the desired size. The coordinates listed in Table II provide a

good starting point for simulations of the TIP4P/2005 model as well.

TABLE I: Equilibrium parameters for the Ice 0 crystal. For the mW potential we report

the equilibrium parameters at T = 187 K and P = 0 bar, and for the TIP4P/2005 potential at

T = 220 K and P = 1 bar. a and c are the sizes of the tetragonal unit cell. x and z are the

parameters that define the Wyckoff position 8i.

mW TIP4P/2005

c/a 1.81 1.81

a 5.93 A 6.04 A

c 10.74 A 10.94 A

x 0.84 0.84

z 0.36 0.36

2

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TABLE II: Unit cell of the Ice 0 crystal for the mW potential at T = 187 K and P = 0

bar. The unit cell has size Lx = 5.93 A, Ly = 5.93 A, Lz = 10.74 A. The first three columns are the

coordinates for the oxygen atoms (A), while the remaining six columns represent the coordinates

for the hydrogens atoms for a particular configuration with proton ordering. We warn that the Ice

0 phase is a proton disordered phase, and that the configuration provided here should be properly

equilibrated to obtain a disordered hydrogen network.

xO yO zO xH1 yH1 zH1 xH2 yH2 zH2

0.00 0.00 0.00 0.34 0.71 -0.54 0.71 -0.34 0.54

2.96 2.97 5.37 2.26 3.32 4.83 2.62 2.26 5.91

2.96 2.97 0.00 3.67 3.31 -0.54 3.31 2.26 0.55

0.00 0.00 5.37 0.35 -0.70 4.83 -0.71 -0.34 5.92

4.96 2.00 3.85 4.25 2.34 4.39 5.31 1.29 4.39

0.97 3.94 3.85 0.29 3.27 3.85 1.33 4.30 3.03

0.97 1.99 9.22 1.67 2.33 9.76 0.29 2.67 9.22

4.95 3.93 9.22 5.30 4.64 9.76 4.60 4.29 8.41

3.94 4.96 6.90 3.59 4.26 6.35 3.26 5.64 6.90

1.99 0.97 6.90 1.28 0.63 6.35 1.63 1.33 7.71

1.99 4.97 1.53 2.34 4.25 0.98 2.67 5.64 1.53

3.94 0.98 1.53 4.65 0.63 0.99 4.29 1.33 2.34

To help the experimental detection of Ice 0, we plot in Fig. S1 the oxygen-oxygen structure

factor S(q) defined as

SOO(k) =

⟨1

N

∑i,j

e−ik·(ri−rj)

⟩,

where N is the number of oxygen atoms in the simulation box, k are the scattering vectors

(k = |k|), and ri is the position of oxygen atom i. The figure displays the principal Bragg

peaks and the Miller indices for the corresponding scattering planes.

3





1 1.5 2 2.5 3 3.5 4

k (Å-1)

0

10

20

S OO

(k)

mW TIP4P/2005

002110

012 111112

022

121

113

114

123

122

Fig. S1: Oxygen-oxygen structure factor S(q) of the Ice 0 crystal for the mW potential.

We show S(q) at T = 187 K and P = 0 bar (solid line) and that for the TIP4P/2005 potential

at T = 220 K and P = 1 bar (dashed line). For visualization convenience, the structure factor of

the TIP4P/2005 model is shifted vertically. The red labels indicate the Miller indices for some

relevant Bragg planes of the Ice 0 crystal structure.

II. METASTABILITY OF ICE 0 IN TIP4P/2005 WATER

In our Article, we calculated the phase diagram of the mW model of water from free-energy

calculations. For the TIP4P/2005 we will use instead a more direct approach by measuring

directly the melting temperature of ice from computer simulations, taking advantage of the

fact that crystalline phases cannot be superheated in presence of a free surface [2], unlike in

bulk simulations where the absence of defects kinetically stabilizes the solid phase.

Molecular dynamics simulations were run using the Gromacs (v.4.5) simulation package.

