Solid−liquid critical behavior of water in nanoporesKenji Mochizuki1 and Kenichiro Koga1

Department of Chemistry, Faculty of Science, Okayama University, Okayama 700-8530, Japan

Edited by Srikanth Sastry, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore, India, and accepted by the Editorial Board May 25, 2015(received for review December 2, 2014)

Nanoconfined liquid water can transform into low-dimensional iceswhose crystalline structures are dissimilar to any bulk ices andwhose melting point may significantly rise with reducing the poresize, as revealed by computer simulation and confirmed by exper-iment. One of the intriguing, and as yet unresolved, questionsconcerns the observation that the liquid water may transform into alow-dimensional ice either via a first-order phase change or withoutany discontinuity in thermodynamic and dynamic properties, whichsuggests the existence of solid−liquid critical points in this class ofnanoconfined systems. Here we explore the phase behavior of amodel of water in carbon nanotubes in the temperature−pressure−diameter space by molecular dynamics simulation and provide un-ambiguous evidence to support solid−liquid critical phenomena ofnanoconfined water. Solid−liquid first-order phase boundaries aredetermined by tracing spontaneous phase separation at varioustemperatures. All of the boundaries eventually cease to exist atthe critical points and there appear loci of response function max-ima, or the Widom lines, extending to the supercritical region. Thefinite-size scaling analysis of the density distribution supports thepresence of both first-order and continuous phase changes betweensolid and liquid. At around the Widom line, there are microscopicdomains of two phases, and continuous solid−liquid phase changesoccur in such a way that the domains of one phase grow and thoseof the other evanesce as the thermodynamic state departs from theWidom line.

water | solid−liquid critical point | carbon nanotube | ice | Widom line

The possibility of the solid–liquid critical point has been re-ported by computer simulation studies of various systems in

quasi-one, quasi-two, and three dimensions that exhibit bothcontinuous and discontinuous changes in thermodynamic func-tions and other order parameters (1–7). However, the idea that asolid–liquid phase boundary never terminates at the critical pointis still commonly accepted as a law of nature, largely because ofthe famous symmetry argument (8, 9) together with the lack ofexperimental observations. Furthermore, critical phenomena inquasi-1D systems are often considered impossible from a differentpoint of view; that is, to begin with, there is no first-order phasetransition in 1D systems as proved for solvable models (10) orshown by the phenomenological argument (9). Therefore, athorough investigation is much needed to support or reject thepossibility of the solid–liquid critical point. We examine the phasebehavior of a model system of water confined in a quasi-1Dnanopore (1, 11–15) and provide evidence to support the existenceof first-order phase transitions and solid–liquid critical points.

Results and DiscussionsFirst, we explore possible solid–liquid critical points of the con-fined water by calculating isotherms in the “pressure–volume”plane, where the pressure is actually Pzz, a component of pres-sure tensor along the tube axis, or simply the axial pressure, andthe volume is ℓz, the length of simulation cell in the axial di-rection per molecule. The isotherms are obtained by extensivecanonical ensemble [constant number of molecules, volume, andtemperature (NVT)] molecular dynamics (MD) simulations ofthe TIP4P model of N = 720, 900, and 1,080 water moleculesencapsulated in model single-walled carbon nanotubes (see

