Solid−liquid critical behavior of water in nanopores

Significance

It is commonly believed that the solid−liquid critical point does not exist, because of the famous symmetry argument and the lack of experimental observation so far. However, recently, the intriguing possibility of the critical point has been suggested for strongly confined substances. Here we perform molecular dynamics simulations of water confined in carbon nanotubes and provide unambiguous evidence of the solid−liquid critical point for water in the hydrophobic nanopore: macroscopic solid–liquid phase separation below a critical temperature T_c, diverging heat capacity and isothermal compressibility at around T_c, and the loci of response function maxima above T_c. We also give a molecular level explanation for how liquid water continuously freezes to ice along a thermodynamic path avoiding the first-order phase boundary.

Abstract

Nanoconfined liquid water can transform into low-dimensional ices whose crystalline structures are dissimilar to any bulk ices and whose melting point may significantly rise with reducing the pore size, as revealed by computer simulation and confirmed by experiment. One of the intriguing, and as yet unresolved, questions concerns the observation that the liquid water may transform into a low-dimensional ice either via a first-order phase change or without any discontinuity in thermodynamic and dynamic properties, which suggests the existence of solid−liquid critical points in this class of nanoconfined systems. Here we explore the phase behavior of a model of water in carbon nanotubes in the temperature−pressure−diameter space by molecular dynamics simulation and provide unambiguous evidence to support solid−liquid critical phenomena of nanoconfined water. Solid−liquid first-order phase boundaries are determined by tracing spontaneous phase separation at various temperatures. All of the boundaries eventually cease to exist at the critical points and there appear loci of response function maxima, or the Widom lines, extending to the supercritical region. The finite-size scaling analysis of the density distribution supports the presence of both first-order and continuous phase changes between solid and liquid. At around the Widom line, there are microscopic domains of two phases, and continuous solid−liquid phase changes occur in such a way that the domains of one phase grow and those of the other evanesce as the thermodynamic state departs from the Widom line.

The possibility of the solid–liquid critical point has been reported by computer simulation studies of various systems in quasi-one, quasi-two, and three dimensions that exhibit both continuous and discontinuous changes in thermodynamic functions and other order parameters (1–7). However, the idea that a solid–liquid phase boundary never terminates at the critical point is still commonly accepted as a law of nature, largely because of the famous symmetry argument (8, 9) together with the lack of experimental observations. Furthermore, critical phenomena in quasi-1D systems are often considered impossible from a different point of view; that is, to begin with, there is no first-order phase transition in 1D systems as proved for solvable models (10) or shown by the phenomenological argument (9). Therefore, a thorough investigation is much needed to support or reject the possibility of the solid–liquid critical point. We examine the phase behavior of a model system of water confined in a quasi-1D nanopore (1, 11–15) and provide evidence to support the existence of first-order phase transitions and solid–liquid critical points.

Results and Discussions

First, we explore possible solid–liquid critical points of the confined water by calculating isotherms in the “pressure–volume” plane, where the pressure is actually $P_{zz}$, a component of pressure tensor along the tube axis, or simply the axial pressure, and the volume is ${\ell }_z$, the length of simulation cell in the axial direction per molecule. The isotherms are obtained by extensive canonical ensemble [constant number of molecules, volume, and temperature ($NVT$)] molecular dynamics (MD) simulations of the TIP4P model of N = 720, 900, and 1,080 water molecules encapsulated in model single-walled carbon nanotubes (see Methods for details). Note that direct calculations of isotherms have provided compelling evidence of the liquid–liquid phase transition in models of supercooled water (16–18). Plotted in Fig. 1 A, B, D, and E are the isotherms in the $P_{zz},{\ell }_z$ plane at the nanotube diameter $D=$ 11.1 Å and 12.5 Å. In any cases examined, the isotherms at low temperatures have a horizontal portion in which $P_{zz}$ does not change with ${\ell }_z$, i.e., $d{P}_{zz}/d{\ell }_z=0$. This indicates that the system undergoes a phase separation under these conditions (19). [If the system size in the axial direction is small, a van der Waals loop will appear (17, 18).] At high temperatures, the slope of the isotherms is always negative, i.e., $d{P}_{zz}/d{\ell }_z<0$. A critical point, if it exists, must be located between the highest-temperature isotherm with $d{P}_{zz}/d{\ell }_z=0$ and the lowest-temperature isotherm without the horizontal portion. In addition, we can judge whether two phases coexist or not from the local density profile defined below. Using this approach, we locate two solid–liquid critical points for water in a nanotube of $D=$11.1 Å at ($T_c$/K, $P_c$/GPa) = (265 ± 5, 0.13 ± 0.02) and (325 ± 5, 2.52 ± 0.02) and four critical points for $D=$12.5 Å at ($T_c$/K, $P_c$/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 red marks in Fig. 1.