The isothermal-isobaric NPT ensemble was sampled through a Nose-Hoover thermostat

and an anisotropic Parrinello-Rahman barostat. Lennard-Jones interactions have a cutoff

at 0.95 nm, and cutoff corrections are applied to both energy and pressure. Electrostatic

interactions are calculated through Ewald summations, with the real part being truncated

at 0.95 nm, and the reciprocal part evaluated using the particle mesh method. The timestep

used was ∆t = 0.001 ps.

An Ice 0 crystal is prepared at density ρ = 0.9 g/cm3 with 1, 500 oxygens in the positions

4





0 20000 40000 60000 80000 100000t (ps)

-82000

-80000

-78000

-76000

E (k

J/m

ol)

T = 220 K T = 215 K

T = 225 K

T = 230 K

T = 235 K

T = 240 K

Fig. S2: Direct determination of Tm of Ice 0 at zero pressure. The energy as a function of

time at different temperatures is shown for simulations of an Ice 0 in contact with an empty gas.

These simulations provide an estimate of the melting temperature at zero pressure of 222.5±2.5K.

of the t12 allotrope [1] (corresponding to five unit cells in each direction), and hydrogens

positioned according to the Bernal-Fowler rules [3]. First, crystals are equilibrated at P = 1

bar (effectively zero) and at different temperatures (T = 215, 220, 225, 230, 235, 240 K), then

the size of the simulation box along the (001) direction is increased ten times, leaving the

crystals in contact with a free surface in both the (001) and (001) directions. The energy as

a function of time for the simulations with the free surface is shown in Fig. S2. While for

T = 215 K and T = 220 K the crystals are mechanically stable, for temperatures T > 225 K

the surface progressively melts. We confirm the same trend with several independent runs.

We thus determine the melting temperature of Ice 0 for the TIP4P/2005 model potential

to be Tm = 222.5± 2.5 K.

Starting from the melting temperature at zero pressure, we can trace the coexistence line

by integration of the Clausius-Clapeyron equation [5]. The results for the melting line of

Ice 0 at different pressures are reported in the phase diagram of Fig. S3, together with the

melting line calculated from the melting point of Ih reported in Ref. [4].

5





140 160 180 220 240 260200 T (K)

0

1

2

3

4

P (k

bar)

liq

Ih

Ice 0

Fig. S3: Virtual melting line of Ice IT for the TIP4P/2005 potential. The melting line

of Ice Ih, the most stable phase at low P , is also shown taken from Ref. 4. The slope of the Ice 0

melting line is steeper than that of Ice Ih also for this more realistic water model.

III. NEW METHODS

A. Order parameter for tetrahedral crystals

In our Article we have shown that metastable crystalline phases play an important role in

the crystallization pathway of water. Previous studies have largely ignored this role, and the

order parameters that are used for the study of water are designed to target only the stable

crystalline phases, Ice Ic and Ih. We develop here a novel scheme that, besides the stable

structures, includes also Ice 0 and the clathrate phases CS (results are displayed here for

the HS− III clathrate structure, but results for the CS− II are almost indistinguishable),

among several tetrahedral high pressure phases.

A (2l + 1) dimensional complex vector (ql) is defined for each particle i as qlm(i) =

1Nb(i)

∑Nb(i)j=1 Ylm(rij), where we set l = 12, and m is an integer that runs from m = −l

to m = l. The functions Ylm are the spherical harmonics and rij is the normalised vec-

tor from the oxygen of molecule i to the one of molecule j. The sum goes over the

first Nb(i) = 16 neighbours of molecule i. This choice accounts for the first two coor-

dination shells of tetrahedral crystals. We then introduce a spatial coarse-graining step

Q12,m(i) = 1Nb(i)

∑Nb(i)k=0 q12,m(k) [6]. The scalar product between Q12,m of two particles is

defined as Q12(i) · Q12(j) =∑

m Q12,m(i)Q12,m(j). Figure S4 shows the scalar product

6





(Q12(i)/|Q12(i)|) · (Q12(j)/|Q12(j)|) for different crystal structures and for the fluid phase at

the melting point. The figure shows that the symmetry underlying the Q12 order parameter

is able to distinguish crystalline structures (having high values of the scalar product) and

disordered configurations (having low values of the scalar product).