Methods for details). Note that direct calculations of isothermshave provided compelling evidence of the liquid–liquid phasetransition in models of supercooled water (16–18). Plotted in Fig.1 A, B, D, and E are the isotherms in the Pzz, ℓz plane at thenanotube diameter D= 11.1 Å and 12.5 Å. In any cases exam-ined, the isotherms at low temperatures have a horizontal por-tion in which Pzz does not change with ℓz, i.e., dPzz=dℓz = 0. Thisindicates that the system undergoes a phase separation underthese conditions (19). [If the system size in the axial direction issmall, a van der Waals loop will appear (17, 18).] At high tem-peratures, the slope of the isotherms is always negative, i.e.,dPzz=dℓz < 0. A critical point, if it exists, must be located betweenthe highest-temperature isotherm with dPzz=dℓz = 0 and thelowest-temperature isotherm without the horizontal portion. Inaddition, we can judge whether two phases coexist or not fromthe local density profile defined below. Using this approach, welocate two solid–liquid critical points for water in a nanotube ofD=11.1 Å at (Tc/K, Pc/GPa) = (265 ± 5, 0.13 ± 0.02) and (325 ±5, 2.52 ± 0.02) and four critical points for D=12.5 Å at (Tc/K,Pc/GPa) = (287 ± 2, 0.12 ± 0.03), (287 ± 2, 0.33 ± 0.04), (337 ± 2,1.49 ± 0.03), and (345 ± 5, 5.16 ± 0.01), as indicated by redmarks in Fig. 1.Fig. 1 C and F shows the T–Pzz phase diagram at D=11.1 Å

and 12.5 Å, respectively. The coexistence lines (first-order phaseboundaries) in the diagram are determined from the average Pzzof the horizontal portion of each isotherm at a given tempera-ture. The solid–liquid coexistence lines as obtained this way ei-ther start from the low-temperature limit, where they are solid–solid phase boundaries (Fig. S1), or branch from a solid–solid–liquid triple point, but all of them ultimately terminate at one ofthe solid–liquid critical points. We identify five ice phases[square, pentagonal, hexagonal, heptagonal, and octagonal icenanotubes whose structures are specified by the roll-up vectors

Significance

It is commonly believed that the solid−liquid critical point doesnot exist, because of the famous symmetry argument and thelack of experimental observation so far. However, recently, theintriguing possibility of the critical point has been suggestedfor strongly confined substances. Here we perform moleculardynamics simulations of water confined in carbon nanotubesand provide unambiguous evidence of the solid−liquid criticalpoint for water in the hydrophobic nanopore: macroscopicsolid–liquid phase separation below a critical temperature Tc,diverging heat capacity and isothermal compressibility ataround Tc, and the loci of response function maxima above Tc.We also give a molecular level explanation for how liquidwater continuously freezes to ice along a thermodynamic pathavoiding the first-order phase boundary.

Author contributions: K.M. and K.K. designed research; K.M. and K.K. performed re-search; K.M. analyzed data; and K.M. and K.K. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. S.S. is a guest editor invited by the Editorial Board.1To whom correspondence may be addressed. Email: mochizuki@okayama-u.ac.jp orkoga@okayama-u.ac.jp.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1422829112/-/DCSupplemental.

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(4,0), (5,0), (6,0), (7,0), and (8,0), respectively (11)] and a liquidphase from the hydrogen bond structures (Fig. 2), and we obtainthe structure factor of oxygen atoms in the axial direction, themean square displacement of molecules, and the reorientationalcorrelation function (Fig. S2). At high pressures around 1 GPaand above, in both nanotubes of D=11.1 Å and 12.5 Å, the innerspace of (6,0), (7,0), and (8,0) ice nanotubes is filled with watermolecules; these are indicated as “filled” ice in Fig. 1 and shown inFig. 2 C, E, and F. The additional water molecules intrude theseice nanotubes at high pressures because otherwise the externaln-membered rings (n= 6, 7, 8) of water molecules would havecollapsed due to strong repulsive force acting on them from thenanotube wall (Fig. S3).Fig. 1F shows how the phase boundaries shift when the nano-

tube diameter D is slightly reduced or enlarged from 12.5 Å (theisotherms in the Pzz, ℓz plane are shown in Fig. S4). Each ice regionin the T–Pzz phase diagram moves upward (to higher pressure) asD decreases. Reducing D further makes narrower ice nanotubes,the (5,0) and (4,0) ice phases shown in Fig. 1C, emerge in a low-pressure region under the (6,0) ice phase. The hollow space insidethe (6,0) ice nanotube is empty when D=12.5 Å, partially filledwhen D=12.3 Å, and fully filled when D=11.1 Å, which suggeststhat the empty (6,0) ice can continuously transform into filled(6,0) ice (Fig. S5). In principle, the phase behavior of water con-fined to the cylindrical nanopores is characterized by a 3D phase