Fig. 1.

Simulation results for pressure–volume isotherms and phase diagrams of water confined in a quasi-1D hydrophobic nanopore (a model pore of carbon nanotube). (A and B) $P_{zz}‐{\ell }_z$ isotherms and (C) T-$P_{zz}$ phase diagram at $D=$11.1 Å. (D and E) $P_{zz}‐{\ell }_z$ isotherms and (F) T-$P_{zz}$ phase diagram at $D=$12.5 Å along with those at $D=$ 12.3 Å and 12.7 Å for $P_{zz}<$ 2.5 GPa. The first-order phase boundaries (solid lines) in C and F are determined by the isotherms in A and B 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 temperature at $P_{zz}=$ 0.1 MPa (11). The Widom lines drawn here are loci of maximum heat capacity $c_p^{*}$ and maximum isothermal compressibility ${\kappa }_T^{*}$. The color of the Widom lines indicates the relative magnitude of $c_p^{*}$: the larger the $c_p^{*}$, the darker the green color.

Fig. 1 C and F shows the T–$P_{zz}$ phase diagram at $D=$11.1 Å and 12.5 Å, respectively. The coexistence lines (first-order phase boundaries) in the diagram are determined from the average $P_{zz}$ of the horizontal portion of each isotherm at a given temperature. The solid–liquid coexistence lines as obtained this way either 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 of the solid–liquid critical points. We identify five ice phases [square, pentagonal, hexagonal, heptagonal, and octagonal ice nanotubes whose structures are specified by the roll-up vectors (4,0), (5,0), (6,0), (7,0), and (8,0), respectively (11)] and a liquid phase from the hydrogen bond structures (Fig. 2), and we obtain the structure factor of oxygen atoms in the axial direction, the mean square displacement of molecules, and the reorientational correlation function (Fig. S2). At high pressures around 1 GPa and above, in both nanotubes of $D=$11.1 Å and 12.5 Å, the inner space of (6,0), (7,0), and (8,0) ice nanotubes is filled with water molecules; these are indicated as “filled” ice in Fig. 1 and shown in Fig. 2 C, E, and F. The additional water molecules intrude these ice nanotubes at high pressures because otherwise the external n-membered rings ($n=$ 6, 7, 8) of water molecules would have collapsed due to strong repulsive force acting on them from the nanotube wall (Fig. S3).

Fig. S1.

Plots of enthalpy against $P_{zz}$ at 0 K for ices confined in the nanotubes of (A) $D=$11.1 Å and (B) $D=$12.5 Å. Instantaneous crystal structures with no defect given by $NPT$ MD simulations are quenched to the energy-minimum structures ($T=0$ K) by the steepest descent method at fixed pressures (inherent structure). Pressures at the intersection of the enthalpies are indicated.

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, $P_{zz}$/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 are drawn abreast. Central water molecules forming a chain in the filled ices are colored red to distinguish them from the exterior rings.

Fig. S2.

Dynamical and statical physical properties under eight thermodynamic conditions; (T/K, $P_{zz}$/MPa) = (250, 50), (250, 500), (250, 3000), and (320, 50) in the nanotube of $D=11.1$ Å corresponding to (4,0) ice, filled (5,0) ice, filled (6,0) ice, and liquid water. In addition, (250, 200), (250, 3,000), (250, 6,000), and (320, 200) in the nanotube of $D=12.5$ Å correspond to (6,0) ice, filled (7,0) ice, filled (8,0) ice, and liquid water. The trajectories are obtained from $NPT$ MD simulations for $N=200$. (A and B) The mean square displacement along z axis (${\text{MSD}}_{\text{z}}$) obtained from $\overline{{\left|z_j\left(t\right)-z_j\left(0\right)\right|}^2}$, where $z_j\left(t\right)$ is the axial coordination of jth water molecule at time t. (C and D) The reorientational correlation function (RCF) defined as $\overline{{\mathit{u}}_j\left(t\right)\cdot {\mathit{u}}_j\left(0\right)}$, where ${\mathit{u}}_j\left(t\right)$ is the self-dipole vector. (E and F) The static structure factor of oxygen atoms along z axis; $S\left(q\right)=\overline{\left|{\displaystyle {\sum }_{j=1}^N\exp \left(-2\pi \hspace{0.17em}iq{z}_j\left(t\right)\right)}\right|}/N$, where $q=k/L_z$ $(k=\mathrm{1,2,3},\mathrm{...})$.