For each pair i and j of neighbouring water molecules (within a distance rc) we define a

connection if the scalar product (Q12(i)/|Q12(i)|) · (Q12(j)/|Q12(j)|) > 0.75. A molecule is

then identified as crystalline if it has at least 12 connected neighbours. In Fig. S5 we show

the probability distribution for the same structures of Fig. S4. We see that by adopting

the threshold of twelve connected neighbours, a very good separation between fluid and

crystalline environments is achieved.

Once particles in crystal-like environments are detected we can apply a further analysis to

distinguish to which of the ices each local environment belongs to. To this aim, the following

two order parameters can be used: Q4 and W4, defined as follows.

0 0.2 0.4 0.6 0.8 1

Q12

*Q12

0.1

1

10

100

P (Q

12*Q

12 )

Ice Ic (TIP4P/2005)

Ice Ic (mW)

Ice Ih (TIP4P/2005)

Ice Ih

CS-II (mW)

liquid (TIP4P/2005)liquid (mW)

Ice 0 (TIP4P/2005)Ice 0

Fig. S4: Distribution functions of the scalar product of the 12-fold bond orientational

order parameter. Scalar product Q12∗Q12 = (Q12(i)/|Q12(i)|)·(Q12(j)/|Q12(j)|) for the crystal

structures considered in this Article, and for a fluid configuration at the melting point, both for the

TIP4P/2005 model (symbols) and the mW model (continuous lines). The crystalline structures

are sampled at the state point T = 187 K and P = 0 bar for mW water, and T = 200 K and P = 0

bar for TIP4P/2005 water. The vertical dashed lines indicates the threshold chosen to identify

connected neighrbours.

7NATURE MATERIALS | www.nature.com/naturematerials 7




0 2 4 6 8 10 12 14 16

connections

0.001

0.01

0.1

1

P (

con

nec

tio

ns)

Ice Ic (TIP4P/2005)

Ice Ic (mW)

Ice Ih (TIP4P/2005)

Ice Ih (mW)

CS-II (mW)

liquid (TIP4P/2005) liquid (mW)

Ice 0 (TIP4P/2005) Ice 0 (mW)

Fig. S5: Distinction of crystals from a liquid. Probability distribution for the number of

connections for the same structures considered in Fig. S4. The vertical dashed line indicates the

number of connection that we adopt to identify crystalline structures.

CS liquid Ice 0

Ice Ic

Ice Ih

Fig. S6: Order-parameter correlation map for all the relevant ice crystals and a liquid.

Q4-W4 map for all the relevant crystalline structures and for the liquid state at the melting point.

Contour lines define the area within which 80% of the probability distribution of the corresponding

structure is located.

8





• Q4

q4 ≡ q4,m(i) =1

Nb(i)

Nb(i)∑j=1

Y4m(rij)

Q4 ≡ Q4,m(i) =1

Nb(i)

Nb(i)∑k=0

q4,m(k)

Q4(i) =√4π/(2 ∗ 4 + 1)|Q4(i)|

• W4

W4(i) =l∑

m1,m2,m3=0

l l l

m1 m2 m3

Q4,m1(i)Q4,m2(i)Q4,m3(i)

|Q4(i)|3

Figure S6 shows that the distributions of the different crystalline structures are well

separated in the Q4-W4 map. Based on this map we can use the following criteria for

distinguishing the different crystal structures.

1. Apply the Q12 scalar product criteria to detect whether the particle is in a crystalline

environment or not.

2. For each crystalline particle apply the conditions in the following order:

3. if Q4 < 0.05 the particle is a clathrate (this is the region where also high pressure

tetrahedral structures are also found, so for high pressure simulations these structures

should also be taken into account - for details see Ref. [7]).

4. if Q4 < 0.11 the crystalline identity is not clear: these particles are named Ice i

(intermediate ice).

5. if Q4 < 0.11 and W4 > 0 the particle is Ice Ih.

6. if Q4 < 0.145 tand W4 < 0 he particle is Ice 0.

7. if Q4 > 0.145 and W4 < 0 the particle is Ice Ic.

In Section IV we will look at the nucleus composition from a microscopic point of view.

