-
The putative liquid-liquid transition is a liquid-solid
transition in atomistic models ofwaterDavid T. Limmer and David
Chandler
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THE JOURNAL OF CHEMICAL PHYSICS 135, 134503 (2011)
The putative liquid-liquid transition is a liquid-solid
transition in atomisticmodels of water
David T. Limmer and David Chandlera)Department of Chemistry,
University of California, Berkeley, California 94720, USA
(Received 15 July 2011; accepted 6 September 2011; published
online 5 October 2011)
We use numerical simulation to examine the possibility of a
reversible liquid-liquid transition insupercooled water and related
systems. In particular, for two atomistic models of water, we
havecomputed free energies as functions of multiple order
parameters, where one is density and anotherdistinguishes crystal
from liquid. For a range of temperatures and pressures, separate
free energybasins for liquid and crystal are found, conditions of
phase coexistence between these phases aredemonstrated, and time
scales for equilibration are determined. We find that at no range
of temper-atures and pressures is there more than a single liquid
basin, even at conditions where amorphousbehavior is unstable with
respect to the crystal. We find a similar result for a related
model of silicon.This result excludes the possibility of the
proposed liquid-liquid critical point for the models we
havestudied. Further, we argue that behaviors others have
attributed to a liquid-liquid transition in waterand related
systems are in fact reflections of transitions between liquid and
crystal. © 2011 AmericanInstitute of Physics.
[doi:10.1063/1.3643333]
I. INTRODUCTION
This paper reports the results of a numerical study aimedat
elucidating the purported1, 2 liquid-liquid phase transitionin
supercooled liquid water. The results indicate that this
hy-pothesized polyamorphism does not exist in atomistic modelsof
water. While not contradicting the existence of
irreversiblepolyamorphism of the sort observed in non-equilibrium
dis-ordered solids of water,3–7 and not excluding the
possibilitiesof liquid-liquid transitions in liquid mixtures,8
polymerizingfluids,9 and some theoretical models,10–13 the results
do sug-gest that a reversible transition and its putative second
criticalpoint are untenable for one-component liquids, like water,
thatexhibit local tetrahedral order and freeze into crystals
withsimilar but extended order.
The terminology “transition” is used here to refer to dis-tinct
phases, where coexistence implies the formation of in-terfaces that
would spatially separate the coexisting phasesor to response
functions that diverge in the thermodynamiclimit.14 The structural
changes for a transition between twoliquids or between a liquid and
a crystal are distinct from con-tinuous pressure induced changes in
normal liquid water.15
These changes associated with a phase transition are globaland
therefore are also distinct from bi-continuous behaviorsthat do not
persist beyond small length scales.16
Polyamorphism of water has been achieved through var-ious
out-of-equilibrium experimental protocols resulting in amultitude
of thermodynamically unstable, kinetically trappedstructures.17–22
These different disordered structures havebeen generally
partitioned into two general categories knownas either low-density
amorphous solids17–19 or high-densityamorphous solids.20–22 Some
have interpreted changes inthese structural motifs as
non-equilibrium manifestations of
a)Electronic mail: [email protected].
an underlying equilibrium phase transition between two formsof
liquid water. This conjecture forms the basis of some at-tempts to
explain many of the well-known anomalous ther-modynamic properties
of water, e.g., Refs. 1, 23, 24, and 25.There are other ways of
explaining these anomalies,26 but thephase transition hypothesis
seems particularly intriguing, andit is the focus of this
paper.
The hypothesis is impossible to test by natural exper-iments
because the location of the presumed transition isoutside
experimentally accessible conditions.27 In particular,bulk
supercooled water is unstable as a liquid, and it
rapidlycrystallizes in the regime of predicted polyamorphism.
Whilethe properties of non-equilibrium glassy materials can
bestudied in this region, it is uncertain whether inferences
re-garding reversible thermodynamic behavior can be made fromsuch
measurements. Some experiments have studied waterconfined to long
pores with radii no larger than 1 nm.28–31
These experiments avoid the instability and thereby attemptto
detect manifestations of the transition. While water doesnot freeze
in such circumstances, it is questionable whetherproperties of bulk
water can be inferred from behaviors ofthese one-dimensional
systems.32, 33
Molecular simulation provides a means to overcome thisambiguity.
Specifically, sufficiently realistic models can bestudied
computationally while controlling order parametersthat distinguish
liquid from crystal. It is in this way that weexamine the
reversible behavior of models of water. Alongwith establishing
coexistence between liquid and crystal, weare able to study the
dynamics of the transition betweenthese phases. We also locate and
explore the free energy sur-face for the region of the
pressure-temperature phase diagramknown as “no man’s land.”2 This
is the region where amor-phous behavior would be unstable in the
absence of control.Our results indicate that some observations
attributed by oth-ers as manifestations of a liquid-liquid
transition are in fact
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Institute of Physics135, 134503-1
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134503-2 D. T. Limmer and D. Chandler J. Chem. Phys. 135, 134503
(2011)
observations of the temperature-pressure boundary separatingthe
region of amorphous instability from that of a single phaseof
amorphous metastability.