Fig. S3.

Radial density profile of filled (6,0) ice plotted against the radius (r). The trajectory is obtained from $NPT$ MD at (D/Å, T/K, $P_{zz}$/GPa) = (11.1, 270, 2.7). Water molecules forming six-membered rings are located around $r=2.68$ Å, which is outside of the global minimum of $U_{\text{wall}}$ ($r=r_0$). These molecules feel a repulsive force from the wall; σ, at which $U_{\text{wall}}=0$, is also indicated.

Fig. 1F shows how the phase boundaries shift when the nanotube diameter D is slightly reduced or enlarged from 12.5 Å (the isotherms in the $P_{zz},{\ell }_z$ plane are shown in Fig. S4). Each ice region in the T–$P_{zz}$ phase diagram moves upward (to higher pressure) as D 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 inside the (6,0) ice nanotube is empty when $D=$12.5 Å, partially filled when $D=$12.3 Å, and fully filled when $D=$11.1 Å, which suggests that the empty (6,0) ice can continuously transform into filled (6,0) ice (Fig. S5). In principle, the phase behavior of water confined to the cylindrical nanopores is characterized by a 3D phase diagram, e.g., the T–$P_{zz}$–D diagram, in which the solid–liquid phase boundaries and the critical points form surfaces and their edges, respectively, and the heat capacity and the compressibility maxima, or the Widom lines (20), also form surfaces extending from the edges.

Fig. S4.

Isotherms of pressures at (A) $D=12.3$ Å and (B) $D=12.7$ Å. The critical points are located at around ($T_c$/K, $P_c$/GPa) = (305 ± 5, 0.2621 ± 0.0535), (305 ± 5, 0.4622 ± 0.0571), and (335 ± 5, 1.7986 ± 0.0741) for $D=12.3$ Å, and (275 ± 5, 0.0553 ± 0.0050), (275 ± 5, 0.1858 ± 0.0389), and (315 ± 5, 0.9861 ± 0.0584) for $D=12.7$ Å.

Fig. S5.

Inherent hydrogen bond structure of partially filled (6,0) ice obtained from the $NPT$ MD simulation at $T=270$ K and $P_{zz}=700$ MPa in the nanotube of $D=12.3$ Å. The central water molecules are colored red.

To confirm spontaneous phase separations and to observe density fluctuations near the critical points, we use the scaled local density defined as (16) $\rho \left(z,t\right)=\left(\mathrm{\Delta }N\left(z,t\right)/\mathrm{\Delta }z\right){\ell }_z$, where $\mathrm{\Delta }N$ is the number of molecules in a cylindrical slab of width $\mathrm{\Delta }z$ centered at z and ${\ell }_z^{-1}$ is the average number of molecules per unit length. Fig. 3 A–C shows the time evolution of $\rho \left(z,t\right)$ of water in the nanotube of $D=$12.5 Å obtained by the $NVT$ MD simulation at $T=$ 300 K, 290 K, and 280 K and at fixed volume (${\ell }_{\text{z}}=$ 0.50 Å). Note that the density of the system is between those of liquid water and (6,0) ice. The initial configuration at $t=$ 0 is a randomly generated one. At 300 K, a temperature above the (6,0) ice–liquid critical point (CP₁), $\rho \left(z,t\right)$ is almost uniform anywhere in the nanotube, indicating that the system is indeed homogeneous. In contrast, at 280 K, a temperature below ${\text{CP}}_1$, multiple nucleation events 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 and merge with each other, reflected in the initial large decrease in the water–water interaction energy $U_{\text{ww}}$ (Fig. 3D), and spontaneous ice–water phase separation is completed around $t=$ 2 ns. After that, the two phases continue to coexist with their domain sizes nearly unchanged. Fig. 3E shows the typical structure of a water nanotube at 280 K, which clearly demonstrates the coexistence of solid and liquid phases. The spontaneous phase separation observed in the $NVT$ MD simulation is direct evidence of the solid–liquid first-order phase transition.

Fig. 3.

Fluctuations above, near, and below the temperature of the solid–liquid critical point. (A–C) Time evolution of the scaled local density $\rho \left(z,t\right)$ for trajectories at (A) 300 K, (B) 290 K, and (C) 280 K obtained from the $NVT$ MD simulation ($N=900$) with fixed density (${\ell }_z=$0.5 Å) and tube diameter ($D=$ 12.5 Å). The width $\Delta z$ of a bin with which the local density is defined is taken to be 5 Å. The color for $\rho \left(z,t\right)<1$ is the same as that for $\rho \left(z,t\right)=$ 1. [The raw $\rho \left(z,t\right)$ descriptions are shown in Fig. S6.] (D) Time evolution of the potential energy between water molecules ($U_{\text{ww}}$). (E) Inherent hydrogen bond structure at 30 ns of the 280 K trajectory (from C).