In particular we will show that our definition of Ice i refers to structures which are on the

tail of the Ice 0 distribution. For this reason, we also consider in the analysis the sum of Ice

0 and Ice i.





0 5 10 15 20r (Å)

0

0.2

0.4

0.6

0.8

1

Ice IIce Ic Ice Ih Ice 0 + i

Ice iIce 0

T = 215 K

0

0.2

0.4

0.6

0.8

1

Ice IIceIc IceIh Ice0+iIce0Icei

T = 225 K

0 5 10 15 20r (Å)

0

0.2

0.4

0.6

0.8

1

Ice IIceIc IceIh Ice0 + iIce0Ice Ii

T = 235 K

0 5 10 15 20r (Å)

n i(r)/n(r)

n i(r)/n(r)

n i(r)/n(r)

Fig. S7: Radial distribution of polymorphs in a crystal nucleus. Radial fractional composi-

tion ni(r)/n(r) for critical nuclei at ambient pressure and T = 215 K (nc = 90, left panel), T = 225

K (nc = 180, central panel) and T = 235 K (nc = 400, right panel). The left and central panel are

for simulations with N = 2000 mW water molecules, while the right panel is for simulations with

N = 4000. The different phases are detected according to bond-orientational order parameters

and plotted with different lines. Continuous lines are for Ice I (Ic + Ih) and for the sum of Ice 0

and Ice i. r is the distance from the centre of mass of the nucleus, and the centre of mass of the

nucleus is located at r = 0.

The above scheme allows to study the structural composition of crystalline nuclei, as

done in Fig. 2d in the main text. In Fig. S7 we show the composition of critical nuclei

at ambient pressure and T = 215 K (nc = 90, left panel), T = 225 K (nc = 180, central

panel) and T = 235 K (nc = 400, right panel). Crystal nuclei are obtained with Umbrella

Sampling simulations, since at these temperatures crystallization events do not happen

spontaneously in simulation. Comparing the figure with Fig. 2d in the main text, we see

that the composition of nuclei is largely independent of the degree of supercooling, and Ice

0 is always present in a significant fraction.

B. Ring analysis

For each configuration, we can define a hydrogen bond network connecting every oxygen

atom to its neighbours via hydrogen bonding. For the atomistic TIP4P/2005 model of

water, we follow the definition of hydrogen bond given in Ref. [8] and which is widely used

in simulations of liquid water. For the coarse-grained mW model, we instead define the

hydrogen bond network by considering all pairs of nearest neighbours. Since we are studying

10





Fig. S8: Ring analysis of the hydrogen bonding network of water. Here we show as example

of a six membered ring (green) and a five-membered ring (red) spanning from a central oxygen

atom (cyan).

tetrahedrally coordinated crystalline environments, identification of nearest neighbours is

always unambiguous.

The majority of water molecules in the supercooled liquid state and all the molecules

in intermediate-to-low pressure crystals are four-coordinated. For any water molecule i we

can thus define 6 triplets that contain the molecule i and any pair of its nearest neighbours

(according to the hydrogen bond network). A ring is then defined as the shortest closed

path that contains all the molecules in the triplet. The topology around a water molecule is

then described by the number of members in each of these rings. Figure S8 depicts a water

molecule (cyan), a five-membered ring (red atoms), and a six-membered ring (green atoms).

Table III shows the average fraction of 5,6,7-membered rings around a water molecule

in several crystalline structures. A comparison between the crystalline fractions and the

structure of liquid water is given in the final Section. For the moment we just note that Ice

0 is the only phase that contains 7-membered rings, which are also quite abundant in liquid

water.

11





Based on the ring topology the phase of molecules in crystalline nuclei can be assigned.

The phases which we consider here are Ice I (either cubic or hexagonal, since the ring

topology does not distinguish between the two) and Ice 0. In Ice I the 6 rings around

each molecule are all 6-membered, (6,6,6,6,6,6). In Ice 0 we have instead two different

environments: (5,5,5,5,6,6) and (5,5,5,7,6,6). We thus classify a crystalline particle as either

Ice I or Ice 0 if at least five of the six rings have the correct number of members. Otherwise,

particles are assigned to the intermediate Ice i form [9].