Others have used molecular simulation for realistic mod-els of
water34–39 and related liquids40–43 to examine their pos-sible
polyamorphisms. Those cited here34–43 are representa-tive but by no
means comprehensive (we exclude from thislist models that do not
exhibit local tetrahedral order, e.g.,Ref. 10). In all cases, the
methods employed have been lim-ited in at least one of three ways:
time scales that are too short,system sizes that are too small, and
order parameters that failto discriminate order from disorder or
fail to be adequatelycontrolled. By employing multiple order
parameters and free-energy sampling methods, we overcome time scale
issues andare able to discriminate between phases of different
symme-tries. By considering different system sizes and size
scalinganalysis, we overcome uncertainty associated with finite
sys-tem sizes.
Most of the results we present in this paper have beencomputed
with a recently developed model by Molinero,so-called “mW” water.44
We use this model for three reasons.First, it is a computationally
convenient model because it con-tains no long-ranged forces,
relying instead on short-rangedthree-body forces to favor
microscopic structures consistentwith those of water. Second, the
behavior of the model is re-alistic in the sense that in the range
of conditions we wishto study its phase diagram is a reasonable
caricature of thatfor water.36, 45, 46 Third, the results obtained
with this modelwould seem to apply to other systems in addition to
waterin that the model is a variant of one developed by
Stillingerand Weber,47 which has been used to treat behaviors of
Si(Refs. 40, 41, and 43) and SiO2.48
II. PHASE SPACE STUDIED
The mW model44 differs from the Stillinger-Weber inter-particle
potential energy function for silicon in two ways. Thefirst is the
adoption of different values for the length and en-ergy parameters
of the model. This difference is inconsequen-tial to our study
because it amounts to a simple rescaling oftemperature and density.
The second is more substantive butslight. Specifically, to capture
some thermodynamic proper-ties of water, the mW model has a
partitioning between two-and three-body terms that differs by 10%
from the partitioningfor silicon. See Ref. 44 for details
concerning the mW model.
A. Pressure-temperature phase diagramof the mW model
We have studied the phase behavior of the mW modelby computing
free energy surfaces throughout its condensedphases. Figure 1 shows
the state points examined. Each circlerepresents a state point
where the free energy has been calcu-lated as a function of the
global system density and an orderparameter that quantifies broken
orientational symmetry. Thelatter distinguishes an amorphous liquid
phase from a crystal,whereas the former distinguishes two amorphous
phases withdifferent densities. Basins in the free energy surface
establishrelative stabilities of the phases.
To put this diagram in context, we highlight state pointsthat
others have identified as relevant to a liquid-liquid
phasetransition in supercooled water. The temperature of
maximumdensity at low pressure sets the scale of the figure. This
cho-sen reference temperature is T0 = 250 K for the mW model,44and
it is T0 = 277 K for water.51 The phase diagram in Fig. 1shows that
the density maximum of liquid mW occurs atslightly supercooled
conditions while that of experimentalwater occurs at a temperatures
slightly higher than the freez-ing temperature.
The points identified by Liu et al.28 come from
measuredrelaxation times of water confined in silica nanopores with
a7 Å radius. These relaxation times have a temperature depen-dence
that changes from super-Arrhenius to Arrhenius uponcooling below a
crossover temperature, Tx(p). This tempera-ture depends upon
external pressure p, and points on this lineare shown in Fig. 1
with unfilled squares, which are attributedin Ref. 28 to crossing a
“Widom line.” A Widom line refersto a locus of maximum response
that ends at a critical point.37
a
c
d
f
e
b
FIG. 1. Phase space sampled in our calculations for the mW
model. Opencircles refer to states where the amorphous liquid is
found to be unstable,filled red circles refer to states where the
amorphous liquid is found to bemetastable with respect to the
crystal, and filled blue circles refer to stateswhere the liquid is
found to be stable with respect to the crystal. The labeledcircles
(a), (b), and (c) identify state points where explicit free energy
surfacesare shown in Fig. 3. The circles labeled (d) and (e)
identify state points whereexplicit free energy surfaces are shown
in Fig. 4. The black circle labeled(f) is the state point where
crystal-liquid phase coexistence is examined inFig. 5. Lines,
diamonds, and squares locate previous estimates of a liquid-liquid
phase transition inferred from experimental results (see Refs. 24,
28,and 31); see text. A star locates a previous prediction of a
liquid-liquid criticalpoint based upon extrapolation of simulation
results for the mW model (seeRef. 49). The blue triangle and
hexagon are estimates of low temperaturecritical point locations
obtained from interpreting simulation results for themST2 (see Ref.
35) model and the ST2r (see Ref. 38) model, respectively.
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134503-3 Transitions in supercooled water J. Chem. Phys. 135,
134503 (2011)
In cases where a phase transition exists, there are many
suchlines because different response functions have different
linesof extrema. Ambiguity ceases only in the proximity of a
criti-cal point. But whether any such lines can be related to Tx(p)
isunclear because Widom lines refer to time-independent
ther-modynamic behavior and Tx(p) refers to time-dependent
non-equilibrium behavior.