diagram, e.g., the T–Pzz–D diagram, in which the solid–liquidphase boundaries and the critical points form surfaces and theiredges, respectively, and the heat capacity and the compressibilitymaxima, or the Widom lines (20), also form surfaces extendingfrom the edges.To confirm spontaneous phase separations and to observe

density fluctuations near the critical points, we use the scaledlocal density defined as (16) ρðz, tÞ= ðΔNðz, tÞ=ΔzÞℓz, where ΔN isthe number of molecules in a cylindrical slab of width Δz cen-tered at z and ℓ−1z is the average number of molecules per unitlength. Fig. 3 A–C shows the time evolution of ρðz, tÞ of water inthe nanotube of D=12.5 Å obtained by the NVT MD simulationat T = 300 K, 290 K, and 280 K and at fixed volume (ℓz = 0.50 Å).Note that the density of the system is between those of liquidwater and (6,0) ice. The initial configuration at t= 0 is a randomlygenerated one. At 300 K, a temperature above the (6,0) ice–liquidcritical point (CP1), ρðz, tÞ is almost uniform anywhere in thenanotube, indicating that the system is indeed homogeneous. Incontrast, at 280 K, a temperature below CP1, multiple nucleationevents of (6,0) ice take place as soon as the NVT MD run starts(Fig. 3C) (21). Then, the embryos of (6,0) ice quickly grow andmerge with each other, reflected in the initial large decrease in thewater–water interaction energy Uww (Fig. 3D), and spontaneousice–water phase separation is completed around t= 2 ns. Afterthat, the two phases continue to coexist with their domain sizes

A B C

D E F

Fig. 1. Simulation results for pressure–volume isotherms and phase diagrams of water confined in a quasi-1D hydrophobic nanopore (a model pore ofcarbon nanotube). (A and B) Pzz‐ ℓz isotherms and (C) T-Pzz phase diagram at D=11.1 Å. (D and E) Pzz‐ ℓz isotherms and (F) T-Pzz phase diagram at D=12.5 Åalong with those at D= 12.3 Å and 12.7 Å for Pzz < 2.5 GPa. The first-order phase boundaries (solid lines) in C and F are determined by the isotherms in A andB and in D and E; those that extend to lower temperatures (dashed lines) ultimately go to the phase boundaries at T = 0 K determined by the pressure-dependent enthalpy H for the corresponding ices (Fig. S1); and the one extending to lower pressures (another dashed line in F) goes to the melting tem-perature at Pzz = 0.1 MPa (11). The Widom lines drawn here are loci of maximum heat capacity cp* and maximum isothermal compressibility κT*. The color of theWidom lines indicates the relative magnitude of cp*: the larger the cp*, the darker the green color.

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nearly unchanged. Fig. 3E shows the typical structure of a waternanotube at 280 K, which clearly demonstrates the coexistence ofsolid and liquid phases. The spontaneous phase separation ob-served in the NVT MD simulation is direct evidence of the solid–liquid first-order phase transition.At 290 K, a temperature close to CP1, two kinds of domains are

distinguishable (Fig. 3B); however, the domain size of (6,0) icefluctuates with large amplitudes, which is reflected in large fluc-tuations in Uww at 290 K (Fig. 3D). The ice-like domains have alarge variety of sizes, but they do not tend to merge; e.g., see threeice domains at t= 10.5 ns. Instead, the sudden formation anddisappearance of (6,0) ice domains occur intermittently. Even adomain of (6,0) ice as large as 100 Å hardly persists over 20 ns.That is, freezing and melting seem to proceed without free-energybarriers. These behaviors, clearly different from those at the higherand lower temperatures, are characteristic of critical phenomena:See Movie S1 showing the large fluctuations of the hydrogen bondstructure in a trajectory at 290 K. It is also confirmed that thesystem exhibits the large fluctuations of the local density at thattemperature even when the trajectory starts from the solid–liquidcoexisting configuration obtained at a lower temperature (Fig. S7).To verify the presence of the solid–liquid critical point, we im-