Fig. S6.

Time evolution of the scaled local density $\rho \left(z,t\right)$ for trajectories at (A) 300 K, (B) 290 K, and (C) 280 K obtained from the $NVT$ MD simulation ($N=900$) with fixed density (${\ell }_z=$0.5 Å) and tube diameter ($D=$12.5 Å). The width $\mathrm{\Delta }z$ of a bin with which the local density is defined is taken to be 5 Å.

At 290 K, a temperature close to ${\text{CP}}_1$, two kinds of domains are distinguishable (Fig. 3B); however, the domain size of (6,0) ice fluctuates with large amplitudes, which is reflected in large fluctuations in $U_{\text{ww}}$ at 290 K (Fig. 3D). The ice-like domains have a large variety of sizes, but they do not tend to merge; e.g., see three ice domains at $t=$ 10.5 ns. Instead, the sudden formation and disappearance of (6,0) ice domains occur intermittently. Even a domain of (6,0) ice as large as 100 Å hardly persists over 20 ns. That is, freezing and melting seem to proceed without free-energy barriers. These behaviors, clearly different from those at the higher and lower temperatures, are characteristic of critical phenomena: See Movie S1 showing the large fluctuations of the hydrogen bond structure in a trajectory at 290 K. It is also confirmed that the system exhibits the large fluctuations of the local density at that temperature even when the trajectory starts from the solid–liquid coexisting configuration obtained at a lower temperature (Fig. S7).

Fig. S7.

Time evolution of the scaled local density $\rho \left(z,t\right)$ for the trajectory obtained from the $NVT$ MD simulation (N = 900) with fixed density (${\ell }_z=$ 0.5 Å) and tube diameter ($D=$12.5 Å). The width $\mathrm{\Delta }z$ of a bin with which the local density is defined is taken to be 5 Å. The color for $\rho \left(z,t\right)<1.0$ is the same as that for $\rho \left(z,t\right)=1.0$. The temperature is immediately changed from 280 K to 290 K at $t=0$ ns, as the critical point is approached. The trajectory at 280 K is same as the latter half of Fig. 3C. At 290 K, the initial coexisting configuration is vanishing within 15 ns; then the large fluctuation of the local density is observed.

To verify the presence of the solid–liquid critical point, we implement the finite-size analysis of the Challa–Landau–Binder parameter $\mathrm{\Pi }\equiv 1-\langle {\alpha }^4\rangle /3{\langle {\alpha }^2\rangle }^2$ (22–24) of the density $\alpha =N/\pi {\sigma }^2{L}_z$, where σ is the radius at which the potential energy of nanotube–water interaction is zero (Fig. S8) and $L_z$ is the length of the nanotube. The parameter $\mathrm{\Pi }$ quantifies the bimodality in the density distribution function $Q\left(\alpha \right)$. Minimum of $\mathrm{\Pi }$ along an isobar, denoted as ${\mathrm{\Pi }}_{\text{min}}$, approaches $2/3$ as $N\to \infty$ when $Q\left(\alpha \right)$ in the thermodynamic limit is unimodal, whereas ${\mathrm{\Pi }}_{\text{min}}$ approaches a value less than $2/3$ when $Q\left(\alpha \right)$ is bimodal. We calculate $\mathrm{\Pi }$ along two isobars at D = 11.1 Å: the $P_{zz}=130$ MPa isobar that passes in the vicinity of an ice–water critical point inferred by the set of isotherms in Fig. 1A and the 500-MPa isobar that crosses the Widom line identified by the calculations of the heat capacity and the compressibility as shown below. For each state at 130 MPa in a range of $T=250$ K and 280 K, $\mathrm{\Pi }$ is calculated from an isothermal–isobaric [constant number of molecule, pressure, and temperature (NPT)] MD run of 2 μs or longer for the systems of N = 100, 200, and 300. Such a long MD run is required to observe phase flipping between solid and liquid phases. For each state along the 500-MPa isobar in a range of $T=320$ K and 350 K, the $NPT$ MD run of 100 ns is sufficient for the system of N = 100, 200, 300, 400, and 500. The interval of T is set to 2 K near ${\mathrm{\Pi }}_{\text{min}}$. Fig. 4 A and B shows the change of $Q\left(\alpha \right)$ of the system with $N=300$ at selected temperatures along the two isobars. At 500 MPa, the density distribution $Q\left(\alpha \right)$ is unimodal and its center simply moves to a lower density with increasing temperature, whereas, at 130 MPa, $Q\left(\alpha \right)$ becomes bimodal or widespread at temperatures between 262 K and 274 K. Fig. 4C shows the finite-size scaling of ${\mathrm{\Pi }}_{\text{min}}$. At 500 MPa, it approaches $2/3$ linearly with $1/N$, indicating the absence of a first-order phase transition in the thermodynamic limit. At 130 MPa, on the other hand, ${\mathrm{\Pi }}_{\text{min}}$ approaches a value clearly smaller than 2/3, proving the presence of a first-order phase transition. Thus, the finite-size scaling analysis too supports the solid–liquid critical point of water confined in the quasi-1D hydrophobic nanopore.

