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.
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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.
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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
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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].
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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
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(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.
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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.
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• 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.
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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
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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.
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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
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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
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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
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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
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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-
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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.
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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
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-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.
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