Evolution of the structure of amorphous ice - from low-density amorphous (LDA) through high-density amorphous (HDA) to very high-density amorphous (VHDA) ice
ABSTRACT We report results of molecular dynamics simulations of amorphous ice for pressures up to 22.5 kbar. The high-density amorphous ice (HDA) as prepared by pressure-induced amorphization of Ih ice at T=80 K is annealed to T=170 K at various pressures to allow for relaxation. Upon increase of pressure, relaxed amorphous ice undergoes a pronounced change of structure, ranging from the low-density amorphous ice (LDA) at p=0, through a continuum of HDA states to the limiting very high-density amorphous ice (VHDA) regime above 10 kbar. The main part of the overall structural change takes place within the HDA megabasin, which includes a variety of structures with quite different local and medium-range order as well as network topology and spans a broad range of densities. The VHDA represents the limit to densification by adapting the hydrogen-bonded network topology, without creating interpenetrating networks. The connection between structure and metastability of various forms upon decompression and heating is studied and discussed. We also discuss the analogy with amorphous and crystalline silica. Finally, some conclusions concerning the relation between amorphous ice and supercooled water are drawn. Comment: 11 pages, 12 postscript figures. To be published in The Journal of Chemical Physics
arXiv:cond-mat/0501543v1 [cond-mat.stat-mech] 22 Jan 2005
Evolution of the structure of amorphous ice - from low-density amorphous (LDA)
through high-density amorphous (HDA) to very high-density amorphous (VHDA) ice.
R. Martoˇ n´ ak,∗D. Donadio, and M. Parrinello
Computational Science, Department of Chemistry and Applied Biosciences,
ETH Zurich, USI Campus, Via Giuseppe Buffi 13, CH-6900 Lugano, Switzerland
(Dated: February 2, 2008)
We report results of molecular dynamics simulations of amorphous ice for pressures up to 22.5
kbar. The high-density amorphous ice (HDA) as prepared by pressure-induced amorphization of Ih
ice at T = 80 K is annealed to T = 170 K at various pressures to allow for relaxation. Upon increase
of pressure, relaxed amorphous ice undergoes a pronounced change of structure, ranging from the
low-density amorphous ice (LDA) at p = 0, through a continuum of HDA states to the limiting
very high-density amorphous ice (VHDA) regime above 10 kbar. The main part of the overall
structural change takes place within the HDA megabasin, which includes a variety of structures
with quite different local and medium-range order as well as network topology and spans a broad
range of densities. The VHDA represents the limit to densification by adapting the hydrogen-bonded
network topology, without creating interpenetrating networks. The connection between structure
and metastability of various forms upon decompression and heating is studied and discussed. We
also discuss the analogy with amorphous and crystalline silica. Finally, some conclusions concerning
the relation between amorphous ice and supercooled water are drawn.
PACS numbers: 64.70.Kb, 61.43.Er, 02.70.Ns, 07.05.Tp
Properties of amorphous solid water at low tempera-
tures continue to attract the attention of both theory and
experiment. It has been known for a long time that at
least two distinct amorphous forms of ice exist. High-
density amorphous ice (HDA) is prepared by compres-
sion of ordinary Ih ice to 12 kbar1and when recovered
at ambient pressure it has a density of ∼ 1.17 g/cm3.
Upon isobaric heating to 117 K, the density drops con-
siderably and a second form is found, called low-density
amorphous ice (LDA)1with a density ∼ 0.94 g/cm3. The
transition between LDA and HDA can also be induced
by pressure2,3. Interest in this system is enhanced by the
possible existence of a second critical point in water, pro-
posed originally in Ref.4and later elaborated in several
variants (Refs.5,6,7). According to this hypothesis, in the
deeply supercooled region a second critical point should
exist, below which two kinds of water, high-density liquid
(HDL) and low-density liquid (LDL) are separated via a
first-order phase transition. Experimentally, however, it
has not yet been possible to access the deeply supercooled
region and directly investigate its properties. In the lack
of direct evidence, the existence of two different amor-
phous forms of ice has been used as an indirect support
for this hypothesis, assuming that HDA and LDA, appar-
ently separated by a sharp transition, are simply glassy
forms of HDL and LDL8.
raised new questions about
the nature of amorphous ices as well as about their rela-
tion to the speculated second critical point of water. A
new amorphous form of ice has been reported9, prepared
by heating HDA under pressure of 11 kbar to T ∼ 170
K and cooling it back to T = 80 K; when recovered
at ambient pressure it has a density of ∼ 1.25 g/cm3.
It has been called very high-density amorphous ice
(VHDA) and characterized experimentally by neutron
diffraction11. More detailed structural measurements of
VHDA ice were performed very recently17and showed
that the radial distribution function (RDF) of VHDA
is in fact more structured than that of HDA and LDA,
revealing the presence of an enhanced medium-range
order. Other experiments12,13,14,16focused on the HDA-
LDA transition induced by heating at ambient or low
pressure. In Ref.12it was shown that by heating HDA
to temperatures intermediate between 80 K and 110 K
the sample can be trapped in an apparent continuum
of metastable structures between HDA and LDA. This
suggests that there might be no sharp transition between
the two forms.In Ref.13the kinetics of the HDA to
LDA transition was studied; this revealed three stages of
the transformation, consisting of the annealing of HDA,
followed by an accelerated transition to the LDA form
and subsequent annealing of the LDA. The experiment
in Ref.15, while also finding a continuum of HDA states,
observed also a propagation of the LDA-HDA interface,
thus providing evidence in favor of a sharp transition
between the two forms. Possible implications of the new
experiments have been discussed16,18,19. Several review
papers on amorphous and supercooled water have also
appeared recently, see Refs.20,21,22.
Simulations can complement the experiment by pro-
viding access to shorter time scales, not easily accessed
in experiment. At the same time they can provide de-
tailed structural information which might not be easily
extracted from the experimental data. The basic phe-
nomenology of amorphous ices has been reproduced in
earlier works8,23,24. New simulations have also been per-
Here we present the results of extensive constant-
pressure MD simulations of amorphous ice at tempera-
tures 80 - 170 K and pressures up to 22.5 kbar. We focus
on the structure of the annealed form of amorphous ice
and study its evolution under increasing pressure. We
identify the LDA and HDA regimes, possibly separated
by a transition. VHDA on the other hand does not ap-
pear to be a new phase, but rather a particular high-
pressure regime of HDA. In particular, we suggest that
both new phenomena, the VHDA and the continuum of
metastable HDA densities at ambient pressure, originate
from the relationship between the density and the topol-
ogy of the hydrogen-bonded network of the HDA phase.