C. CNT-US scheme and free energy barriers

Umbrella Sampling is a rare event simulation technique that allows overcoming large free

energy barriers along a path in phase space described by an order parameter n(q), where n is

the order parameter (that we consider here to be a scalar) and q are phase-space coordinates.

It works by introducing a bias η in the system’s Hamiltonian

H′ = H + η(n)

and then recovering the probability distribution of the order parameter in the unbiased

system P (n) by removing the bias as

P (n) = P ′(n) exp {βη(n)} . (1)

TABLE III: Average fraction of 5,6,7-membered rings around a water molecule in sev-

eral crystalline structures. Note that the number of m-membered rings per molecule is given

by nm = fm/m.

crystal f5 f6 f7

Ice Ic 0 1 0

Ice Ih 0 1 0

Ice 0 5/9 3/9 1/9

Clathrate CS − I 20/23 3/23 0

Clathrate CS − II 5/17 2/17 0

Clathrate HS − III 15/17 2/17 0





Here P ′(n) =∫exp (−βH′)δ(n−n(q)) dq/Z, where Z =

∫exp (−βH′) dq, is the probability

distribution of the order parameter in the biased ensemble.

In the study of crystal nucleation, the order parameter n is chosen as the size of the

largest crystalline cluster in the system. The bias is instead a quadratic function of the type

η = kn(n− n0)2 (kn: constant), that enforces the system to sample a region where n ∼ n0.

Several simulation windows are then prepared at different values of n0 and the barrier is

reconstructed by the Weighted Hystogram Analysis Method.

The above procedure suffers from a fundamental problem. Starting from a nucleus of size

n ∼ n0, an independent configuration is obtained only when the structure of the nucleus

changes, and such structural modifications require big changes in the nucleus size. But these

are usually suppressed bye the use of an harmonic potential. The problem can be mitigated

with a Parallel Tempering scheme, in which the different windows are allowed to exchange

configurations. But this requires a large number of simulations windows with big overlaps,

increasing the error in the barrier reconstruction step. In other words, the essential difficulty

comes from the fact that this method is valid only when the free energy barrier (∆F ) change

as a function of n over |n − n0| ∼√kBT/kn is reasonably smaller than kBT . Thus, this

method is not efficient for the case when |∂∆F (n)/∂n| is large. This is generally the case

of a high nucleation barrier.

These problems can be avoided by using the free-energy barrier as obtained from Classical

Nucleation Theory (CNT) to bias the simulations. It has been shown that CNT predictions

are relatively accurate for supercooled liquid water, at least at mild supercooling [10]. If

CNT is reasonably correct, then the bias flattens the free energy profile F (n) allowing for

a uniform sampling in the order parameter space. We thus introduce the following biasing

function

η(n) = |∆µ|n2/3(n1/3 − 3n1/3c /2) (2)

where |∆µ| is the chemical potential difference between the fluid and the stable crystalline

phase, and nc is the critical nucleus size. |∆µ| can be calculated from thermodynamic

integration, and nc is thus the only tunable parameter of the scheme.

The simulation procedure is the following. First several simulations are started with

different values of nc. Simulations with nc close to the real critical nucleus size will be

characterized by large fluctuations in the order parameter space. This is shown in Fig. S9

for the state point T = 249 K and P = 0 bar: the driving force is β|∆µ| = 0.237 and we





0 106 2x106 3x106MC sweeps

0

20

40

60

80

100

n

Fig. S9: An example of CNT-US trajectory. The graph shows the fluctuations in the order

parameter n for 2000 mW water molecules at T = 249 K and P = 0 bar. A reflective boundary

is set at n = 100 to constrain the trajectory inside the window n ∈ [0, 100]. For this state point

β|∆µ| = 0.237 and nc is set to nc = 1150. The sampling is taken over several independent replicas,

each much longer than the one shown.

observe that a value of nc = 1150 produces fluctuations which span the whole sampling

window. Using these simulations we can measure P ′(n), and then use Eq. (1) and Eq. (2)

to obtain the unbiased P (n) function, and the free energy barrier βF (n) = − logP (n).

Simulations can be parallelized simply by launching independent trajectories, and by using

reflective boundaries to sample different regions of the order parameter space independently.