A different basis for identifying relevant points is madeby
Zhang et al. considering the same system.31 In this caseit is the
density of the water that is measured. This observeddensity
exhibits hysteresis upon alternating heating and cool-ing scans,
and the hysteresis grows upon increasing pressure.We have already
noted that it is questionable whether thephase behavior of bulk
water can be related to that of wa-ter confined to narrow pores.
Virtually all molecules in thosepores are influenced by interfaces.
Nevertheless, points ofmaximum hysteresis, denoted by the filled
diamonds in Fig. 1,have been attributed to a line of first-order
liquid-liquid tran-sitions, and points of less significant
hysteresis, marked byopen diamonds in the figure, are attributed to
a continuationof that transition.31
Another proposed line of liquid-liquid transitions is
con-structed by Fuentevilla and Anisimov.24 Here, a
postulatedscaling form is used to extrapolate from experimentally
ac-cessible equilibrium thermodynamic data. The resulting
pre-diction and its analytic continuation are drawn as solid
anddashed lines in Fig. 1. Even if a critical point is present
thispredicted line is questionable because it is generally
impossi-ble to identify critical divergences from a small rise in
non-critical background fluctuations of the sort contributing to
theheat capacity at standard conditions. This fact is illustratedby
Moore and Molinero’s predicted critical point for mWwater.49 Its
location, the star in Fig. 1, is found by extrapo-lation from a
small rise in a response function computed atdistant thermodynamic
conditions. We find no evidence for aliquid-liquid transition
anywhere near this predicted criticalpoint. Rather, it and all
other estimates pertaining to a pur-ported liquid-liquid transition
lie close to a spinodal associ-ated with crystallization. This
finding is not inconsistent withMoore and Molinero’s more recent
report that the amorphousphase of the mW model seems to be forever
changing and im-possible to equilibrate at a point in the phase
diagram whereliquid-liquid transitions have been suggested.36
Two additional marked points in Fig. 1, the blue trian-gle and
hexagon, refer to other estimates of a location for aliquid-liquid
critical point. These are estimates obtained fromextrapolating
simulation results for variants of the ST2 watermodel,52 about
which we have more to say later.
B. Anomalous thermodynamics of the mW model
Water exhibits anomalous thermodynamic properties atlow
temperatures, properties that are are non-singular butnonetheless
unusual. Because these behaviors have been pro-posed as indicators
of a liquid-liquid transition, it is importantto show that the mW
model exhibits such behaviors. Specif-ically, we focus on the
density maximum as a function oftemperature, and the relatively
large rate of increases upon
lowering temperature of both isothermal compressibility
andisobaric heat capacity.50
Thus, we have used a constant pressure ensembleto compute ρ =
N/〈V 〉, κT = 〈(δV )2〉/kBT 〈V 〉 and Cp= 〈(δH )2〉/kBT 2 for the mW
model. Here, N , V , and Hdenote number of molecules, volume, and
enthalpy, respec-tively; δV and δH denote deviations from mean
values of Vand H , respectively; the pointed brackets denote an
ensembleaverage; kB is Boltzmann’s constant.
Figure 2 compares our computed results at ambientpressure with
those found from experimental observation ofwater.50 As in Fig. 1,
we use the low pressure point of den-sity maximum as our reference
state for these comparisons.Figure 2 shows that the qualitative
trends and magnitude ofanomalies of mW water agree with those of
experimental wa-ter. The low-temperature end of the displayed
graphs occurat the point where the liquid becomes unstable. Down to
thattemperature, the growths of κT and Cp are notable but modestin
size and far from the sort of divergent behavior one
wouldordinarily associate with a critical point or phase boundary.
Atall stable and metastable liquid phase states we have studied,see
Fig. 1, we find similar nonsingular behavior.
The lower panels of Fig. 2 show that the trends observedat 1 bar
persist to higher pressures in the mW model, with thedensity
maximum temperature decreasing slightly as pressureincreases. These
trends are consistent with experiment.50
III. ORDER PARAMETERS AND FREE ENERGIES
In this section, we define order parameters and presentfree
energy functions of those order parameters.
A. Measures of crystalline order
We use two types of order parameters. One is bulk den-sity, the
other quantifies orientational order. For the latter, weuse
Steinhardt, Nelson, and Ronchetti’s Q6 and ψ6.53 For afinite system
analyzed with computer simulation, these vari-ables prove more
convenient than Fourier the components ofthe density. They also
prove more useful than dynamic mea-sures, which cannot distinguish
liquid from crystal at super-cooled conditions, where diffusion is
slow in the liquid dueto glassy dynamics and nonzero in the crystal
due to defectmotion.
Both Q6 and ψ6 are functions of a projection of the den-sity
field into averaged spherical harmonic components. Toevaluate Q�,
for each water molecule i, we calculate the setof quantities,
qi�,m =1
4
4∑j∈ni
Ym� (φij , θij ) , −� � m � � , (1)
where the sum is over those nearest 4 neighbors, ni .Ym� (φij ,
θij ) is the �,m spherical harmonic function associ-ated with of
the angular coordinates of the vector �ri − �rj join-ing molecules
i and j , measured with respect to an arbitraryexternal frame.