A preliminary account of this work has already been pub-
lished in Ref.31. Here we present a more detailed account
of the results as well as new simulations and new anal-
ysis. The paper is organized as follows. In section II
we present the model and details of our simulation tech-
nique. In section III we present our results. First we
describe the protocol used to prepare various forms of
amorphous ice and then discuss in detail their properties,
comparing with experiment as well as with other simu-
lations. In section IV we draw an analogy of another
important tetrahedrally-bonded system, amorphous sil-
ica. Finally, in the last section V we summarize our find-
ings and draw some conclusions concerning the relation
between amorphous ice and supercooled water.
II. MODEL AND TECHNIQUE
Our tool is constant-pressure molecular dynamics
(MD) simulation. We employed the GROMACS code32
using the Parrinello-Rahman constant-pressure MD33,
the Berendsen thermostat34and a time step of 2 fs. Elec-
trostatic interactions were treated by the particle-mesh
Ewald method35. An initial proton-disordered configu-
ration of Ihice with 360 H2O molecules and zero dipole
moment in an orthorhombic box was prepared using the
Monte Carlo procedure of Ref.36.
The interaction between water molecules was described
by the classical TIP4P model37.
tions this was found to reproduce well the transitions
Ih – HDA and LDA – HDA, both qualitatively and
quantitatively8,23,24. Although it is known to system-
atically overestimate the experimental ice densities by
about 2 %, the TIP4P model has recently been shown to
be capable of predicting qualitatively correctly the entire
phase diagram of water38.
In previous simula-
III.RESULTS AND DISCUSSION
A. Preparation and annealing of amorphous ice
First we shall describe the simulation protocol that
we applied. We started by compressing the ice Ih at
T = 80 K, increasing the pressure in steps of 1.5 kbar.
At 13.5 kbar a sharp transition occurs and the density
increases by almost 30 % to 1.37 g/cm3(Fig.1). The
sample was then further compressed at T = 80 K to 22.5
kbar and from 15 kbar decompressed to p = 0. This par-
ticular HDA form obtained by pressure-induced amor-
phization of Ihice at T = 80 K and subsequent compres-
sion or decompression, without any thermal treatment,
will be in the following denoted by HDA’ (as-prepared
HDA phase). During decompression the HDA’ density
gradually decreased and at p = 0 reached the value of
1.19 g/cm3, close to the HDA experimental value of 1.17
g/cm31. The radial distribution function (RDF) of HDA’
at p = 0 is shown in Fig.2; it has a broad second peak
between 3.3 and 4.6˚ A, very similar to that of HDA at
p = 010. Inspired by the experiments9that led to the
discovery of the VHDA we decided to anneal the HDA’
phase at each intermediate pressure between 22.5 kbar
and zero (a similar treatment was applied in Ref.9at
p = 8.4,11 and 19 kbar) in order to search for possible
new structures. Annealing was performed by heating up
to T = 170 K and subsequent cooling down to 80 K; the
temperature was always changed in steps of 10 K. The
phase obtained in this way will be called relaxed phase
(RP). We note here that in Ref.39a slow molecular dif-
fusion was observed in the MD simulation of the TIP4P
model at T = 173 K; therefore at 170 K it should be pos-
sible to equilibrate the system within accessible simula-
tion times. While in experiments a HDA’ sample heated
at an arbitrary pressure might recrystallize8,18,40, in the
time scale of a simulation this is not likely to happen.
We are therefore restricted to exploring the (metastable)
disordered structures. In order to check the metastability
of the RP phase upon decompression, the RP prepared
at each pressure was finally decompressed at T = 80 K
down to p = 0, decreasing the pressure in steps of 1.5
18 20 22
the relaxed phase (RP)
RP decompression (VHDA)
80 K for the various amorphous phases during compres-
sion/decompression. The triangles point in the direction of
pressure change, the lines are just a guide for the eye.
1: Densityvs. pressure dependence at
RP p=0.75 kbar
RP p=1.5 kbar
RP p=2.25 kbar
RP p=3 kbar
RP p=4.5 kbar
RP p=7.5 kbar
RP p=10.5 kbar
RP p=22.5 kbar
HDA’ at p=0
RP from 4.5 kbar at p=0
RP from 15 kbar at p=0 (VHDA)
FIG. 2: Oxygen – oxygen radial distribution function of var-
ious amorphous phases at T = 80 K and p = 0 − 22.5 kbar.
The curves in the upper part of the figure have been shifted
by 8 for clarity.
After each change of pressure or temperature the sys-
tem was equilibrated for 5 ns and observable quantities
were averaged over 0.5 ns. An additional equilibration
for 25 – 50 ns was performed during the annealing at the
highest temperature of T = 170 K. We stress here that
the long equilibration times are indeed necessary in order
to allow the system to undergo structural changes; e.g at
pressure p = 0.75 kbar the equilibration of the system at
T = 170 K takes about 20 ns. A comprehensive summary
of the density vs. pressure dependence of Ih, HDA’, RP
and decompressed RP phases at T = 80 K is presented in
Fig.1 and will be discussed in detail in the next section.
In Fig.3 we show the relaxation of the density during
annealing of the HDA’ ice at p = 16.5 kbar to T = 170 K.
It can be seen that upon annealing at 90 and 100 K the
density grows while between 110 and 130 K the sample
undergoes a thermal expansion. At 140 K and 150 K
the density grows further but no substantial change is
observed above T = 150 K.
We have verified that the enthalpy of the relaxed phase
is lower than that of HDA’ ice at any pressure (Fig.4).
Assuming that the entropy contribution to the Gibbs po-
tential can be neglected at T = 80 K (and entropy differ-
ences between amorphous phases are anyway expected to
0 10 2030 40
90 100 110 120 130 140 150 160T=170 K
FIG. 3: Time dependence of the density during annealing
of the HDA’ ice at p = 16.5 kbar to T = 170 K. Points
where the temperature is increased by 10 K are marked by
vertical dashed lines. The temperature (in K) during each
time interval is shown in the top part of the figure.
be small41) this demonstrates that upon annealing at any
pressure HDA’ ice irreversibly relaxes to a state with a
lower free energy. This a posteriori justifies the necessity
of annealing in order to reach a metastable equilibrium
corresponding at each pressure to a well-defined ther-
modynamic phase. We note that the lowest difference
between the enthalpy of the HDA’ phase and that of the
RP is found at p = 4.5 kbar, suggesting that at the lat-
ter pressure HDA’ is closest to its corresponding relaxed
amorphous form (see also next subsection).