Fig. S8.

Potential energy profile of the model carbon nanotubes ($U_{\text{wall}}$) of $D=11.1$ Å, 12.3 Å, 12.5 Å, and 12.7 Å plotted against the radius (r); $r_0$, at which $U_{\text{wall}}$ has the global minimum, is 2.20 Å, 2.82 Å, 2.92 Å, and 3.02 Å, respectively; σ, at which $U_{\text{wall}}=0$, is 2.75 Å, 3.35 Å, 3.45 Å, and 3.55 Å, respectively. The locations of $r_0$ and σ of the nanotube of $D=11.1$ Å are indicated.

Fig. 4.

Finite-size scaling analysis of the Challa–Landau–Binder parameter of the density distribution function $Q\left(\alpha \right)$. (A and B) $Q\left(\alpha \right)$ at selected temperatures for (A) 500 MPa and (B) 130 MPa. The temperature at which the $Q\left(\alpha \right)$ gives a minimum $\mathrm{\Pi }$ (${\mathrm{\Pi }}_{\text{min}}$) is 336 K at 500 MPa and 264 K at 130 MPa. (C) Finite-size scaling of ${\mathrm{\Pi }}_{\text{min}}$ along isobars of 130 MPa and 500 MPa: ${\mathrm{\Pi }}_{\text{min}}$ vs. $1/N$. At 500 MPa, ${\mathrm{\Pi }}_{\text{min}}$ approaches 2/3 + 0.0001 (a linear fit to the data of N = 200, 300, 400, and 500). At 130 MPa, ${\mathrm{\Pi }}_{\text{min}}$ approaches 0.660 (a linear fit to the data of N = 200 and 300).

To investigate the nature of continuous solid–liquid phase changes, we perform long-time $NPT$ MD simulations for the systems of $N=$ 200 at states on isobaric paths along which continuous phase changes are observed; the production run at each state is 80 ns. First, we focus on the (configurational part of) isobaric heat capacity $c_p$ obtained from $\left(\overline{H^2}-{\overline{H}}^2\right)/k_{\text{B}}{T}^2$, the fluctuations in H defined as $U+P_{zz}\pi {\sigma }^2{L}_z$ with U as the potential energy of the system. It is confirmed that there is a maximum heat capacity $c_p^{*}$ for each isobaric path (Fig. S9). Plotted in Fig. 5 A and B are $H\left(T\right)$ and $c_p\left(T\right)$ at $P_{zz}$= 500 MPa, 1,000 MPa, and 1,500 MPa in the pore of $D=$11.1 Å, where the continuous freezing to (5,0) ice is observed. The lower the pressure $P_{zz}$, the larger the heat capacity maximum $c_p^{*}$ and the lower the temperature $T*$ of the maximum heat capacity, which is consistent with the maximum slope of $H\left(T\right)$ at $T*$ being steeper as $P_{zz}$ is reduced. The loci $T*\left(P_{zz}\right)$ of the heat capacity maxima $c_p^{*}$ are plotted in the T–$P_{zz}$ phase diagram (Fig. 1 C and F). Each locus of $c_p^{*}$ is smoothly connected with a locus of the solid–liquid first-order transitions. An exception is the locus between (4,0) ice and liquid phases, which extends up to a lowest pressure examined for D = 11.1 Å but would be connected with the first-order boundary in a phase diagram for smaller D. In one case, two loci of $c_p^{*}$ and one first-order phase boundary seem to meet at a critical point (Fig. 1C); in the other, endpoints of two first-order boundaries are connected with a locus of the heat capacity maxima (Fig. 1F). Next, the isothermal compressibility ${\kappa }_T\equiv -{\left(\partial {\ell }_z/\partial P_{zz}\right)}_T/{\ell }_z$ in the axial direction is obtained along isotherms in Fig. 1 A, B, D, and E by fitting the third-order polynomial function to each $P_{zz}$−${\ell }_z$ curve, and the loci $P_{zz}^{*}\left(T\right)$ of the maximum isothermal compressibility ${\kappa }_T^{*}$ are determined. The result is shown in the T–$P_{zz}$ phase diagram (Fig. 1 C and F). Each locus of the maximum compressibility is smoothly connected with a first-order phase boundary, as in the case of $c_p^{*}$. It is confirmed that the values of maxima ${\kappa }_T^{*}$ and $c_p^{*}$ increase as the endpoint of the first-order phase boundary is approached. There are at least three loci of ${\kappa }_T^{*}$ that coincide with loci of $c_p^{*}$. Recently, Luo et al. (25) have shown that near the critical point, in general, the loci of maximum $c_p$ and ${\kappa }_T$ converge into a single line, the Widom line (20). Thus, the behaviors of the response function maxima presented here suggest the existence of the solid–liquid critical points and the Widom lines. Note that each ice phase [except (8,0) ice] is enclosed by the coexistence lines and the Widom lines, or the loci of $c_p^{*}$ and ${\kappa }_T^{*}$.