plement the finite-size analysis of the Challa–Landau–Binder pa-rameter Π≡ 1− hα4i=3hα2i2 (22–24) of the density α=N=πσ2Lz,where σ is the radius at which the potential energy of nanotube–water interaction is zero (Fig. S8) and Lz is the length of thenanotube. The parameter Π quantifies the bimodality in the densitydistribution functionQðαÞ. Minimum of Π along an isobar, denotedas Πmin, approaches 2=3 as N→∞ when QðαÞ in the thermody-namic limit is unimodal, whereas Πmin approaches a value less than2=3 whenQðαÞ is bimodal. We calculate Π along two isobars atD =11.1 Å: the Pzz = 130 MPa isobar that passes in the vicinity of anice–water critical point inferred by the set of isotherms in Fig. 1Aand the 500-MPa isobar that crosses the Widom line identified bythe calculations of the heat capacity and the compressibility asshown below. For each state at 130 MPa in a range of T = 250 Kand 280 K, Π is calculated from an isothermal–isobaric [constantnumber of molecule, pressure, and temperature (NPT)] MD run of2 μs or longer for the systems of N = 100, 200, and 300. Such a longMD run is required to observe phase flipping between solid and

liquid phases. For each state along the 500-MPa isobar in a rangeof T = 320 K and 350 K, the NPT MD run of 100 ns is sufficient forthe system of N = 100, 200, 300, 400, and 500. The interval of T isset to 2 K near Πmin. Fig. 4 A and B shows the change of QðαÞ ofthe system with N = 300 at selected temperatures along the twoisobars. At 500 MPa, the density distribution QðαÞ is unimodal andits center simply moves to a lower density with increasing tem-perature, whereas, at 130 MPa, QðαÞ becomes bimodal or wide-spread at temperatures between 262 K and 274 K. Fig. 4C showsthe finite-size scaling of Πmin. At 500 MPa, it approaches 2=3 lin-early with 1=N, indicating the absence of a first-order phase tran-sition in the thermodynamic limit. At 130 MPa, on the other hand,Πmin approaches a value clearly smaller than 2/3, proving thepresence of a first-order phase transition. Thus, the finite-sizescaling analysis too supports the solid–liquid critical point of waterconfined in the quasi-1D hydrophobic nanopore.To investigate the nature of continuous solid–liquid phase

changes, we perform long-time NPT MD simulations for thesystems of N =200 at states on isobaric paths along which con-tinuous phase changes are observed; the production run at eachstate is 80 ns. First, we focus on the (configurational part of)isobaric heat capacity cp obtained from ðH2 −H

2Þ=kBT2, the

Fig. 2. Inherent hydrogen bond structures of six ices formed in the nanotubes(A) (4,0) ice obtained from the NPT MD trajectory at (T/K, Pzz/MPa) = (250, 50);(B) (5,0) ice at (230, 200) and (C) filled (6,0) ice at (270, 2,700) at D=11.1 Å; and(D) (6,0) ice at (280, 10), (E) filled (7,0) ice at (280, 800) and (F) filled (8,0) ice at(280, 5,100) at D=12.5 Å. Top views and the corresponding side views aredrawn abreast. Central water molecules forming a chain in the filled ices arecolored red to distinguish them from the exterior rings.