HRP - HHDA’ [kcal/mol]
FIG. 4: Decrease of enthalpy upon annealing: difference be-
tween the enthalpy of the RP phase and that of the HDA’
phase at T=80 K.
B. Evolution of the RP with increasing pressure
In this section we analyze the properties of the RP and
their dependence on pressure, with the focus on struc-
ture. The density vs. pressure dependence of the RP
(Fig. 1) can be considered as the equation of state of
amorphous ice. We note first that the HDA’ and RP
curves cross at about 7 kbar; for lower pressure anneal-
ing results in lower density while for higher pressures the
At p = 0 the density after annealing reaches a value
of 0.97 g/cm3, which agrees well with the experimental
LDA value of 0.94 g/cm3. The remarkable feature of
the RP curve is the narrow region between 1.5 and 2.25
kbar where the density increases by about 9 %, from 1.04
g/cm3to 1.13 g/cm3. Upon further increase of pressure
the density grows fast and reaches at p = 3 kbar the value
of 1.19 g/cm3. Beyond that point the density growth
slows down progressively and at the highest pressure of
22.5 kbar the density reaches a value of 1.49 g/cm3.
The O-O RDF’s of RP at different pressures are shown
in Fig.2. We also calculated the O-H RDF (not shown)
for RP at p = 0,2.25,6 and 15 kbar. Integrating be-
tween 1.5 and 2.25˚ A we found at all pressures a coordi-
nation number of 2, indicating a fully hydrogen-bonded
network. In order to characterize the evolution of the net-
work topology we calculated the ring statistics for RP at
all pressures. This reveals information on medium-range
order that otherwise might not be easily extracted from
the RDF42,43. We applied the ring counting algorithm44
using the criterion from Ref.45to identify the hydrogen
bonds (we used for the O-O distance a cutoff parameter
of rcut = 3.05˚ A and a tolerance of ∆ = 0.2˚ A). Our
aim is to draw qualitative conclusions, so we did not try
to bring down the statistical error by repeating several
times the quench from 170 K to 80 K and averaging over
the resulting structures. We now discuss the evolution
of the RP at increasing pressure in terms of RDF and
network ring statistics (Fig.5) and show that there are 3
The RDF of the RP at p = 0 (Fig.2) exhibits at r = 3.1
˚ A a very deep minimum between the first and second
shell and a well-defined second shell peak at r = 4.4
˚ A, very similar to that found experimentally for LDA
in Ref.46. The phase thus coincides with the LDA as
expected. The same is true at 0.75 kbar at a density
of 0.99 g/cm3, where the RDF within the second peak
is practically indistinguishable from the one at p = 0,
and only beyond 5˚ A can small differences be seen. At
1.5 kbar the density increases to 1.04 g/cm3while the
RDF becomes slightly different from that of LDA at p =
0; the very deep minimum between the first and second
shell is still present. At all these pressures the network
is dominated by 6-membered rings.
The properties of the RP at p = 2.25 kbar are rather
different, correlating with the sharp increase of density.
The second shell peak of the RDF drops and shifts to
lower r and at the same time the RDF grows substan-
tially in the region around r = 3.3˚ A, revealing the pres-
ence of interstitial molecules10. A dramatic change is
seen in the ring statistics: the number of 6-membered
rings now starts to drop and at the same time the num-
8 1012 14
FIG. 5: Number of n-membered network rings in the RP as
a function of pressure at T = 80 K and p = 0 − 22.5 kbar.
ber of 8-membered rings grows fast. This behavior is
compatible with a transition from LDA to HDA occur-
ring between 1.5 and 2.25 kbar. Around p = 4.5 kbar the
RDF develops a broad second peak between 3.2 and 4.4
˚ A, similar to that of HDA’ at p = 0 (Fig.2). This agrees
with the observation based on the enthalpy difference and
shows that the HDA’ is indeed closest to the RP at this
pressure. Approaching p ∼ 10 kbar the ring statistics
are definitely dominated by 8 and 9-membered rings; the
network has thus undergone a substantial reconstruction
(see also section IV). For p > 10 kbar, the ring statis-
tics almost stabilize, revealing that the reconstruction of
the network is practically completed; at the same time
the density growth slows down further and the compress-
ibility approaches that of ice Ih. This indicates that the
increase of density due to the more efficient packing of
the molecules has at p ∼ 10 kbar reached its limit at
the value of ρlim∼ 1.39 g/cm3, and further compression
proceeds mainly by elastic compression. In this limiting
regime the RDF develops a pronounced second peak at
r = 3.2˚ A (Fig.2) while the original second shell peak at
r = 4.4˚ A disappears completely. In the subsection IIIE
we will identify this regime with the VHDA form.
The above analysis provides quite a clear picture of
the evolution of the network topology when going from
LDA to VHDA. Concerning this point, rather contra-
dictory opinions have been presented in the literature,
based on the indirect information provided by the de-
tailed analysis of radial and spatial distribution functions
obtained from diffraction experiments and the empirical
potential structure refinement procedure47. First we note
that there is no sign of any discontinuity upon evolution
of the HDA phase into the limiting VHDA regime. In
Ref.11it was speculated that there is no need to postu-
late any significant reorganization of the network struc-
ture in moving between HDA and VHDA and that they
appear topologically isomorphous. Our results, however,
show clearly that the evolution of HDA between 2.25 kbar
and 10 kbar must inevitably involve a substantial recon-
struction of the network. On the other hand, in Ref.10
it was suggested that HDA’ under pressure shows some
characteristics of interpenetrating networks, similar to
those of high-pressure crystalline ices VII and VIII. We
checked this feature in the RP up to the highest pres-
sure investigated, using the following algorithm. Start-
ing from each molecule we considered a sphere with ra-
dius rcut= 5˚ A and tested whether the molecules within
the sphere were connected to the central one by a path
containing no more than 4 hydrogen bonds. We found
that practically all molecules fulfilled this criterion; this
is clearly incompatible with the presence of interpene-
trating networks31. The VHDA structure can therefore
be considered as the upper limit to efficient amorphous
packing of the molecules without creating interpenetrat-
ing networks, as suggested in Ref.11. Still, it is an in-
teresting question whether amorphous ice with interpen-
etrating networks can exist. Very recently, a study of
VHDA was performed48in the region of densities rang-
ing up to 1.9 g/cm3, thus much higher than those stud-
ied here. Based on the analysis of bond angle distri-
butions, they showed that VHDA at very high densities
approaches a disordered close-packed structure, with lo-
cal order more similar to the fcc/hcp than to the bcc
crystal. From this they concluded that VHDA does not
represent a disordered version of ice VII and therefore
does not have interpenetrating networks. The algorithm
applied here in fact provides direct evidence. We have
also generated a sample of the RP at p = 42 kbar, with
density ρ = 1.61 g/cm3, and found that the ring statis-
tics were practically equal to the average of the RP in
the interval 10 - 22.5 kbar. This shows that the network
topology stabilizes already for densities ρ > ρlim∼ 1.39
g/cm3, although the local order converges only at much
C. Analysis of the shape of the rings
In order to obtain a deeper insight into the structural
response to pressure of the RP in the three regimes (LDA,
HDA, VHDA), we also performed a more detailed analy-
sis of the network structure, separating the contributions
coming from different rings.