Fig. S9.

The temperature T dependence of the isobaric specific heat capacity $c_p$. (A–C) The trajectories are obtained from the $NPT$ MD simulations of $N=200$ at $D=11.1$ Å and (D and E) at $D=12.5$ Å. Every isobar shows a peak ($c_p^{*}$) at $T=T^{*}$.

Fig. 5.

The temperature (T) dependence of static and dynamical properties along the paths of continuous freezing of water into (5,0) ice in the nanotube of $D=11.1$ Å at $P_{zz}=$ 500 MPa, 1,000 MPa, and 1,500 MPa. Shown here are those obtained from $NPT$ MD simulations of 200 water molecules. (A) The configurational part of enthalpy. (B) The isobaric specific heat capacity ($c_p$). The temperature of maximum heat capacity $T*$ is 332 K for 500 MPa, 380 K for 1,000 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 ($D_z$). (D) The normalized peak intensity $\widehat{S}\left(T\right)/\widehat{S}\left(T_0\right)$ of the static structure factor of oxygen atoms in the z direction. The reference temperature $T_0$ is 270 K for 500 MPa, 290 K for 1,000 MPa, and 310 K for 1,500 MPa. (E) The typical inherent hydrogen bond structure obtained at $T*$ (332 K at 500 MPa). The islands of the ordered and the disordered structures are clearly seen. The stacked pentagonal rings, indicated by yellow, represent microscopic domains of the (5,0) ice.

Now we examine how the dynamic and structural properties of water change as it continuously freezes in (5, 0) ice, by crossing the Widom lines or the loci of $c_p^{*}$ and ${\kappa }_T^{*}$. Fig. 5C shows temperature dependence of the diffusion coefficient of water along the tube axis ($D_z={\text{lim}}_{t\to \infty }\overline{{\left|z\left(t\right)-z\left(0\right)\right|}^2}/2t$). There are three stages through which the dynamic property $D_z$ changes: The most rapid change in $D_z$ is observed in the intermediate range, which includes the temperature $T*$ of maximum heat capacity. For limited ranges at higher or lower T, one may find an Arrhenius behavior in $D_z$, but, overall, $D_z$ is non-Arrhenius. The static structure factor $S\left(q\right)$ of oxygen atoms along the z axis also shows a three-stage variation with T. The first peak of $S\left(q\right)$ appears at around $q=$0.35 Å⁻¹, corresponding to the lattice spacing of ice nanotubes in the axial direction. The peak intensity $\widehat{S}\left(T\right)$ given by integrating the local spectrum $S\left(q\right)$ around the first peak is plotted in Fig. 5D, which indicates that with increasing T, the ordered crystalline structure disappears continuously, and it happens most rapidly in the intermediate range of T. That intermediate range coincides with the range in which $D_z$ varies most rapidly and $c_p$ has a maximum. The temperature range of the 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 there is 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 nanoconfined water does not mean that there are homogeneous intermediate structures between an ice crystal and liquid water; rather, the ice-like and liquid-like domains in microscopically inhomogeneous states grow and disappear continuously. The existence of interfaces between ice and liquid water in the intermediate stage means that the associated interfacial tension is too small to promote macroscopic phase separation. This, in turn, indicates that microscopic structures of liquid water confined in nanopores can be similar to those of ice under certain conditions (1, 3). The three-stage transformation found here is different from the dynamic crossover observed in supercooled water (a sharp change from an Arrhenius to a non-Arrhenius behavior of the diffusion constant at $T*$) (20). The behavior found here is rather similar 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 symmetry exists or does not exist (9). The argument is, however, not unassailable, since, for example, one may conceive that defects in a solid increase in number continuously upon heating until the whole system is rendered disordered. This is exactly what we observe for the quasi-1D ices. Fig. S10 shows the fraction of defects in a pentagonal ice nanotube as a function of T and the density distributions of water molecules in the cross section of the pore at four 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 from 150 K to 360 K, the fraction of defects increases continuously and exceeds 60%, and the fivefold symmetry of a pentagonal ice nanotube disappears gradually. In contrast, when a bulk solid is heated, a small number of defects immediately lead to a collapse of the entire crystalline structure at a specific temperature (27–29). Given the fact that simulated water and simple liquids may be continuously transformed to ordered solids in quasi-1D (1, 4) and quasi-2D nanopores (3, 6) while such gradual transformation is not observed for the bulk systems, we suspect that a decisive factor enabling water and other simple molecules to exhibit continuous freezing is confinement.