Since qi�,m is defined in terms of sphericalharmonics, it
transforms simply under rotations of the sys-tem or the arbitrary
external frame. These quantities are then
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134503-4 D. T. Limmer and D. Chandler J. Chem. Phys. 135, 134503
(2011)
FIG. 2. Average thermodynamic properties as a function of
temperature. Upper panels compare mW model results with those of
experiment at p = 1 bar.The filled blue circles are the calculated
results for the mW model, where error estimates are one standard
deviation. The empty black circles are experimentalresults taken
from Ref. 50. Lower panels show pressure dependence of the mW model
results.
summed over all particles to obtain a global metric
Q�,m =N∑
i=1qi�,m , (2)
and then contracted along the m axis to produce a parameterthat
is invariant with respect to the orientation of the
arbitraryexternal frame,
Q� = 1N
(�∑
m=−�Q�,mQ
∗�,m
)1/2. (3)
The other orientation order parameter we consider, ψ�,
isevaluated by first defining bond variables through local
con-tractions of the q�,m, which are reference frame
independent,
bij =∑�
m=−� qi�,mq
j∗�,m(∑�
m=−� qi�,mq
i∗�,m
)1/2 (∑�m=−� q
j
�,mqj∗�,m
)1/2 , (4)and then summing over all of the bonds made
betweenmolecule i and its nearest 4 neighbors,
ψi� =1
4
4∑j∈ni
bij . (5)
Finally, the global parameter is obtained by summing over
allmolecules,
ψ� = 1N
N∑i=1
ψi�. (6)
The mean or most probable value of Q� for an amorphousphase
approaches zero in the thermodynamic limit, while it isfinite for a
crystalline phase. As such, Q� is a distinguishingorder parameter
for amorphous and crystalline phases. In con-trast, because its
contractions occur locally and not over theentire system, ψ� is
non-vanishing in the thermodynamic limitfor both disordered and
ordered states. Nevertheless it is a use-ful measure of
orientational order because the distributions ofψi� for the low
temperature liquid are sensitive to the amountof crystallization in
the system, and their mean values at lowtemperatures differ
significantly between liquid and crystal.Further, as ψ� retains
local information, it is useful in deter-mining the existence of
grain boundaries and defects. We havetaken the � = 6 multipole
because we have found empiricallythat it is particularly sensitive
to distinguishing liquid waterand ice.
B. Free energy surfaces at conditions of metastability
Free energies of density ρ, orientational order parametersQ6 and
ψ6, and so forth, are related to the probabilities of theorder
parameters in the usual way. Specifically,
F (ρ,Q6, ...) = −kBT ln P (ρ,Q6, ...) + const., (7)where the
probability P (ρ,Q6, ...) is proportional to the par-tition
function for micro-states with the specified values of theorder
parameters. The irrelevant additive constant in Eq. (7)refers to
normalization and standard state conventions.
To evaluate the probabilities and their associated free
en-ergies, we have adopted a hybrid Monte Carlo simulationapproach
as used by Duane et al.54 We consider ensembleswith N , p, and T
fixed. Two different moves are made within
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134503-5 Transitions in supercooled water J. Chem. Phys. 135,
134503 (2011)
FIG. 3. Free energy surfaces for mW water as a function of ρ and
Q6. The system is periodically replicated and contains N = 216
particles. As shown in (a),the liquid is metastable with respect to
the crystal. As the system is cooled, the barrier disappears, as
illustrated in (b). Finally in (c) the free energy obtains alarge
gradient along the Q6 direction and fluctuations in density are
damped out. Adjacent contour lines are spaced by 1 kBT , and
statistical uncertainties aresmaller than that energy.
this framework: random changes in volume, and short molec-ular
dynamics trajectories. These are made with a ratio of1:5. Maximum
volume displacement and maximum molec-ular dynamics trajectory
length are adjusted to yield a 30%acceptance. This technique
produces suitably swift equilibra-tion even within the supercooled
regime.
The order parameters ρ, Q6 or ψ6, are controlled withumbrella
sampling, by propagating the system under itsunbiased hamiltonian
and computing the order parametersonly when determining Metropolis
acceptance probabilities.All molecular dynamics propagation was
done using theLAMMPS molecular dynamics simulation package.55
Mostof the free energy calculations were accomplished with216
particles. For each window in the umbrella sampling,the simulations
ran long enough to obtain at least 1000independent samples of each
of the biased observables. Theumbrella biasing potentials employed
were
U = k(ρ − ρ∗)2 + κ(Q6 − Q∗6)2, (8)
or the same formula with Q6 replaced by ψ6. Adopting κ inthe
range of 500 to 2000 kBT and k in the range of 1000 to2000 kBT
cm3/g proved satisfactory. Statistics gathered inthese biased
ensembles were unweighted and the free energydifferences between
each ensemble were estimated usingthe multi-state Bennett
acceptance ratio (MBAR).56 Errorestimates for the free energies we
have calculated in this wayare less than kBT .