The shape of the rings has been characterized through
the eigenvalues of the inertia tensor. Given the three
eigenvalues I1, I2and I3, sorted by increasing magnitude,
we define an elongation parameter ǫ as ǫ = (I2− I1)/I3
and the asphericity (α) as the root mean square devia-
tion of Iinormalized to I3. According to the definitions,
ǫ can vary from 0, for a circular ring or a sphere, to 1
for a linear arrangement, while α is zero for a sphere,
0.236 for a circular ring and 0.58 for a linear arrange-
ment. The distribution of α for all the rings considered
is peaked around ∼ 0.2 at p = 0. Increasing the pres-
sure increases the spreads of the distribution and shifts
the peak toward higher α. The tail of the distribution
never extends below 0.15, which means that the rings are
mainly planar structures.
The probability distribution of the elongation param-
eter provides a clear indication of the evolution of the
shape of different rings as a function of the pressure.
In HDA at p = 2.25 kbar (Fig.6a) the P(ǫ) shows that
the water molecules in five and six-membered rings ar-
range themselves in circular rings. The larger the ring
the broader the distribution, indicating that larger rings
can arrange into elongated structures without paying too
much in terms of strain energy. P(ǫ) at different pres-
sures for six and nine-membered rings are shown in Fig.6
panels (b) and (c), respectively. Six and nine-membered
rings, the quantity of which is most affected by pressure-
induced phase transitions, are shown as representative of
the behavior of small and large rings under pressure, re-
spectively. At low pressures (up to 2.25 kbar) the shape
of the small rings (Fig.6b) remains unchanged but when
the transition to HDA occurs their number rapidly de-
creases, and elongated large eight and nine-membered
rings are formed. In fact, already at p = 1.5 kbar the
main peak of the P(ǫ) of nine-membered rings (Fig.6c)
shifts from 0.1 to 0.48. The broad P(ǫ)’s of large rings at
higher pressures show that such rings can easily adapt to
any shape so as to achieve a more efficient compaction. In
the HDA region the morphology of small rings evolves to-
ward more elongated shapes and their amount decreases
continuously. When the onset pressure for the VHDA
regime is reached, the topology of the amorphous net-
work stops changing and also the P(ǫ)’s stabilize. The
residual small rings in VHDA are arranged into elongated
structures (the P(ǫ) is peaked at 0.4), whereas the P(ǫ)
of the nine-membered rings has no sharp peaks. At this
regime the response to further compression consists in
the deformation of short-range structural features, such
as the bond angle distributions, as no more important
rebonding is induced by increasing the pressure up to 42
We define a ring-restricted radial distribution function
n-rRDF(r) as the probability of finding two atoms at
a distance r within the same n-membered ring.
quantity allows us both to recognize the separate contri-
bution of different rings to the g(r) and to characterize
the response of different rings to compression. In Fig.7
the oxygen-oxygen n-rRDF(r) at different pressures are
shown for six and nine-membered rings. The position
of the first peak of both rRDF’s is unaffected by com-
pression up to 15 kbar and even a compression to 42
kbar induces a shift as small as 0.04˚ A. Low pressures
up to 2.25 kbar do not affect significantly the 6-rRDF,
whereas in the same range of pressures nine-membered
rings already provide a sizable contribution to the inter-
stitial region between the first and the second peak of
the g(r). In the region of stability of the HDA (6 kbar)
six-membered rings are strained and contribute to the
interstitial region of the g(r) through a broadening and a
shifting to the left of the second peak. On the other hand
the second peak of the 9-rRDF(r) spreads partly in the
0 0.10.20.3 0.4
0 0.10.20.3 0.4
P (ε) (6-membered rings)
p = 0 kbar
p = 1.5 kbar
p = 2.25 kbar
p = 6 kbar
p = 9 kbar
p = 15 kbar
p = 42 kbar
0 0.10.2 0.30.4
P(ε) (9-membered rings)
p = 0 kbar
p = 1.5 kbar
p = 2.25 kbar
p = 6 kbar
p = 9 kbar
p = 15 kbar
p = 42 kbar
FIG. 6: (a) Probability distribution of the elongation param-
eter of the rings in HDA at 2.25 kbar. (b) and (c) P(ǫ) at sev-
eral pressures from LDA to VHDA for six and nine-membered
rings, respectively. We note that all the curves are normal-
ized to one and therefore do not reflect the change of the total
number of rings with a given size.
p = 0 kbar
p = 1.5 kbar
p = 2.25 kbar
p = 6 kbar
p = 9 kbar
p = 15 kbar
p = 42 kbar
p = 0 kbar
p = 1.5 kbar
p = 2.25 kbar
p = 6 kbar
p = 9 kbar
p = 15 kbar
p = 42 kbar
FIG. 7: Ring-restricted radial distribution function at several
pressures for six (a) and nine-membered (b) rings.
interstitial region and gives rise to a shallow peak at 4.9
˚ A. Such a peak becomes more pronounced as the pressure
increases and is a feature of VHDA17. It is worth not-
ing that even at 42 kbar the six-membered rings do not
provide a sharp contribution to this feature. We note
that the rRDFs fully account for the interstitial peak
in the g(r) of VHDA, showing that this peak originates
from contributions within the same ring. This consti-
tutes independent evidence that there is no formation of
D. Transition LDA-HDA
The question of whether there is indeed a transition
between LDA and HDA and, if so, what its character is,
is of great importance. From the ring statistics shown
in Fig.5 it can be seen that the response of the num-
ber of rings to the applied pressure is rather different in
the regions below 1.5 and above 2.25 kbar. Because of
the limited accuracy of the ring statistics calculated in
the frozen states, we can conclude that this behavior is
compatible with the existence of a phase transition as
required by the second critical point scenario4.