We note that one of the advantages of the method is that during the initial stages of

the simulations one gets an estimate of the critical size. The estimated critical size is equal

to the value of nc that best obtains a uniform sampling of the order parameter n. As an

example, Fig. S10 shows the comparison between four barriers at ambient pressure: T = 215

K (nc = 90, black line), T = 225 K (nc = 180, blue line), T = 235 K (nc = 400, green line)

and T = 249 K (nc ∼ 1100, red line), together with the extrapolations predicted from the

CNT-US bias (dashed lines). The figure contains an important physical information. The

state point T = 249 K is close to the melting point of Ice 0 (Tm = 245 K), and has a barrier

significantly higher than 100 kBT . This means that the homogeneous nucleation of ice never

occurs at these temperatures, a result also recently shown in Ref. [11]. The steep increase of

14





0 100 200 300 400 500 600 700 800n

0

20

40

60

80

100

β ∆

F

T = 215 K T = 225 K T = 235 K T = 249 K

Fig. S10: Estimation of the nucleation barrier by the new CNT-US scheme. Here we

compare the free energy barrier for 2000 mW water molecules at ambient pressure and T = 215 K

(black line), T = 225 K (blue line), T = 235 K (green line, for N = 4000) and T = 249 K (red line).

The dashed lines are the barrier predictions according to the CNT-US bias with the nc estimate.

the barrier causes a drop in the nucleation rate by more than 20 orders of magnitude from

T = 215.1 K to T = 235 K, as reported in Table IV.

The shape of the barriers also points to the importance of metastable phases. Figure 1d in

the main text shows that the CNT functional form provides a poor description of the shape

of the barriers. Usually this is amended by introducing size-dependent surface tensions, or

various other terms to the CNT free energy. Here we follow instead a different approach.

We consider the nucleus as composed of the stable phase in its core (Ice Ic or Ice Ih), and

a surface composed of the metastable Ice 0 phase. We then write the free energy cost of

forming a nucleus as ∆F ≈ −|∆µI |NI − |∆µ0|N0 + 4πR2Iγ0/I + 4πγ0(RI +∆R)2, where the

TABLE IV: Nucleation rates for mW water. Nucleation rates for state points at ambient

pressure. The corresponding free energy barriers are shown in Fig. S10.

T (K) P (bar) nc ρ (g/cm3) f∗n (fs−1) Z β∆F (nc) κ (nm−3fs−1)

215.1 0 81 0.99 0.008 0.018 23.5 3.0× 10−13

225 0 180 1.00 0.026 0.0115 40.1 3.8× 10−20

235 0 400 1.00 0.047 0.0077 72.0 6.3× 10−34

15





subscripts 0 and I refer to the respective Ice phases, R is the radius of the nucleus, ∆R is

the thickness of the metastable phase shell, and N is the number of molecules in each phase.

By ignoring second order terms in ∆R we can write ∆F ≈ aN + bN2/3 + cN1/3. The best

fits according to this model are shown as continuous lines in Fig. 1d, and fully capture the

data from simulations (symbols) at all pressures.

IV. MICROSCOPIC ANALYSIS OF THE NUCLEATION PATHWAY

In Fig. S11(left panel) we plot the Q4-W4 probability distribution map at T = 235 K

and ambient pressure for crystalline particles belonging to the critical nucleus. Comparing

this figure to the one obtained for bulk crystalline phases (Fig. S6), we note that even for

relatively big nuclei (the critical size is nc = 400) the probability distribution functions for

crystalline particles are considerably broadened by finite size effects and for the high con-

centration of crystalline defects. Still (at least) three distribution can be found in proximity

of their bulk counterparts. For W4 > 0 the distribution belongs to the population of the

Ice Ih phase. For negative values of W4 the two distributions for Ice Ic and Ice 0 overlap.