Figure 3 depicts representative free energies for three
dif-ferent state points. The locations of those state points
arenoted in Fig. 1. Each free energy surface includes the rangeof
densities where liquid and crystal basins are located. Withthe
variable Q6, we see a significant separation between liq-uid and
crystal basins. For the state points considered inFig. 3, with N =
216, the crystal basin is centered aroundQ6 ≈ 0.5, while the liquid
basin, when it exists, is centeredaround Q6 ≈ 0.05. As N increases,
the former changes little,but the latter tends to zero. This
behavior is illustrated explic-itly in Sec. IV.
The state points considered in Fig. 3 show how the freeenergy
surfaces evolve as the pressure or temperature arechanged. In Fig.
3(a), a barrier separates the liquid phasefrom the crystal.
Therefore, at that state point the liquid ismetastable. Lowering
the temperature and increasing the pres-sure, Fig. 3(b) shows the
barrier to crystallization has van-ished. Further decreasing
temperature, Fig. 3(c) shows in-creasing driving force towards the
stable crystal. At thosestate points, the liquid is unstable.
Similar behavior is found with the free energy of ρ andψ6. This
function, F (ρ,ψ6), is shown in Fig. 4 at two differ-ent state
points. Figure 4(a) shows this free energy at a tem-perature and
pressure, labeled (d) in Fig. 1, where the liquidis metastable with
respect to the crystal. At this state point,the mean value, 〈ψ6〉 is
about 0.27, a value that reflects therelatively small amount of
local ordering present in the super-cooled liquid. In contrast, for
the crystal we find 〈ψ6〉 ≈ 0.9.Figure 4(b) shows the free energy
for a temperature and pres-sure in the region of the phase diagram
where the amorphousphase is unstable, the so-called no man’s land.
This point, la-beled (e) in Fig. 1, is close to a proposed location
of a liquid-liquid critical point.49 We see, however, that it is
not a point
FIG. 4. Free energy for mW water as a function of ρ vs ψ6 at
conditionswhere the liquid is (a) metastable, and (b) unstable. The
system is periodicallyreplicated and contains N = 216 particles.
Adjacent contour lines are spacedby 1 kBT , and statistical
uncertainties are smaller than that energy.
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134503-6 D. T. Limmer and D. Chandler J. Chem. Phys. 135, 134503
(2011)
of criticality. The behavior of ψ6 is strongly correlated to
thepotential energy. This fact follows from the functional formof
ψ6 and the three-body potential of the mW model. Thus,the behavior
of F (ρ,ψ6) should be similar to that of F (ρ,U )where U denotes
the total potential energy of the mW model.
For all of the state points considered, which includes abroad
swath of no-man’s land, there is no evidence of a bi-furcation of
the free energy along the density direction withinthe liquid region
(i.e., where Q6 and ψ6 are small). What bi-furcation does exist is
associated with a transition between anamorphous phase and a
crystal. We now consider whether itis a first order transition.
IV. FREEZING TRANSITION
A. mW model
The character of the transition can be analyzed bystudying the
system-size dependence of the contracted freeenergy,57
F (Q6) = −kBT ln(∫
dρ exp [−βF (ρ,Q6)])
. (9)
This function is shown in Fig. 5 for the mW model at one ofthe
pressures and temperatures where an amorphous phase isin
coexistence with the crystal. Such points are at the bound-ary
between the blue and red regions in Fig. 1. The quan-tity F (Q6) =
F (Q6) − min[F (Q6)] reaches its maximumvalue when an interface
separating amorphous and crystalphases extends across the entire
system. This maximum valueis the interfacial free energy.
Accordingly, for a first-ordertransition, it should be proportional
to N2/3. This scaling issatisfied to a good approximation for the
system sizes consid-ered in Fig. 5.
Nonzero values of Q6 in an amorphous phase are dueto
fluctuations. As such, the mean value of Q6 for the amor-phous
phase should disappear as 1/N1/2. This scaling is alsofound for the
system sizes studied and is illustrated in Fig. 5.In contrast, for
a crystal Q6 will have a nonzero mean that
FIG. 5. Free energies as a function of Q6 for N = 216 (blue),
512 (green),and 1000 (red), calculated at T/T0 = 1.09 and p = 1
bar. (Left inset) Inter-facial free energy for different system
sizes. For comparison a line of slope2/3 is also shown. (Right
inset) The mean value of Q6 for liquid (circles) andcrystal
(squares) for different system sizes. Error estimates are shown in
themain figure, but are smaller than the symbols in the insets.
remains finite as N → ∞. This behavior is consistent withour
numerical results, as also illustrated in Fig. 5.
Thus, the transition between liquid and crystal in mWwater
appears to be a standard freezing transition which is firstorder
and between phases with different orientational sym-metry. The
analysis used here to reach that conclusion canbe applied to other
models. For example, we have carried outthis analysis to study the
Stillinger-Weber model of silicon.Here too, we find that the model
exhibits a freezing transi-tion, and contrary to recent
suggestions43 there is no evidencefor an equilibrium liquid-liquid
transition. We also arrive atthis same conclusion for another model
of water, which weturn to now.