In order to determine whether the density of the RP
changes discontinuously between 1.5 and 2.25 kbar, we
performed at T = 170 K also a ∼ 30 ns simulation at an
intermediate pressure of 1.875 kbar. We did not observe
any metastability or hysteresis effects but instead rather
large and slow density fluctuations. Since our system
is relatively small, such behavior is compatible with a
weak first order transition, occurring just below the crit-
ical temperature, where the system oscillates over a low
barrier between two states. In principle it is, however,
also possible that the change of density with pressure is
genuinely continuous, with a highly compressible region
between 1.5 and 2.25 kbar. In order to shed more light
on this important issue, it would be necessary to perform
simulations with larger systems, including several system
sizes, and apply standard finite-size scaling techniques49.
It might also be useful to perform free energy calcu-
lations, e.g. umbrella sampling, with a suitable order
parameter, similar to that performed in Ref.50.
techniques would, however, require considerable CPU re-
sources to achieve a good equilibration and sampling,
since in the interesting region the free energy surface is
very rough, resulting in very long autocorrelation times.
E. Decompression of RP to p = 0
Most experimental data on amorphous ices have been
gathered on samples decompressed to ambient pressure.
To our knowledge, there are no experimental data for the
RP under pressure to which we could directly compare
our results of the subsection IIIB. We discuss now the
interesting behavior of the RP upon decompression.
8 1012 14
FIG. 8: Density of the RP after decompression at T = 80 K
from pressure p.
The density of the LDA phase from 0.75 kbar relaxes at
p = 0 to ρ = 0.98 g/cm3, very close to the ρ = 0.97 g/cm3
of LDA prepared at p = 0. The LDA phase from 1.5 kbar
on the other hand reaches at p = 0 a somewhat higher
value of ρ = 1.025 g/cm3, suggesting the possibility that
at T = 80 K and p = 0 there might not be a unique struc-
ture of the LDA (in agreement with Refs.26,13) and this
phase can actually span a narrow density interval. In the
pressure interval p = 2.25 − 10 kbar, all decompression
curves are roughly parallel (Fig.1) and the faster growing
RP density results upon decompression in an increasing
density at p = 0, spanning the interval from 1.10 to 1.26
g/cm3. The picture changes remarkably for p > 10 kbar.
Here, the slope of the RP curve becomes close to that
of the decompression curves which lie close to each other
and initially almost follow the RP curve. Decompression
from almost all pressures results at p = 0 in a narrow
interval of densities around ρV HDA∼ 1.29 g/cm3, corre-
sponding to the decompression from the limiting density
ρlim. For convenience, in Fig.8 we show the dependence
of the final p = 0 density on the original pressure p at
which the RP was prepared, where the saturation can
be clearly seen. This agrees very well with the obser-
vation in Ref.9where the samples annealed at 11 and
19 kbar reached upon decompression the same VHDA
density of 1.25 g/cm3; in fact, this was the reason why
VHDA was originally suspected to represent a new ther-
modynamic phase. The RDF of RP decompressed from
15 kbar (Fig.2) is clearly similar to that of VHDA recov-
ered at p = 0 in experiment11, showing the presence of
the distinct peak at r = 3.37˚ A, very close to the first shell
peak. This allows us to identify this form as VHDA. The
spectrum of metastable states at p = 0 (Fig.8) is thus
compatible with the existence of a narrow LDA region
and a broad continuum of metastable HDA states with a
density below that of VHDA (as found experimentally in
Ref.15). We note that the density spectrum of the HDA
states extends both above and below that of the HDA’,
and there is no reason to consider HDA’ as a particu-
lar state representative of the HDA phase. While the
LDA states might be separated from the HDA ones by a
gap, it must certainly be much smaller than the density
difference between LDA and HDA’ at p = 0.
We now suggest an explanation for the existence of an
upper limit ρV HDAto the density of metastable HDA at
T = 80 K and p = 0. In the HDA phase with ρ < ρlim,
the system with increasing pressure achieves a better
packing of molecules due to network reconstruction. This
must involve the breaking and remaking of bonds, which
at any density requires crossing a free energy barrier, and
is only possible due to the annealing at higher temper-
ature, in our case 170 K. During cooling to 80 K, the
network topology becomes frozen. Upon subsequent de-
compression at 80 K, the barriers cannot be crossed and
therefore the system cannot relax to a lower density via
reconstruction of the network. This is illustrated in Fig.9
where the evolution of the ring statistics upon decompres-
sion of the RP from p = 13.5 kbar is shown; it can be seen
that no pronounced change in the network topology oc-
curs. The largest change is seen in the number of 9 and
8-membered rings, which decrease by about 20 % and
9%, respectively, while the number of other rings stays
practically constant. The decompression thus proceeds
dominantly via relaxation of the elastic compression and
that is why the HDA states with ρ < ρlim, which have
a variety of different topologies, relax elastically to dif-
ferent states, spanning a range in density. On the other
hand, all HDA states with ρ > ρlim, which have almost
the same network topology, relax upon decompression to
the same density ρV HDA. This accounts for the behav-
ior observed in Ref.9, with no need to postulate VHDA
to be a new phase, and is also consistent with the fact
that we did not observe any discontinuity during the evo-
lution from HDA to VHDA. The origin of the behavior
of VHDA upon decompression therefore appears to be
kinetic rather than thermodynamic.
8 1012 14
FIG. 9: Ring statistics during the decompression of RP at
T = 80 K from p = 13.5 kbar to zero.
The two points from the highest pressures of 21 and
22.5 kbar reach upon decompression a lower density, 1.21
and 1.23 g/cm3, respectively.
related to the fact that our decompression is performed
too fast, resulting in excessive elastic energy at p = 0.
