INTRODUCTION
The carbon nanotube provides a well-defined quasi one-dimensional (1-D) space where one can prepare a quasi 1-D material and explore its new properties that have not been found for the corresponding bulk material. Water is one of the substances that may actually be confined to the carbon nanotube, but its properties in the quasi 1-D space had been as little studied as, or even less studied than, those of other quasi 1-D materials despite the fact that water itself has been more studied than any other substance. However, recent theoretical studies and the following experimental studies of water confined in the carbon nanotube showed that the confined water freezes into crystalline structures that were never found in the bulk water and exhibits the phase behavior of melting and freezing that no bulk system has ever shown.[1-3] This article is a review of these theoretical studies of the confined water. There are other theoretical or experimental studies on water in the carbon nanotube, focusing on the dynamic properties of water,[4] and more studies of confined water in general.[5]
Computer simulation studies[1] show that the structure of solid water in the zigzag (‘, 0), single-walled, carbon nanotube (SWCN) (with ‘ =13, 14, …, or 17) is quite different from the bulk ice structure; it is a 1-D array of n-gonal ”rings” consisting of n molecules. As in ordinary ice—in fact, as in any one of the 12 bulk ice polymorphs excluding ice X—every H2O molecule in the solid water in the carbon nanotube is hydrogen bonded to its four neighboring molecules. Because it has a hollow-tube structure and its width is of the order of 1 nm (=10~9 m), this quasi 1-D ice is called the ice nanotube. Simulations show that liquid water confined in carbon nanotubes freezes into a square, pentagonal, hexagonal, and heptag-onal form of the ice nanotube, each corresponding to n=4, 5, 6, and 7, respectively. Which structure is selected on freezing is dependent on the diameter of the carbon nanotube and the external conditions such as pressure and temperature. It is also found from the simulations and free-energy calculations that the phase behavior of confined water is qualitatively different from the bulk counterpart. Melting and freezing in any bulk system occur as an infinitely sharp change of the state of matter, i.e., as the first-order phase transition; however, freezing into and melting from the ice nanotube are either continuous with no well-defined melting point or discontinuous as in any bulk system.
In 2002, it was reported[3] that the structure of water inside the carbon nanotube at low temperatures determined by X-ray diffraction is consistent with that of the ice nanotube found in the computer simulation.
STRUCTURE OF ICE NANOTUBE
In any known crystalline ice every water molecule is hydrogen bonded to its four neighboring molecules. This is because with this condition the potential energy of the system would be significantly lower than otherwise. Thus, all the 12 different forms of bulk ice have a perfect hydrogen bond network structure. If this condition were possible in a quasi 1-D space, water would freeze into a quasi 1-D ice with a perfect hydrogen bond network. The ice nanotube is one type of such a quasi 1-D ice. It refers to a set of crystalline ice forms that extend in only one direction with a constant diameter of the order of 1 nm. The structure of the ice nanotube, which will be described in the following, was speculated and was analyzed by the classical and quantum calculations of the potential energy in vacuum at 0 K[6] before the actual demonstration by the computer simulation that liquid water freezes into the ice nanotube in the carbon nanotube.[1]
The ice nanotube is a 1-D array of n-gonal rings of water molecules. Imagine a strip of the 2-D square lattice with some finite width and infinite height (Fig. 1a). There is a water molecule at each intersection of the vertical and horizontal lines. In this way, every molecule, except those along the two edges of the strip, is connected to its four neighbors by hydrogen bonds as denoted by the solid line segments in the figure. Now if one rolls up the strip and joins the two long edges such that each molecule along one edge forms the fourth bond with a molecule along the other edge at the same height, then one would obtain a hollow-tube structure of water with a perfect hydrogen bond network (Fig. 1b), which is the basic structure of the ice nanotube. This is analogous to the SWCN, which may also be viewed as a rolled graphite sheet. The difference between this ice and the carbon nanotube is that, when unrolled, the carbon nanotube has a honeycomb lattice structure, whereas the ice nanotube has a square lattice structure. Also, while the carbon nanotube may take many different forms each being specified by the index (‘, m), the ice nanotube seems to have only a small number of different forms; only four forms, square (n=4), pentagonal (n=5), hexagonal (n = 6), and heptagonal (n = 7) ice nanotubes, were found in the computer simulation,[1] and even less structures were found in the experiment.[3] The diameter of the narrowest square ice nanotube is about 4 A and that of the widest heptagonal ice nanotube is about 6 A.
Fig. 1 (A) Square lattice with finite width and infinite height; (B) The basic structure of a perfect hydrogen bond network in the ice nanotube.