B. mST2 model
We have considered molecules interacting by a modifiedform of
Stillinger and Rahman’s pair potential.52 The modifi-cation
incorporates long-ranged electrostatics rather than thesimple
spherical truncation of the original model. The result-ing system,
which we call “mST2 water,” has been studiedby Liu et al.,35 who
report finding evidence of a liquid-liquidtransition and an
associated critical point. The mST2 modelis more difficult to
simulate than the mW model because theformer contains long-ranged
interactions and the latter doesnot. As such, our investigation of
its behavior is more limitedthan those we have preformed for mW
water. Nevertheless,our investigation seems sufficient to challenge
the finding ofa liquid-liquid transition at the conditions examined
by Liuet al. It also seems sufficient to discount an assortment of
lessdirect simulation studies that also report evidence of a
liquid-liquid transition in ST2 water.37–39, 58 Indeed, we have
con-sidered this particular model of water because it is so
oftenexamined in publications supporting the hypothesized
liquid-liquid transition, most recently in a paper59 motivated by
pre-liminary reports of our work.
Figure 6 shows free energies we have computed for themST2 model
using N = 216 molecules. The procedures weemployed are identical to
those used for the mW model, ex-cept for the technical detail that
we modify LAMMPS to han-dle the specific mST2 potential. We focus
on the region ofthe p-T plane where Liu et al. report bifurcation
in the freeenergy as a function of density. In that region, we too
find a bi-furcation, but not between two amorphous phases. The
grandcanonical Monte Carlo simulation method of Ref. 35 is
suffi-cient to detect a phase boundary for the liquid, but it
cannotdistinguish liquid from crystal because it does not control
dis-tinguishing order parameters. In our calculations, where bothρ
and Q6 are controlled, we find that a boundary does in factexist
between liquid and crystal. But at the thermodynamicconditions
considered by Liu et al., there is no evidence of asecond liquid
basin in the free energy F (ρ,Q6).
The specific free energies shown are found by first com-puting F
(ρ,Q6) from our simulations at T = 235 K andp = 2.2 kbar, i.e., we
compute F (ρ,Q6) = F (ρ,Q6; p, T ).The free energy shows that for
this point in the phase diagram,the crystal is stable with respect
to the liquid. A specific statepoint considered by Liu et al. is at
the same temperature but
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134503-7 Transitions in supercooled water J. Chem. Phys. 135,
134503 (2011)
FIG. 6. Free energies for the mST2 model of water. The system is
periodically replicated and contains N = 216 molecules. Panel (a)
is the contracted F̃ (ρ).Panels (b) and (c) are the surfaces F̃
(ρ,Q6). Phase coexistence between amorphous and crystal phases
occurs at μ = 0.27 kBT , where μ is the chemicalpotential relative
to that of phase space point (T , p)=(235 K, 2.2 kbar). Adjacent
contour lines in (b) and (c) are spaced by 1 kBT and statistical
uncertaintiesare of the order of, or less than, that energy. Error
bars in (a) are one standard deviation.
a different pressure or chemical potential for which the
freeenergy can be reached by a shift in chemical potential,
F̃ (ρ,Q6; T ,μ) = F (ρ,Q6) − ρV̄ μ, (10)with μ ≈ 0.55 kBT .
Here, μ is the chemical potential rel-ative to that at (T , p) =
(235 K, 2.2 kbar), and V̄ is the av-erage volume of the system at
that temperature and pressure.A lower value of μ brings the system
to a point of coex-istence between the liquid and the crystal.
These free energysurfaces are shown in panels (b) and (c) of Fig.
6. The freeenergy computed by Liu et al. is the contraction
F̃ (ρ) = −kBT ln(∫
dQ6 exp[−βF (ρ,Q6) − βρV̄ μ]
).
(11)This function is shown in panel (a) of Fig. 6.
Like Liu et al., we find a bistable free energy at this
tem-perature and for this size system. The locations for the
minimawe find for F̃ (ρ) are in good accord with those found by
Liuet al. But our free energy has a large barrier between the
twobasins, reflecting a finite crystal-liquid surface tension,
whilethat reported by Liu et al. exhibits a small barrier. Liu et
al.suggest that their result is indicative of a liquid-liquid
tran-sition and the proximity of a critical point. However, our
freeenergy surface shows no such phase transition behavior. Thereis
only a crystal-liquid first-order transition. We suggest thatthe
Liu et al. result is a non-equilibrium phenomenon, where along
molecular dynamics run at constant T − p and initiatedfrom their
low-density amorphous phase will eventually equi-librate in either
the low density crystal or in the higher densitymetastable liquid.
The time scale for this equilibration is long,as we discuss in Sec.
V.
Whatever the cause for the Liu et al. results, the bista-bility
cannot be attributed to a liquid-liquid transition with-out also
showing that the barrier separating presumed liquid-phase basins
satisfies the requisite growth with N , scaling asN2/3. This
demonstration has not been done, and from ourresults, it is
unlikely that it can be done.