This may, in turn, allow some barriers to be crossed and
enable a transition to a density ρ < ρV HDA.
We comment now on the experiments12,14, where
HDA’ was heated to intermediate temperatures between
80 and 120 K and at each temperature annealed for sev-
eral hours. On this time scale every increase of tempera-
ture resulted in an initial slow drop of the density which
afterwards gradually stabilized; an additional drop of the
density could be observed only by further increase of tem-
perature. This behavior clearly points to the fact that
as the HDA phase approaches its low density limit, the
barriers increase and can be overcome only at a higher
temperature. Our picture of the structural evolution of
the HDA phase is compatible with these experimental
findings. It is plausible to assume that the height of the
barriers is correlated to the amount of network recon-
struction necessary to change the volume, and is there-
fore related to the pressure derivatives of the number of
rings. As shown in section IIIB, the reconstruction of the
We believe that this is
RP upon increasing pressure is most dramatic at the low
density limit of the HDA spectrum, and with increasing
density becomes gradually less pronounced, until it prac-
tically vanishes for ρ > ρlim. We explicitly checked the
degree of metastability of HDA forms with different den-
sities (RP decompressed from several different pressures)
by heating at p = 0; the results are shown in Fig.10.
The temperature was increased in steps of 10 K, spend-
ing 5.5 ns at each temperature, and we note that also
here it would be useful to perform the heating several
times and average in order to improve the statistical er-
ror. Nevertheless, it can be clearly seen that the most
stable structure under heating is actually the lowest den-
sity HDA (RP decompressed from 2.25 kbar) which un-
dergoes only a small drop in density up to 130 K. With in-
creasing initial density the samples start to relax at lower
temperature, which confirms the above relation between
the network topology and barriers. It is also seen that
HDA’ is the least stable of all the samples, as may be ex-
pected for an insufficiently equilibrated phase possessing
an excess free energy. We stress that at T = 170 K all
curves reach practically the same density, lower than 0.99
g/cm3, which also agrees with the experimental finding
in Refs.9,11that VHDA upon heating converts to LDA.
This is different from what found in Ref.28where it was
argued that VHDA upon heating converts to a different
form of LDA, with a higher density of about 1.04 g/cm3.
RP from 2.25 kbar
RP from 4.5 kbar
RP from 10.5 kbar (VHDA)
RP from 15 kbar (VHDA)
FIG. 10: Density as a function of temperature during heating
of various decompressed RP phases as well as HDA’ at p = 0.
In Ref.28various HDA forms were prepared by an al-
ternative procedure, consisting of the pressure-induced
amorphization of cubic ice at different temperatures
ranging from 50 to 300 K. It was concluded that the
HDA’ produced by pressure-induced amorphization at
liquid nitrogen temperature and below represents a lim-
iting form of HDA and thus the phase spans a density
interval between HDA’ and VHDA. This procedure does
not cover the part of the HDA spectrum that has a den-
sity below that of HDA’ and can exist at p = 0 and
T = 80 K in metastable form and therefore we do not
consider the imperfectly equilibrated HDA’ to be a lim-
iting form of HDA.
In Ref.11the interstitial occupancy in HDA’ and
VHDA at ambient pressure was calculated by integrating
the O-O RDF between 2.3 and 3.3˚ A. The values of 5.0
and 5.8 were found, respectively, corresponding to 50 %
or almost 100 % occupancy of the “lynch pin location”. It
was speculated that due to some unknown specific mech-
anism only these values and no intermediate ones can be
made stably at ambient pressure. We calculated the same
occupancy at T = 80 K and p = 0 in our HDA’ as well as
in the RP decompressed from all pressures (Fig.11). In
HDA’ we found a value of 4.9 while in the HDA branch
of decompressed RP we found an apparently continuous
spectrum ranging from 4.3 (from 2.25 kbar) to about 6
(from pressures above 15 kbar). This again clearly shows
that while VHDA indeed represents a limiting structure,
this is not the case for HDA’ whose occupancy close to
the value of 5 can be regarded as accidental.
8 1012 14
FIG. 11: Oxygen occupation number within 3.3˚ A in the RP
decompressed from pressure p at T = 80 K.
Recently, a neutron and X-ray diffraction study of the
VHDA structure was reported in Ref.17, showing the
presence of at least seven well-defined shells in the RDF
of the VHDA, extending almost to the distance of ∼ 20˚ A.
This reveals the presence of an enhanced medium-range
order in the VHDA. In order to check this property, we
also prepared a bigger VHDA sample consisting of 2880
water molecules, allowing us to calculate the RDF up to a
distance of 20˚ A. We followed basically the same protocol
as for the 360-molecule sample and annealed the HDA’
at p = 15 kbar, but using shorter simulation times. The
radial distribution function DOO(r) = 4πρr(g(r) − 1) in
the decompressed sample is shown in Fig.12. Apart from
the height of the first peak, our result agrees well with the
experimental one (Fig.2(b)) in Ref.17, and also shows at
least seven coordination shells extending beyond ∼ 16˚ A.
The presence of such enhanced medium-range order is
likely to be related to the fact that the network topol-
ogy of VHDA is dominated by large rings. In Ref.17the
existence of a well-defined shell at 5˚ A was pointed out;
this is also clearly present in our RDF and in subsection
IIIC we have identified its origin in the contribution of
9-membered rings (see Fig.7(b)).
FIG. 12: Oxygen-oxygen radial distribution function DOO(r)
for the VHDA prepared by annealing at p = 15 kbar and
decompressing to zero pressure.
IV. ANALOGY TO SILICA
In spite of the different nature of the bonds between
water molecules in amorphous ice and between silicon
and oxygen atoms in amorphous silica, both systems con-
sist of a continuous random network of corner-sharing
tetrahedra51and in some windows of their phase dia-
grams display analogous phenomenologies when pressure
is applied. Each tetrahedral unit in silica is made of a
four-fold coordinated silicon atom in the center and four
bridging oxygen atoms at the corners. The size distri-
bution of the primitive rings44,52is peaked at six silicon
atoms per ring and presents a sizable amount of four
to ten-fold rings42,44,52. A high-density (HD) phase of
amorphous silica was discovered by Grimsditch53about
20 years ago. It was shown that upon compression above
8 GPa a-SiO2undergoes a permanent densification which
amounts to about 20% when the system is released to
ambient pressure and the transition to the HD phase is
accompanied by irreversible structural changes, observ-
able by Brillouin and Raman measurement53,54.