To distinguish them we thus plot in Fig. S11(right panel) the reduced probability function

P (Q4|W4 < −0.05) (thus excluding the Ice Ih population), showing that the distribution

function is indeed bimodal. The different curves in Fig. S11(right panel) are obtained by

considering only particles within a certain distance r from the center of mass of the nucleus,

showing that particles close to the nucleus center mainly belong to the Ice Ic population,

and that moving towards the surface there is a rapid increase in the Ice 0 population. The

presence of a small Ice 0 peak also close to the nucleus center (r < 7 A) is due to deviations

from the spherical shape caused by the surface fluctuations of the nucleus. The large overlap

between the Ic and Ice 0 does not allow for an unambiguous distinction of the phase at the

particle level, a problem which is common to most uses of order parameters for the phase

detection of small crystalline units. In our study we follow the standard approach of assign-

ing the particle phase according to the values of the order parameters in the bulk [12–14].

Our choice of the parameters (see Fig. S6) is very conservative with respect to the Ice 0

phase, so that our results should provide a lower bound on the actual amount of this phase.

For the same reasons we have retained the definition of an intermediate ice phase (Ice i).

Ice i is defined as defected or partially formed structures that act as precursors of the crys-





0.05 0.1 0.15 0.2 0.25Q4

0

5

10

15

20

25

P(Q 4)

r < 7 År < 9 Å r < 11 År < 13 År < 15 Å

r

r}Fig. S11: Order parameter probability distribution functions for the critical nucleus at

T = 235 K and ambient pressure. (left panel) Q4-W4 map for crystalline particles belonging

to the critical nucleus at T = 235 K and ambient pressure (its radial profile is shown in the right

panel of Fig. S7). (right panel) Reduced probability distribution P (Q4) for crystalline particles

with W4 < −0.05: the different curves show P (Q4) for particles within a specified distances from

the centre of mass of the nucleus.

tallization process [9]. As shown in Fig. S11(right panel) there is no discernible peak that

can be attributed to Ice i (which in our map is defined whenever Q4 < 0.11), but instead

the population of Ice i can be interpreted as coming from the tails of the distribution of Ice

0. This is to be expected, as Ice i is not a crystalline phase of water, contrary to Ice 0 and

Ice I. Ice i was shown to play an important role in the crystallization pathway of water [9],

and in the following we show that the same applies for Ice 0, which we believe is its true

parent distribution.

V. STRUCTURAL SIMILARITIES BETWEEN SUPERCOOLED WATER AND

THE ICE 0 PHASE

The idea of structural similarity between a fluid phase and the crystal phase which nucle-

ates from it was already found to be important in several systems, as for example carbon [15],

hard spheres [16, 17] and ultra-soft potentials [18]. The peculiar thermodynamic properties

of water stem from the strength and directionality of the hydrogen bonds. To study the

structure of water we will thus consider the topology of rings and the distribution of dipole

moments between neighbouring particles.

17





0 1 2 3 4 5 6 7 8 9 10 11 12

n

0

0.1

0.2

0.3

0.4

0.5

f nT = 220 K T = 300 K Ice 0

0 1 2 3 4 5 6m

0

0.1

0.2

0.3

0.4

0.5

P (m

)

n = 5 T = 220 K n = 6 T = 220 K n = 7 T = 220 K n = 5 T = 300 K n = 6 T = 300 K n = 7 T = 300 K

Fig. S12: Structural similarity between a supercooled liquid water and Ice 0. (left panel)

Average fraction of n-membered rings around a water molecule, for mW water at both T = 220

K (circles) and T = 300 K (squares). Diamond symbols represent the distribution in Ice 0. (right

panel) Probability distribution to find m rings with 5 (circles), 6 (squares) and 7 (diamonds)

members, among the six rings that surround each four-coordinated water molecule at T = 220 K

(filled symbols) and at T = 300 K (empty symbols).

The ring topology is considered in Fig. S12. In the left panel we show the average fraction

fn of n-membered rings for mW water at both T = 220 K (circles) and T = 300 K (squares).

The three most abundant ring configurations are the 5,6,7-membered rings. Among all

crystalline phases at intermediate and low pressure, the Ice 0 phase is the only one containing

7-membered rings, while the stable crystals (Ice Ic and Ih) contain only 6-membered rings.