Another variant of the ST2 model, considered by Pooleet al.,38
uses a reaction field approximation to estimate the
effects of long-ranged forces. We call it the “ST2r” model.One
expects similar phase behaviors from the mST2 and ST2rmodels.60
Based on an extrapolation from the equation ofstate computed for
ST2r model, Poole et al. predict the pres-ence of a liquid-liquid
transition, and the critical point lo-cation obtained from that
estimate is shown in Fig. 1. Thedensity-maximum reference
temperature for both mST2 andST2r is T0 ≈ 330 K. Poole et al.’s
estimate the critical tem-perature to be Tc = 245 K. Our
calculations for mST2, shownin Fig. 6, are at the lower
temperature, T = 235 K. Accord-ingly, at some pressure, we should
find bistable liquid behav-ior if indeed a critical point existed
at the higher tempera-ture. But we find that upon adding p V to our
computedF (ρ,Q6; 2.2 kbar, 235 K), where p = p − 2.2 kbar, no
sec-ond liquid basin can be discerned for any reasonable value ofp.
Therefore, and similar the to behavior found with the mWmodel,
extrapolation from the behavior of a one-phase systemas done in
Ref. 38 proves to be a poor indicator of a phasetransition.
V. DYNAMIC METASTABILITY
To arrive at the results of Secs. IV A and IV B, equi-libration
is achieved with umbrella sampling. Various otherreweighting Monte
Carlo procedures could be used.57, 61
Some researchers, however, attempt to learn about a possi-ble
reversible phase transition in supercooled water
throughstraightforward molecular dynamics simulation. This
ap-proach is limited to cases where relaxation is swift comparedto
computationally feasible trajectory lengths, but
relaxationassociated with phase transitions is generally not swift,
es-pecially at supercooled conditions. To judge the feasibilityof
such an approach, it is therefore useful to estimate perti-nent
relaxation times. For supercooled water, there are twoimportant
classes: times required to nucleate and grow a crys-tal, and times
required to reorganize atomic arrangements inthe liquid. We have
estimated both with molecular dynam-ics of mW water. We use the
equilibrium sampling describedin Secs. IV A and IV B to prepare
initial configurations
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134503-8 D. T. Limmer and D. Chandler J. Chem. Phys. 135, 134503
(2011)
from which we carry out Newtonian trajectories to
computedynamical properties. These trajectories evolve with a
Nose-Hoover integrator62 with a thermostat time constant of 5 psand
a barostat time constant of 5 ps.
At conditions of liquid metastability, where a free en-ergy
barrier separates liquid and crystal basins, nucleationis the
rate-determining step to form the equilibrium phase.For those
conditions, we have computed this rate constantfollowing a standard
Bennett-Chandler procedure for rare-event sampling.61 Specifically,
we take Q6 as the reactioncoordinate, so that the rate constant for
nucleation is knuc= ν exp[−F (Q∗6)/kBT ], where Q∗6 is the point of
maximumF (Q6) between liquid and crystal basins, and the
prefactor,ν, includes the transmission coefficient. This prefactor
is de-termined by sampling trajectories initialized at the top
ofthe free energy barrier, i.e., initialized at configurations
withQ6 = Q∗6.63 Other choices of transition state are possible,
butthe net result is invariant to that choice.64 The mean time
tonucleate the crystal is then 1/knuc = τxtl. Results obtained
inthat way with N = 216 mW particles are shown in Fig. 7.
Thereference time used to represent these results, τ0, is the
struc-tural relaxation time at the reference liquid state used
through-out this paper, T0 = 250 K. For mW this time is τ0 ≈ 0.5ps;
for experimental water, this time is larger by a factor ofabout
5.
For conditions of liquid instability, (i.e., the no-man’sland
where there is no barrier between liquid and crystal), themethod of
rare-event sampling is no longer appropriate. Forthose conditions,
we compute first-passage times.65 The re-sults obtained depend upon
the initial preparation of the sys-tem because the unstable system
is far from equilibrium. Inthe particular preparation we employ, we
equilibrate the sys-tem in the liquid region at T/T0 = 0.84 where
the liquid ismetastable. Then at time t = 0, the system is quenched
to the
FIG. 7. Time scales of the supercooled liquid mW water at 1 bar.
Computedstructural relaxation times, τliq, are shown with blue
points. Computed crys-tallization times, τxtl, are shown with red
points. Statistical uncertainties aresmaller than the symbols.
Dashed lines are drawn as guides to the eye. Thegrey region is
where the liquid is unstable.
target temperature and allowed to evolve towards the
crystalstate. The first-passage time is taken as the first time a
tra-jectory with initial conditions prepared in that way reaches
aconfiguration with Q6 = 0.2. We find an exponential distribu-tion
of first-passage times. Mean values from that distributionare the
values of τxtl shown in Fig. 7 for no-man’s land statepoints.