As in the case of amorphous ice, the nature of the phase
transition is still debated, since no discontinuous volume
change is observed in compression experiments53,54,55
while computer simulations do not clarify whether there
is a kinetically hindered first order transition56or a pres-
sure window where there is a balance between two den-
croscopic mechanisms cooperate in accommodating the
amorphous silica network in a smaller volume42,57,58,59
and make the phase transition between LDA and HDA
apparently smoother than in amorphous ice. In the low
pressure regime (≃3 GPa according to Refs.42,57) the
In fact three different mi-
volume diminishes only by the buckling of the network,
which results in a shift toward smaller values of the Si-
O-Si bond angle distribution. In this process the tetra-
hedral units are not deformed and no bonds are broken.
At higher pressures the buckling mechanism is supple-
mented by a substantial rebonding in the network, which
mainly affects the medium-range order features: the local
tetrahedral order is preserved but the ring-size distribu-
tion broadens and its peak shifts to larger rings59. In
the response of silica to pressure the buckling and the
rebonding mechanisms correspond to elastic and plastic
regimes, respectively, as observed by D´ avila et al. in MD
simulations43. On the other hand, in both regimes short-
range structural properties such as the Si-O-Si and the
O-Si-O angles vary continuously. In addition, coordina-
tion defects may be formed and contribute to densifica-
tion at even higher pressures (e.g. > 5 GPa in Ref.42). In
ice the nature of the hydrogen bond inhibits this latter
mechanism, as no more than four hydrogen bonds per
water molecule can be formed: in fact the average co-
ordination number of the RP is ∼4 at all the explored
pressures. Consequently, when the limit for the topologi-
cal densification is reached, amorphous ice turns back to
an elastic regime (VHDA).
The increasing of the characteristic size of the
rings upon densification is a general feature of both
amorphous52,60,61and crystalline62tetrahedral networks.
Among the tetrahedrally coordinated crystalline silica
polymorphs, the lower density forms (cristobalite and
tridimite) consist of six-membered rings only.
denser silica crystals the average size of the rings in-
creases accordingly to the density. For example coesite,
which is 30% denser than α-cristobalite has an average
ring size of ten and contains rings of size up to twelve. It
is worth noting that unlike the HD crystalline phases of
ice, it is impossible to form silica polymorphs with inter-
penetrating networks, because of sterical interactions.
Upon increase of pressure, relaxed amorphous ice un-
dergoes a pronounced change of structure, from LDA at
p = 0 to VHDA at p > 10 kbar. During this transfor-
mation, there is possibly a discontinuity between LDA
and HDA, although from our simulations performed on
a relatively small system we cannot distinguish between
a weak first-order transition and a continuous change.
Nevertheless, we can clearly distinguish the LDA and
HDA regimes. This identification is based on the ex-
istence of a transition region characterized by a rapid
change of density, on the presence of interstitial molecules
and the behavior of the ring statistics. It is important
to note that the main part of the overall change of the
network topology does not occur during the LDA-HDA
transition (similar conclusion was also made in Ref.13)
but rather within the HDA phase, between p ∼ 2 and
10 kbar. Concerning the as-prepared HDA’, initially be-
lieved to be the only possible form of HDA, it does not
seem to have any special importance and represents just
one particular structure within the HDA megabasin. It
is not in equilibrium even within the space of amorphous
structures and its properties are determined by the condi-
tions of preparation28. As shown above, when prepared
at T = 80 K, HDA’ is rather close to RP at 4.5 kbar.
The HDA megabasin includes a broad range of struc-
tures with different local and medium-range order and
also spans a broad interval of densities. On the high-
density side, the onset of the VHDA regime marks the
limit to which the densification can be pushed by adapt-
ing the network topology, without creating interpenetrat-
ing networks. The low density limit of HDA stability is
more difficult to assess with precision. However, as shown
experimentally by Refs.12,15,13and also in our simula-
tions, the HDA region reaches substantially below HDA’
From the above it is clear that many forms of HDA ex-
ist and are metastable at T = 80 K upon decompression
to p = 0. The important question then is, to what extent
does this affect the use of the LDA/HDA phenomenology
as support for the conjecture for the second critical point
in water. In Ref.16it is claimed that since no unique HDA
exists, it is difficult to justify the conjecture of a second
critical point for water. We do not actually think that
this is necessarily the case. In our picture, the variety of
HDA ices which exist as metastable forms at p = 0 cor-
responds to the variety of topologically different HDL at
different pressures. These various HDA forms cannot in-
terconvert upon decompression at T = 80 K because this
would involve rebonding and require overcoming free en-
ergy barriers. In Ref.29it was suggested that VHDA is
a better candidate for the glassy phase continuous with
the HDL, rather than HDA’. Since we reserve the use of
VHDA for the particular high-pressure regime, we can
say that all forms of the HDA branch of the RP appear
to be equally good candidates for the glassy phase that is
the putative continuation of the HDL. Consequently, the
large density variation of HDA ice at p = 0 might reflect
the existence of a region of high compressibility of the
supercooled HDL just below the critical point25. As sug-
gested in Refs.13,26, various forms of LDA ice may also
exist at T = 80 K and p = 0, but their density variation
is likely to be much smaller. A sharp transition may well
exist between the upper limit of LDA and lower limit of
HDA continua, as suggested by the experiments in Ref.13.
To shed more light on this issue, simulations on larger
systems combined with finite-size scaling techniques, as
well as free energy calculations could be helpful.
In the discussion of the polyamorphism of ice its anal-
ogy with the properties of silica glasses can be useful,
although attention should also be paid to important dif-
We should like to acknowledge stimulating discussions
with V. Buch, D. Chandler, D. D. Klug, M. M. Koza and
J. S. Tse.
1O. Mishima, L. D. Calvert, and E. Whalley, Nature (Lon-
don) 310, 393 (1984).
2O. Mishima, L. D. Calvert, and E. Whalley, Nature (Lon-
don) 314, 76 (1985).
3O. Mishima, J. Chem. Phys. 100, 5910 (1994).
4P. H. Poole, F. Sciortino, U. Essmann, and H. E. Stanley,
Nature 360, 324 (1992).
5H. Tanaka, Nature 380, 328 (1996).
6H. Tanaka, J. Chem. Phys. 105, 5099 (1996).
7R. C. Dougherty, Chem. Phys. 298, 307 (2004).