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fluctuations in H defined as U +Pzzπσ2Lz with U as the potentialenergy of the system. It is confirmed that there is a maximumheat capacity cp* for each isobaric path (Fig. S9). Plotted in Fig. 5A and B are HðTÞ and cpðTÞ at Pzz= 500 MPa, 1,000 MPa, and1,500 MPa in the pore of D=11.1 Å, where the continuousfreezing to (5,0) ice is observed. The lower the pressure Pzz, thelarger the heat capacity maximum cp* and the lower the tem-perature T* of the maximum heat capacity, which is consistentwith the maximum slope of HðTÞ at T* being steeper as Pzz isreduced. The loci T*ðPzzÞ of the heat capacity maxima cp* areplotted in the T–Pzz phase diagram (Fig. 1 C and F). Each locusof cp* is smoothly connected with a locus of the solid–liquid first-order transitions. An exception is the locus between (4,0) ice andliquid phases, which extends up to a lowest pressure examined forD = 11.1 Å but would be connected with the first-order boundaryin a phase diagram for smaller D. In one case, two loci of cp* andone first-order phase boundary seem to meet at a critical point(Fig. 1C); in the other, endpoints of two first-order boundaries areconnected with a locus of the heat capacity maxima (Fig. 1F).Next, the isothermal compressibility κT ≡ − ð∂ℓz=∂PzzÞT=ℓz in theaxial direction is obtained along isotherms in Fig. 1 A, B, D, and Eby fitting the third-order polynomial function to each Pzz−ℓz curve,and the loci Pzz*ðTÞ of the maximum isothermal compressibility κT*are determined. The result is shown in the T–Pzz phase diagram(Fig. 1 C and F). Each locus of the maximum compressibility issmoothly connected with a first-order phase boundary, as in thecase of cp*. It is confirmed that the values of maxima κT* and cp*increase as the endpoint of the first-order phase boundary isapproached. There are at least three loci of κT* that coincide withloci of cp*. Recently, Luo et al. (25) have shown that near thecritical point, in general, the loci of maximum cp and κT convergeinto a single line, the Widom line (20). Thus, the behaviors of theresponse function maxima presented here suggest the existence ofthe solid–liquid critical points and the Widom lines. Note thateach ice phase [except (8,0) ice] is enclosed by the coexistencelines and the Widom lines, or the loci of cp* and κT* .Now we examine how the dynamic and structural properties

of water change as it continuously freezes in (5, 0) ice, bycrossing the Widom lines or the loci of cp* and κT*. Fig. 5C shows

temperature dependence of the diffusion coefficient of wateralong the tube axis (Dz = limt→∞jzðtÞ− zð0Þj2=2t). There are threestages through which the dynamic property Dz changes: The mostrapid change in Dz is observed in the intermediate range, whichincludes the temperature T* of maximum heat capacity. Forlimited ranges at higher or lower T, one may find an Arrheniusbehavior in Dz, but, overall, Dz is non-Arrhenius. The staticstructure factor SðqÞ of oxygen atoms along the z axis also showsa three-stage variation with T. The first peak of SðqÞ appearsat around q=0.35 Å−1, corresponding to the lattice spacingof ice nanotubes in the axial direction. The peak intensity SðTÞgiven by integrating the local spectrum SðqÞ around the first peakis plotted in Fig. 5D, which indicates that with increasing T, theordered crystalline structure disappears continuously, and ithappens most rapidly in the intermediate range of T. That in-termediate range coincides with the range in which Dz variesmost rapidly and cp has a maximum. The temperature range ofthe intermediate stage shrinks as the critical point is approached.How can the crystalline ices gradually transform to liquid water?