Fig. S10.

Symmetry change with increasing of defective water molecules in D = 11.1 Å nanotube at 500 MPa, obtained by $NPT$ MD simulations for a system of N = 200. The density distribution (arbitrary unit) on the sectional plane for (A) 150 K, (B) 300 K, (C) 332 K, and (D) 360 K. (E) The average ratio of defective molecules in a system. The defect is defined as a water molecule that does not have four hydrogen bonds with its neighbors. Here, two water molecules are considered to be hydrogen-bonded if their intermolecular hydrogen–oxygen distance is shorter than 2.5 Å, which is determined by the location of the first minimum of the hydrogen–oxygen radial distribution function. It is seen that the fivefold symmetry at 150 K gradually disappears and high symmetry of liquid state appears in company with a continuous increment of defective molecules.

Although the ice nanotubes at sufficiently low temperatures have perfect crystalline structures as observed in our molecular simulation, there is a possibility that the quasi-1D ices are intrinsically polycrystalline solids whose grain size is larger than a typical cell size of molecular simulation, say, 10 nm in length. Then there would be no contradiction between the solid–liquid critical phenomena and the symmetry argument.

Another commonly accepted view that seems to contradict the phase behavior reported here is that there cannot be phase transitions in 1D systems with short-range interactions. It is based on either conclusions derived from exactly solvable models (10), Landau’s phenomenological argument (9), or van Hove’s theorem (30). However, it is established with counterexamples (31–36) that van Hove’s theorem is valid for a limited class of 1D systems (30, 37), and the model system we studied here 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, intermolecular interactions in a transverse dimension in addition to those along the tube axis, and an infinite number of states for each molecule at given position z.

Among the 1D models, the Chui–Weeks model of interfaces (34) and the Dauxois–Peyrard–Bishop model of DNA denaturation (35, 36) seem to be most relevant to the confined water studied here: Both have an infinite number of states for each site in a row, assume Hamiltonian with potentials for each site and for the nearest-neighbor site–site interaction, and exhibit first-order phase transitions. Here we consider the model of DNA, as it is physically more similar to water confined in a nanopore, and try to find a correspondence between the solvable model and the realistic model of confined water, which may help us better understand the observed solid–liquid phase transition. The model of DNA is a row of stacked base pairs: The ith base pair with a stretching $y_i$ of the hydrogen bond has a potential $V\left(y_i\right)$, and two neighboring base pairs are coupled by a potential $W\left(y_i,y_{i-1}\right)$ whose anharmonicity may be tuned. With the short-range interaction W between base pairs, the model of DNA exhibits first-order melting transition. Water confined in a nanopore takes a form of stacked clusters of molecules. The potential energy of the system may be expressed as the sum of the energy $V^{\prime }\left(Y_i\right)$ of the ith cluster and the sum of the neighboring cluster–cluster interaction energy $W^{\prime }\left(Y_i,Y_{i-1}\right)$, with $Y_i$ the configuration of the ith cluster. In the lowest-energy state (n-gonal ice nanotube), each cluster forms the n-membered ring of H₂O and the rings are stacked to form the n-gonal tube, while, in a high-energy state, clusters with dangling OH bonds are randomly stacked. Because of the apparent correspondences between V and $V^{\prime }$ and between W and $W^{\prime }$, the phase transition of confined water may well be of first order. Furthermore, the model of DNA exhibits both continuous and first-order transitions, depending on the anharmonicity in the potential W (36). In the model water, the form of the potential function $W^{\prime }$ should change with the nanotube diameter and the axial pressure. Then, the variation of $W^{\prime }$ caused by the tube diameter or pressure might be responsible for the confined water undergoing the first-order or continuous solid-liquid phase change. As remarked above, there are other confined systems that exhibit both continuous and discontinuous solid–liquid phase transitions (3, 4, 6). Phase behaviors of these quasi-1D and quasi-2D systems, too, may be understood by examining a correspondence with the solvable models (34, 35) or the 2D versions of such models.