8P. H. Poole, U. Essmann, F. Sciortino, and H. E. Stanley,
Phys. Rev. E 48, 4605 (1993).
9T. Loerting, C. Salzmann, I. Kohl, E. Mayer, and A. Hall-
brucker, Phys. Chem. Chem. Phys. 3, 5355 (2001).
10S. Klotz, G. Hamel, J. S. Loveday, R. J. Nelmes,
M. Guthrie, and A. K. Soper, Phys. Rev. Lett. 89, 285502
11J. L. Finney, D. T. Bowron, A. K. Soper, T. Loerting,
E. Mayer, and A. Hallbrucker, Phys. Rev. Lett. 89, 205503
12C. A. Tulk, C. J. Benmore, J. Urquidi, D. D. Klug, J. Neue-
feind, B. Tomberli, and P. A. Egelstaff, Science 297, 1320
13M. M. Koza, H. Schober, H. E. Fischer, T. Hansen, and
F. Fujara, J. Phys.: Condens. Matter 15, 321 (2003).
14M. Guthrie, J. Urquidi, C. A. Tulk, C. J. Benmore, D. D.
Klug, and J. Neuefeind, Phys. Rev. B 68, 184110 (2003).
15O. Mishima and Y. Suzuki, Nature 419, 599 (2002).
16G. Johari and O. Andersson, J. Chem. Phys. 120, 6207
17M. Guthrie, C. A. Tulk, C. J. Benmore, and D. D. Klug,
Chem. Phys. Lett. 397, 335 (2004).
18D. D. Klug, Nature 420, 749 (2002).
19A. K. Soper, Science 297, 1288 (2002).
20P. Debenedetti and H. Stanley, Phys. Today 56, 40 (2003).
21P. Debenedetti, J. Phys.: Condens. Matter 15, R1669
22C. Angell, Annu. Rev. Phys. Chem. 55, 559 (2004).
23J. S. Tse and M. L. Klein, Phys. Rev. Lett. 58, 1672 (1987).
24I. Okabe, H. Tanaka, and K. Nakanishi, Phys. Rev. E 53,
25M. Yamada, H. E. Stanley, and F. Sciortino, Phys. Rev. E
67, 010202(R) (2003).
26N. Giovambattista, H. E. Stanley, and F. Sciortino, Phys.
Rev. Lett. 91, 115504 (2003).
27I. Brovchenko, A. Geiger, and A. Oleinikova, J. Chem.
Phys. 118, 9473 (2003).
28B. Guillot and Y. Guissani, J. Chem. Phys. 119, 11740
29N. Giovambattista, H. E. Stanley, and F. Sciortino (2004),
30C. McBride, C. Vega, E. Sanz, and J. L. F. Abascal, J.
Chem. Phys. 121, 11907 (2004).
31R. Martoˇ n´ ak, D. Donadio, and M. Parrinello, Phys. Rev.
Lett. 92, 225702 (2004).
32E. Lindahl, B. Hess, and D. van der Spoel, J. Mol. Mod.
7, 306 (2001).
33M. Parrinello and A. Rahman, Phys. Rev. Lett. 45, 1196
34H. J. C. Berendsen, J. P. M. Postma, W. F. van Gun-
steren, A. DiNola, and J. R. Haak, J. Chem. Phys. 81,
35U. Essmann, L. Perera, M. L. Berkowitz, T. Darden,
H. Lee, and L. G. Pedersen, J. Chem. Phys. 103, 8577
36V. Buch, P. Sandler, and J. Sadlej, J. Phys. Chem. B 102,
37W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W.
Impey, and M. L. Klein, J. Chem. Phys. 79, 926 (1983).
38E. Sanz, C. Vega, J. L. F. Abascal, and L. G. MacDowell,
Phys. Rev. Lett. 92, 255701 (2004).
39H. Tanaka, J. Chem. Phys. 113, 11202 (2000).
40S. Klotz,G. Hamel, J. Loveday,
M. Guthrie, Z. Kristallogr. 218, 117 (2003).
41O. Mishima and H. E. Stanley, Nature (London) 396, 329
42K. Trachenko and M. T. Dove, Phys. Rev. B 67, 064107
43L. P. D´ avila, M.-J. Caturla, A. Kubota, B. Sadigh, T. D.
de la Rubia, J. F. Shackelford, S. H. Risbud, and S. H.
Garofalini, Phys. Rev. Lett. 91, 205501 (2003).
44X. L. Yuan and A. N. Cormack, Comp. Mat. Sci. 24, 343
45P. Raiteri, A. Laio, and M. Parrinello, Phys. Rev. Lett.
93, 87801 (2004).
46J. L. Finney, A. Hallbrucker, I. Kohl, A. K. Soper, and
D. T. Bowron, Phys. Rev. Lett. 88, 225503 (2002).
47A. K. Soper, Chem. Phys. 202, 295 (1996).
48A. M. Saitta, T. Str¨ assle, G. Rousse, G. Hamel, S. Klotz,
R. J. Nelmes, and J. S. Loveday, J. Chem. Phys. 121, 8430
49N. B. Wilding, J. Phys.: Condens. Matter 9, 585 (1997).
50J. S. van Duijneveldt and D. Frenkel, J. Chem. Phys. 96,
51W. H. Zachariasen, J. Amer. Chem. Soc. 54, 3841 (1932).
52C. S. Marians and L. W. Hobbs, J. non-Cryst. Solids 119,
53M. Grimsditch, Phys. Rev. Lett. 52, 2379 (1984).
54R. J. Hemley, H. K. Mao, P. M. Bell, and B. O. Mysen,
Phys. Rev. Lett. 57, 747 (1986).
55Q. Williams and R. Jeanloz, Science 239, 902 (1988).
56D. Lacks, Phys. Rev. Lett. 84, 4629 (2000).
57K. Trachenko and M. T. Dove, Phys. Rev. B 67, 212203
58L. Huang and J. Kieffer, Phys. Rev. B 69, 224203 (2004).
59L. Huang and J. Kieffer, Phys. Rev. B 69, 224204 (2004).
60L. Stixrude and M. S. T. Bukowinski, Phys. Rev. B 44,
61M. Grimsditch, A. Polian, and A. C. Wright, Phys. Rev.
B 54, 152 (1996).
62C. S. Marians and L. W. Hobbs, J. Non-Cryst. Solids 124,