The line in the p-T plane separating filled and unfilledcircles
in Fig. 1 is the boundary between metastable and un-stable liquid
conditions. At metastable conditions close to thatboundary, we have
checked that the τxtl found from the first-passage method agrees
with that found from the rare-eventsampling.
For the structural relaxation time of the supercooledmetastable
liquid, we consider trajectories initiated from equi-librated
configurations in the liquid region and observe thetime t it takes
an average particle to move one molecular di-ameter, i.e., 3 Å=
(1/N ) ∑i |�ri(t) − �ri(0)|. For temperaturesbelow the limit of
liquid stability, we use the same procedurefor generating initial
conditions as we used in the calculationof the first passage times.
At the higher temperatures we con-sidered, we find an exponential
distribution of these times.At temperatures below T/T0 ≈ 0.88 the
distribution deviatesfrom an exponential, and increasingly so as
temperature isfurther lowered. This behavior implies the onset of
glassydynamics66 with an onset temperature of about 0.88 T0.
Themean value of these distributions is graphed as τliq in Fig.
7.
The mean nucleation or mean first-passage time, τxtl,shows
expected non-monotonic temperature dependence.67
At higher temperatures, nucleation rates increase upon cool-ing
because the barrier to nucleation decreases in size. In con-trast,
at lower temperatures, the process of crystallization isslowed by
the onset of glassy dynamics. At conditions wherethe amorphous
phase is unstable, τxtl and τliq are within twoorders of magnitude
of each other. These are average timesfor crystal nucleation and
liquid structural relaxation, respec-tively. The distributions of
these times are broad, with widthsat least as large as the mean
values. Therefore, the distri-butions of possible times for these
respective processes willoverlap, which is why a liquid state is no
longer physicallyrealizable in this region of the phase
diagram.
The boundary to unstable amorphous behavior is oftenreferred to
as the “homogeneous nucleation line.” This termi-nology is possibly
confusing because no significant barrier tonucleation exists in
no-man’s land. Indeed, studying the samemodel with straightforward
molecular dynamics in the regionof no-man’s land, at T/T0 = 0.72,
Moore and Molinero con-clude that the critical nucleus is less than
10 molecules.36
Coarsening times for relaxing defects in the crystal are
neces-sarily longer than τxtl; and for mW water these times seem
tobe at least two orders of magnitude larger.36
The most important point to take from Fig. 7 is thattime scales
for forming crystals are many orders of magni-tude larger than
those to equilibrate the liquid at standardconditions. This fact
explains why straightforward molecu-lar dynamics simulation has
thus far proved to be an unre-liable probe of phase transitions in
supercooled water andrelated materials. In the case of amorphous
phase behavior,close to or within no-man’s land, we see from Fig. 7
that mW
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134503-9 Transitions in supercooled water J. Chem. Phys. 135,
134503 (2011)
water requires time scales to equilibrate that are 3 orders
ofmagnitude larger than those of the normal temperature
liquid.Thus, while Moore and Molinero’s study of freezing at
suchconditions36 is sufficiently long to illustrate likely
trajectoriesleading to a crystal, it is too short to provide
quantitative infor-mation on the underlying probability
distributions that dictatethe instability of the liquid phase. It
is also not possible fromthat study to determine if other
trajectories exist that mightlead to a second metastable liquid
phase.
For real water (or more elaborate atomistic models)equilibration
times must account for reorganization thatovercomes donor-acceptor
asymmetry of hydrogen bonding,a feature that is absent in mW water.
Barriers to orientationalreorganization will be comparable to those
of translationalreorganization. Thus, near the no-man’s land
boundary oneexpects equilibration times of, say, ST2 water, to be
severalorders of magnitude longer than those of mW water. Mooreand
Molinero36 estimate it to be seven orders of magnitudelonger. This
is an issue that can be examined in future studiesusing methods of
rare-event sampling.61 For now, however,this paper has demonstrated
that time scale issues do notprohibit the systematic study of
reversible phase behavior ofwater and related systems using the
methods of free energysampling,57, 61 and such study draws a
picture contrary toa widely popularized notion of a second critical
point atsupercooled conditions.1, 2, 12, 13, 24, 27, 37
ACKNOWLEDGMENTS
Without implying their agreement with what we write,we are
grateful to C. Austen Angell, Sergey Buldyrev, PabloDebenedetti,
Yael Elmatad, Aaron Keys, Valeria Molinero,Athanassios
Panagiotopoulos, Ulf Pedersen, Peter Poole, Eu-gene Stanley,
Patrick Varilly, Benjamin Widom, and MichaelWidom for helpful
discussions regarding our work. Earlywork on this project was
supported by the Director, Officeof Science, Office of Basic Energy
Sciences, Materials Sci-ences and Engineering Division and Chemical
Sciences, Geo-sciences, and Biosciences Division of the U.S.
Department ofEnergy (DOE) (Contract No. DE-AC02-05CH11231).
Whilefor the later stages the authors gratefully acknowledge the
He-lios Solar Energy Research Center, which is supported by
theDirector, Office of Science, Office of Basic Energy Sciencesof
the U.S. DOE (Contract No. DE-AC02-05CH11231).
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