The figure also shows the fractional composition of Ice 0 (diamonds), suggesting that the

formation of this phase is triggered in regions of the network that are more abundant in 5-

membered rings. In the right panel we plot the probability distribution to find m rings with

5 (circles), 6 (squares) and 7 (diamonds) members, among the six rings that surround each

four-coordinated water molecule. For 7-membered rings, we see that most water molecules

are surrounded either by 0 or 1 rings (the last case being more common). This is the same

as in Ice 0 where for 7-membered rings m = 0, 1. For 6-membered rings we see that the

peak of the distribution is at m = 2, which is the same as in Ice 0, and in stark contrast to

the stable phases, where it is m = 6. Finally 5-membered rings are also peaked at m = 2,

while Ice 0 has its peak at m = 3. Again we see that the Ice 0 precursor should appear

in regions which are more abundant in 5-membered rings. We note that by lowering the

18





-1 -0.5 0.5 10

< cos φ >

0

1

2

3

4

5

P (<

co

s φ >

) liquid

Ice Ic

Ice Ih

Ice 0

first shell

-1 -0.5 0.5 10

< cos φ >

0

0.5

1

1.5

2

P (<

co

s φ >)

liquid

Ice Ic

Ice Ih

Ice 0

second shell

Fig. S13: Similarity of dipole-moment correlations between a supercooled liquid water

and Ice 0. (left panel) Distribution of the average angle < cos θ > between the dipole moment of

a molecule and its hydrogen-bonded neighbours for TIP4P/2005 water at T = 200 K and P = 1

bar in the liquid phase and the Ice Ic, Ih and Ice 0 phases. (right panel) The same as in the left

panel but for second nearest neighbours.

temperature from T = 300 K (empty circles) to T = 220 K (filled circles), the fraction of

m = 3, 4 5-membered rings increases, consistently with the fact that the structural similarity

with Ice 0 increases with supercooling.

Finally we consider the distribution of the dipole moments in both the fluid phase and

the crystals. We employ here the TIP4P/2005 model of water, as the mW model does not

contain information on the orientation of the dipole moment. We consider here the angle

between the dipole moment (d) of molecule i and a neighbouring molecule j as

< cos θ >ij= di · dj

Figure S13 shows the distribution of the average angle between a central water molecule and

its first (left panel) and second (right panel) nearest neighbours in both the liquid phase and

in the Ice Ic, Ih and Ice 0 phases. Both panels show the remarkable similarity in molecular

orientations between the supercooled liquid phase and the Ice 0 phase.

[1] Zhao, Z. et al. Tetragonal Allotrope of Group 14 Elements. J. Am. Chem. Soc. 134, 12362–

12365 (2012).

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[2] Vega, C., Martin-Conde, M. & Patrykiejew, A. Absence of superheating for ice Ih with a free

surface: A new method of determining the melting point of different water models. Mol. Phys.

104, 3583–3592 (2006).

[3] Buch, V., Sandler, P. & Sadlej, J. Simulations of H2O solid, liquid, and clusters, with an

emphasis on ferroelectric ordering transition in hexagonal ice. J. Phys. Chem. B 102, 8641–

8653 (1998).

[4] Abascal, J. L., Sanz, E. & Vega, C. Triple points and coexistence properties of the dense

phases of water calculated using computer simulation. Phys. Chem. Chem. Phys. 11, 556–562

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[5] Vega, C., Sanz, E., Abascal, J. & Noya, E. Determination of phase diagrams via computer sim-

ulation: methodology and applications to water, electrolytes and proteins. J. Phys. Condens.

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[6] Lechner, W. & Dellago, C. Accurate determination of crystal structures based on averaged

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[7] Romano, F., Russo, J. & Tanaka, H. (in preparation).

[8] Luzar, A. & Chandler, D. Effect of environment on hydrogen bond dynamics in liquid water.

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[9] Moore, E. & Molinero, V. Structural transformation in supercooled water controls the crys-

tallization rate of ice. Nature 479, 506–508 (2011).

[10] Reinhardt, A. & Doye, J. P. Note: Homogeneous TIP4P/2005 ice nucleation at low super-

cooling. J. Chem. Phys. 139, 096102 (2013).

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tion. J. Am. Chem. Soc. 135, 15008–15017 (2013).

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U.S.A. 107, 14036–14041 (2010).

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