As in the ordinary ice, each oxygen atom is surrounded by four others, and between the central O and each one of its four neighboring O’s is a hydrogen atom. These four H’s are not at the midpoints between two O’s, but two are closer to the central O (forming H2O with the central O) and two are closer to the neighboring O’s. These conditions, which are referred to as the ice rule, are satisfied in most of the bulk ice forms. In the ordinary crystalline form of ice, i.e., ice Ih, every configuration satisfying the ice rule has nearly the same potential energy. The number of such configurations is (3/2)N, with N being the number of water molecules as obtained by Pauling, so that the residual entropy S per mole because of the degeneracy is, with R the gas constant, S=R ln(3/2)=0.81 cal/(mol K), in essential agreement with experiment.
In the ice nanotube, however, not all the configurations satisfying the ice rule correspond to the ground state; only some of them would have nearly a same lowest potential energy and contribute to the residual entropy. This is because four neighboring molecules of any given molecule are lying in the directions significantly different from the ideal tetrahedral directions as in ice Ih (Fig. 2). Two neighbors are those in the same n-gonal ring, and thus the two directions from the central molecule to such two neighbors form an angle p(1 — Two other neighbors are those in the two adjacent rings, and thus the two directions to such neighbors are both along an edge of the n-gonal prism and just opposite, forming an angle p. In ice Ih, two OH ”arms” in a H2O molecule may be in any two directions, and thus each molecule has six possible orientations. But in the ice nanotube, two OH arms of one molecule would never form hydrogen bonds with two neighbors in the two opposite directions along the edge of the n-gonal prism because the angle p is too large. With this constraint, if the two OH arms formed hydrogen bonds with two neighbors in the same ring, then the formation of a closed loop of the hydrogen bonds along that ring would be impossible. Therefore, one OH arm of a molecule must form a hydrogen bond with one neighbor in the same ring and the other OH arm of the molecule must form a hydrogen bond with one neighbor at the same edge of the prism. This is a condition that every molecule in the ice nanotube must satisfy in addition to the ice rule.
Fig. 2 Configuration of an O atom and its four neighboring O atoms in the ordinary ice (ice Ih) and in the ice nanotube.
It follows from this condition that the OH arms along each ring line up either clockwise or counterclockwise and that the OH arms along each edge of the n-gonal prism are either all up or all down. That is, there are only two ways of arranging a set of all the OH arms along each ring or along each edge. Because the number of such rings is N/n and the number of such edges is n, the total number of possible configurations of OH arms, or H atoms, in the n-gonal ice nanotube of N molecules would be 2n+(N/n).
However, the numerical calculation shows that configurations of an n-gonal ice nanotube would have significantly lower potential energies if they have the smallest number of two neighboring edges both with the same direction of the OH arms (up or down); the smallest number is 0 if n is even and 1 if n is odd. This is a special condition on the configuration of H atoms in the ice nanotube. The possible number of configurations would then be 2(N/n) for the n-gonal ice nanotube. The numerical calculation shows that the potential energy is nearly the same for these configurations. With R=NAk, where NA is Avogadro’s number and k is Boltzmann’s constant, the degeneracy W per mole is then 2(NA/n) and the residual entropy S per mole is S = k ln W=(R/n) ln 2; thus for n=4, 5, 6, and 7, we have S=0.70, 0.56, 0.47, and 0.40 cal/(mol K), respectively, which is smaller than that of ice Ih.
FREEZING AND MELTING BEHAVIOR
Necessary Conditions
The computer simulation and the free-energy calcula-tion[1] show that the following three conditions are essential for the formation of the ice nanotube. First, water may freeze into the ice nanotube only when it is confined to a cylindrical pore; the spontaneous formation of the ice nanotube without confinement has never been observed in experiment or in computer simulation. Second, the diameter of the cylindrical pore must be in a range between 10 and 15 A. The n-gonal ice nanotube with large n, say 10 or 20, would fit a pore with larger diameter, but then all the hydrogen bond angles would be so different from the ideal angle that such a structure becomes energetically less favorable than other possible crystalline or amorphous forms. Third, the surface of cylindrical pore should be hydrophobic in the sense that water molecules do not form hydrogen bonds to the surface atoms. This is because the closed hydrogen bond network of water molecules, i.e., the structure of ice nanotube, would be impossible if water molecules interact with the surface more strongly than with each other. The inner space of the carbon nanotube with an appropriate diameter may satisfy all the conditions described above and thus serve truly an ideal environment for the formation of the ice nanotube.