Further analysis of the hydrogen bond structures reveals that thereis a sort of microscopic phase separation in the intermediate stage,where microscopic ice-like and liquid-like domains coexist(Fig. 5E). That is, the continuous melting observed for the nano-confined water does not mean that there are homogeneous in-termediate structures between an ice crystal and liquid water;rather, the ice-like and liquid-like domains in microscopically

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Fig. 4. Finite-size scaling analysis of the Challa–Landau–Binder parameterof the density distribution function QðαÞ. (A and B) QðαÞ at selected tem-peratures for (A) 500 MPa and (B) 130 MPa. The temperature at which theQðαÞ gives a minimum Π (Πmin) is 336 K at 500 MPa and 264 K at 130 MPa.(C) Finite-size scaling of Πmin along isobars of 130 MPa and 500 MPa: Πmin vs.1=N. At 500 MPa, Πmin approaches 2/3 + 0.0001 (a linear fit to the data of N =200, 300, 400, and 500). At 130 MPa, Πmin approaches 0.660 (a linear fit tothe data of N = 200 and 300).

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Fig. 5. The temperature (T) dependence of static and dynamical propertiesalong the paths of continuous freezing of water into (5,0) ice in the nanotubeof D= 11.1 Å at Pzz =500 MPa, 1,000 MPa, and 1,500 MPa. Shown here arethose obtained from NPT MD simulations of 200 water molecules. (A) Theconfigurational part of enthalpy. (B) The isobaric specific heat capacity (cp).The temperature of maximum heat capacity T* is 332 K for 500 MPa, 380 K for1,000 MPa, and 402 K for 1,500 MPa (as indicated by orange marks in A–D).(C) Arrhenius plot of the diffusion coefficient in the z direction (Dz). (D) Thenormalized peak intensity SðTÞ=SðT0Þ of the static structure factor of oxygenatoms in the z direction. The reference temperature T0 is 270 K for 500 MPa,290 K for 1,000 MPa, and 310 K for 1,500 MPa. (E) The typical inherent hy-drogen bond structure obtained at T* (332 K at 500 MPa). The islands of theordered and the disordered structures are clearly seen. The stacked pentago-nal rings, indicated by yellow, represent microscopic domains of the (5,0) ice.

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inhomogeneous states grow and disappear continuously. The ex-istence of interfaces between ice and liquid water in the in-termediate stage means that the associated interfacial tension is toosmall to promote macroscopic phase separation. This, in turn, in-dicates that microscopic structures of liquid water confined innanopores can be similar to those of ice under certain conditions(1, 3). The three-stage transformation found here is different fromthe dynamic crossover observed in supercooled water (a sharpchange from an Arrhenius to a non-Arrhenius behavior of thediffusion constant at T*) (20). The behavior found here is rathersimilar to that observed in a supercritical Lennard–Jones fluid (26),although there are some qualitative differences.There is a famous argument for the nonexistence of the solid–

liquid critical point based on the assertion that a particular sym-metry exists or does not exist (9). The argument is, however, notunassailable, since, for example, one may conceive that defects ina solid increase in number continuously upon heating until thewhole system is rendered disordered. This is exactly what we ob-serve for the quasi-1D ices. Fig. S10 shows the fraction of defectsin a pentagonal ice nanotube as a function of T and the densitydistributions of water molecules in the cross section of the pore atfour temperatures. The fraction of defects in ice is 1.5% at 150 K,and decreases to 0.7% at 100 K. As the temperature goes up from150 K to 360 K, the fraction of defects increases continuously andexceeds 60%, and the fivefold symmetry of a pentagonal icenanotube disappears gradually. In contrast, when a bulk solid isheated, a small number of defects immediately lead to a collapseof the entire crystalline structure at a specific temperature (27–29). Given the fact that simulated water and simple liquids maybe continuously transformed to ordered solids in quasi-1D (1, 4)and quasi-2D nanopores (3, 6) while such gradual transformationis not observed for the bulk systems, we suspect that a decisivefactor enabling water and other simple molecules to exhibitcontinuous freezing is confinement.Although the ice nanotubes at sufficiently low temperatures

have perfect crystalline structures as observed in our molecularsimulation, there is a possibility that the quasi-1D ices are in-trinsically polycrystalline solids whose grain size is larger than atypical cell size of molecular simulation, say, 10 nm in length. Thenthere would be no contradiction between the solid–liquid criticalphenomena and the symmetry argument.Another commonly accepted view that seems to contradict

