The putative liquid-liquid transition is a liquid-solid transition …J. Chem. Phys. 138, 034505...

The putative liquid-liquid transition is a liquid-solid transition in atomistic models ofwaterDavid T. Limmer and David Chandler

Citation: The Journal of Chemical Physics 135, 134503 (2011); doi: 10.1063/1.3643333 View online: http://dx.doi.org/10.1063/1.3643333 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/135/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Liquid-solid and solid-solid phase transition of monolayer water: High-density rhombic monolayer ice J. Chem. Phys. 140, 184507 (2014); 10.1063/1.4874696 The putative liquid-liquid transition is a liquid-solid transition in atomistic models of water. II J. Chem. Phys. 138, 214504 (2013); 10.1063/1.4807479 Free energy surface of ST2 water near the liquid-liquid phase transition J. Chem. Phys. 138, 034505 (2013); 10.1063/1.4775738 Liquid-liquid transition in ST2 water J. Chem. Phys. 137, 214505 (2012); 10.1063/1.4769126 Determination of the solid-liquid interfacial free energy along a coexistence line by Gibbs–Cahn integration J. Chem. Phys. 131, 114110 (2009); 10.1063/1.3231693

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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

0021-9606/2011/135(13)/134503/10/$30.00 © 2011 American 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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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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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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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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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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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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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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