Freezing into the Ice Nanotube
The structure of the ice nanotube was noted as a possible form that the hydrogen bond network may extend only in one dimension with perfect connectivity, i.e., with every water molecule being hydrogen bonded to each of its four neighbors, and the stability of various forms of the ice nanotube at 0 K in vacuum was studied.[6] However, whether liquid water may really freeze into that structure in the carbon nanotube had remained to be seen until the molecular dynamics simulations1-1-1 demonstrated the spontaneous freezing of water into the ice nanotube at various conditions.
The potential functions and their parameters were chosen to simulate water molecules confined in the zigzag (‘, 0) SWCN. The diameter D of this form of the carbon nanotube is given by D = \f3aj sin(re/’) with a =1.423 A is the distance between two neighboring carbon atoms. Freezing and melting behavior of water was examined for the systems of the (13, 0), (14, 0), (15, 0), (16, 0), (17, 0), and (18, 0) carbon nanotubes with D =10.3, 11.1, 11.9, 12.6, 13.4, and 14.2 A. Equilibrium properties of each system at a given temperature T and a given pressure Pz in the direction parallel to the axis of the carbon nanotube was achieved by the constant NPzT ensemble molecular dynamics simulation. For each system, a series of such simulation was performed along the isobaric path or along the isothermal path. Details of the simulation methods are given elsewhere.[1]
Fig. 3 (A) Square, (B) pentagonal, (C) hexagonal, (D) heptagonal ice nanotubes at 240 K (except d at 230 K) at 50 MPa in the (14, 0), (15, 0), (16, 0), (17, 0) carbon nanotubes, and (E)-(H) the corresponding liquid phases at 320 K (except h at 300 K) at 50 MPa. Shown here for clarity are the structures at the local minima of the potential-energy surface; any instantaneous displacement from these structures because of the thermal vibrations is absent.
At high enough T (e.g., 350 K or 80°C) and low or moderate Pz (e.g., around or below 50 MPa), water confined in any of these carbon nanotubes is in a liquid state; it has no long-range order in its structure and has high diffusivity. The potential energy of the system because of the intermolecular interactions between water molecules at such T and Pz is much higher than that of the bulk water at the corresponding condition. This indicates that the hydrogen bonds in the liquid water are on average not strengthened but weakened by the hydrophobic inner surface of the carbon nanotube, as opposed to the hydrophobic effect on the structure of water around a hydrophobic solute molecule.
However, such a disordered structure of water at high T turns into an ordered structure at sufficiently low T in the (13, 0), (14, 0), (15, 0), (16, 0), and (17, 0) carbon nanotube (Fig. 3). The resulting ordered structure is identical or almost identical to the anticipated structure of the ice nanotube; the only difference, if any, is that some defects may exist in the otherwise ordered structure, and such defects may in turn cause a twisted structure (Fig. 3b). It is unclear as yet whether an ordered structure with a few defects is already in equilibrium at finite T or would be ultimately replaced by a perfect defect-free structure. In any case, water molecules in the ordered phase have extremely small or nearly zero diffusivity, which is also indicative of a solid state.
The diameter of the carbon nanotube plays a crucial role in selecting a crystalline form of the ice nanotube. When Pz is fixed at 50 MPa and T is decreased stepwise, liquid water freezes into the square ice nanotube (n=4) in the (13, 0) and (14, 0) carbon nanotube, whereas it freezes into the pentagonal, hexagonal, and heptagonal ice nanotubes (n = 5, 6, and 7) in the (15, 0), (16, 0), and (17, 0) carbon nanotubes, respectively.
In the (18, 0) carbon nanotube, however, crystallization was not observed, at least within the timescale of simulation. As remarked earlier, the structure of the ice nanotube becomes increasingly unstable as n goes up from 6. Thus, there must be an upper limit for n for the possible forms of the ice nanotube. Then there must also be a corresponding upper limit for the index ‘ or the diameter D of the carbon nanotube in which water may freeze into the ice nanotube. The results of the simulation suggest n=7 (i.e., the heptagonal ice nanotube) to be the upper limit, although it does not exclude the possibility of n=8 (i.e., the octagonal ice nanotube). However, formation of the ice nanotube with n>8 from pure water confined to a wider carbon nanotube seems unlikely.
Fig. 3
Fig. 3
In any ordered structure found at low T in the simulation, the orientation of water molecules indeed satisfies the special condition described in the previous section, as seen in Fig. 3; that is, the configuration of H atoms lying along the n edges parallel to the axis of the n-gonal prism is uniquely determined, and the only disorder is that in the arrangement of the clockwise and counterclockwise configurations of H atoms along each n-gonal ring.