the phase behavior reported here is that there cannot be phasetransitions in 1D systems with short-range interactions. Itis based on either conclusions derived from exactly solvablemodels (10), Landau’s phenomenological argument (9), or vanHove’s theorem (30). However, it is established with counter-examples (31–36) that van Hove’s theorem is valid for a limitedclass of 1D systems (30, 37), and the model system we studiedhere is not included in the class. Rather, ours belongs to“almost 1D” systems, which may well have phase transitions (38):It has an external field due to the cylindrical wall, inter-molecular interactions in a transverse dimension in addition tothose along the tube axis, and an infinite number of states for eachmolecule at given position z.Among the 1D models, the Chui–Weeks model of interfaces

(34) and the Dauxois–Peyrard–Bishop model of DNA denatur-ation (35, 36) seem to be most relevant to the confined waterstudied here: Both have an infinite number of states for each sitein a row, assume Hamiltonian with potentials for each site and forthe nearest-neighbor site–site interaction, and exhibit first-orderphase transitions. Here we consider the model of DNA, as it isphysically more similar to water confined in a nanopore, and try tofind a correspondence between the solvable model and the re-alistic model of confined water, which may help us better un-derstand the observed solid–liquid phase transition. The model ofDNA is a row of stacked base pairs: The ith base pair with astretching yi of the hydrogen bond has a potential V ðyiÞ, and two

neighboring base pairs are coupled by a potential W ðyi, yi−1Þwhose anharmonicity may be tuned. With the short-range in-teraction W between base pairs, the model of DNA exhibits first-order melting transition. Water confined in a nanopore takes aform of stacked clusters of molecules. The potential energy of thesystem may be expressed as the sum of the energy V ′ðYiÞ of the ithcluster and the sum of the neighboring cluster–cluster interactionenergy W ′ðYi,Yi−1Þ, with Yi the configuration of the ith cluster.In the lowest-energy state (n-gonal ice nanotube), each clusterforms the n-membered ring of H2O and the rings are stacked toform the n-gonal tube, while, in a high-energy state, clusters withdangling OH bonds are randomly stacked. Because of the ap-parent correspondences between V and V ′ and between W andW ′, the phase transition of confined water may well be of firstorder. Furthermore, the model of DNA exhibits both continuousand first-order transitions, depending on the anharmonicity in thepotential W (36). In the model water, the form of the potentialfunction W ′ should change with the nanotube diameter and theaxial pressure. Then, the variation of W ′ caused by the tube di-ameter or pressure might be responsible for the confined waterundergoing the first-order or continuous solid-liquid phasechange. As remarked above, there are other confined systems thatexhibit both continuous and discontinuous solid–liquid phasetransitions (3, 4, 6). Phase behaviors of these quasi-1D and quasi-2D systems, too, may be understood by examining a correspon-dence with the solvable models (34, 35) or the 2D versions ofsuch models.To better understand the nature of the phase behavior of a class

of confined fluids that exhibit both continuous and first-ordersolid–liquid phase changes and the associated critical phenomena,there would be several directions to pursue. In particular, wepropose two issues to be resolved in connection with the argu-ments about the solid–liquid critical point and the nonexistence ofphase transitions in one dimension. First, it is important to ex-amine whether the solid phases, e.g., ice nanotubes in the presentstudy, are truly crystalline solids or intrinsically polycrystals whosegrains are not macroscopic. For this purpose, larger-scale molec-ular simulations and perhaps the free-energy calculations of asingle crystal and polycrystals may be required. Second, it is worthexamining in detail a correspondence between the solvable modelssuch as the model of DNA denaturation and the realistic modelstudied here. For example, it is interesting to obtain a phase di-agram for the model of DNA by changing the anharmonicity inthe potential W and examine the nature of the crossover betweenthe first-order and continuous phase changes. Also, it may beworthwhile to attempt to evaluate the effective potential W ′ inthe realistic model and see whether the change in W ′ with thenanotube diameter and the pressure qualitatively corresponds tothe change in W in the model of DNA.Potential applications of the robust solid–liquid critical phe-

nomena in confined water would be to facilitate chemical re-actions, control structural changes of biological molecules, andpromote protein crystallization, using critical fluctuations (39) atdesired temperature by choosing a suitable confining geometryand pressure.