To better understand the nature of the phase behavior of a class of confined fluids that exhibit both continuous and first-order solid–liquid phase changes and the associated critical phenomena, there would be several directions to pursue. In particular, we propose two issues to be resolved in connection with the arguments about the solid–liquid critical point and the nonexistence of phase transitions in one dimension. First, it is important to examine whether the solid phases, e.g., ice nanotubes in the present study, are truly crystalline solids or intrinsically polycrystals whose grains are not macroscopic. For this purpose, larger-scale molecular simulations and perhaps the free-energy calculations of a single crystal and polycrystals may be required. Second, it is worth examining in detail a correspondence between the solvable models such as the model of DNA denaturation and the realistic model studied here. For example, it is interesting to obtain a phase diagram for the model of DNA by changing the anharmonicity in the potential W and examine the nature of the crossover between the first-order and continuous phase changes. Also, it may be worthwhile to attempt to evaluate the effective potential $W^{\prime }$ in the realistic model and see whether the change in $W^{\prime }$ with the nanotube diameter and the pressure qualitatively corresponds to the change in W in the model of DNA.

Potential applications of the robust solid–liquid critical phenomena in confined water would be to facilitate chemical reactions, control structural changes of biological molecules, and promote protein crystallization, using critical fluctuations (39) at desired temperature by choosing a suitable confining geometry and pressure.

Methods

We perform extensive MD simulations in the canonical ensemble ($NVT$ MD) and in the isothermal–isobaric ensemble ($NPT$ MD) for a model system of water confined in a smooth cylindrical nanopore (a model of the single-walled carbon nanotube). A periodic boundary condition is applied in the axial direction (z axis). The temperature T and the internal axial pressure $P_{zz}$ (the pressure tensor parallel to the z axis) are controlled using modified Nośe–Andersen’s method (40). The intermolecular interaction of water is taken to be the TIP4P model (41): Rhe melting temperatures of the model water in carbon nanotubes are in good agreement with experimental results (11, 42, 43). The interaction between a water molecule and the cylindrical wall is described by the Lennard–Jones potential integrated with respect to the positions of carbon atoms over the cylindrical surface with an assumption of uniform distribution of carbon atoms (1). The nanotube diameter 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 diameters 11.1 Å and 12.5 Å are those of the zig-zag (14,0) and (16,0) single-walled carbon nanotubes, respectively.

In the $NVT$ MD simulations, the number N of water molecules is 900 for the nanotubes of length $L_z=$310–520 Å and diameter $D=$12.5 Å, N = 1,080 for $L_z=$312–360 Å and $D=$12.5 Å, $N=900$ for all $L_z$ and $T=T^{*}$ 12.3 Å and 12.7 Å, and $N=200$ for all $r_0$ and $U_{\text{wall}}$ 11.1 Å. The highest density examined in the simulations is $N=900$, which corresponds to $P_{zz}$ up to 7 GPa. Confined water under such an ultra-high pressure is hardly realized in experiment. Nevertheless, the exploration into high-pressure states gives a coherent understanding of the phase behavior in confined spaces. The finite-size scaling analysis is implemented by the NPT MD simulations of the systems with N = 100, 200, 300, 400, and 500. Other NPT MD simulations are performed with N = 200 for most of the thermodynamic states and N = 500 for the states in which we observe the microscopic phase separation as shown in Fig. 5E.

Trajectories are generated by the Gear predictor–corrector method with a 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 coexistence conditions. Then, equilibrium properties are obtained from production runs of 30–90 ns. In the NPT MD simulation, each simulation time is 100 ns for most cases and is extended to a few microsecond for the finite-size scaling analysis. The equilibration run of 20 ns or longer is followed by the production run for analyses. The instantaneous configurations are used for the analyses, except for the snapshots. For the latter, we use the inherent structures, i.e., structures obtained by applying the constant volume steepest descent method to the instantaneous structures visited by the trajectories.