MethodsWe perform extensive MD simulations in the canonical ensemble (NVT MD)and in the isothermal–isobaric ensemble (NPT MD) for a model system ofwater confined in a smooth cylindrical nanopore (a model of the single-walled carbon nanotube). A periodic boundary condition is applied in theaxial direction (z axis). The temperature T and the internal axial pressure Pzz(the pressure tensor parallel to the z axis) are controlled using modifiedNo�se–Andersen’s method (40). The intermolecular interaction of water istaken to be the TIP4P model (41): Rhe melting temperatures of the modelwater in carbon nanotubes are in good agreement with experimental results(11, 42, 43). The interaction between a water molecule and the cylindricalwall is described by the Lennard–Jones potential integrated with respect tothe positions of carbon atoms over the cylindrical surface with an

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assumption of uniform distribution of carbon atoms (1). The nanotubediameter D is fixed to 11.1 Å, 12.3 Å, 12.5 Å, and 12.7 Å; the cross-sectional potential energy profiles are shown in Fig. S8. The diameters11.1 Å and 12.5 Å are those of the zig-zag (14,0) and (16,0) single-walledcarbon nanotubes, respectively.

In the NVT MD simulations, the number N of water molecules is 900 forthe nanotubes of length Lz =310–520 Å and diameter D=12.5 Å, N = 1,080for Lz =312–360 Å and D=12.5 Å, N= 900 for all Lz and T = T* 12.3 Å and12.7 Å, and N= 200 for all r0 and Uwall 11.1 Å. The highest density examinedin the simulations is N= 900, which corresponds to Pzz up to 7 GPa. Confinedwater under such an ultra-high pressure is hardly realized in experiment.Nevertheless, the exploration into high-pressure states gives a coherent un-derstanding of the phase behavior in confined spaces. The finite-size scalinganalysis is implemented by the NPT MD simulations of the systems with N =100, 200, 300, 400, and 500. Other NPT MD simulations are performed withN = 200 for most of the thermodynamic states and N = 500 for the states inwhich we observe the microscopic phase separation as shown in Fig. 5E.

Trajectories are generated by the Gear predictor–corrector method witha time step of 0.5 fs. The equilibration time of each state point in the NVT

MD simulation is at least 20 ns, and 200 ns or longer at some coexistenceconditions. Then, equilibrium properties are obtained from productionruns of 30–90 ns. In the NPT MD simulation, each simulation time is 100 nsfor most cases and is extended to a few microsecond for the finite-sizescaling analysis. The equilibration run of 20 ns or longer is followed by theproduction run for analyses. The instantaneous configurations are usedfor the analyses, except for the snapshots. For the latter, we use the in-herent structures, i.e., structures obtained by applying the constant volumesteepest descent method to the instantaneous structures visited bythe trajectories.

ACKNOWLEDGMENTS.Most of the calculations were performed at the ResearchCenter for Computational Science in Okazaki, Japan. We thank H. Akiyoshi,K. Himoto, T. Yagasaki, T. Sumi, M. Matsumoto, and H. Tanaka for their support.We are grateful to B. Widom for helpful comments. This work was supportedby a Grant-in-Aid for Scientific Research (26888011 and 26287099); ResearchCenter of New Functional Materials for Energy Production, Storage and Trans-port at Okayama University; and the program for promoting the enhancementof research universities, Ministry of Education, Culture, Sports, Science andTechnology (MEXT).

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