Physica A 314 (2002) 462 – 469

www.elsevier.com/locate/physa

How does water freeze inside carbon nanotubes?
Kenichiro Kogaa; ∗ , G.T. Gaob , Hideki Tanakac , X.C. Zengd
a Department of Chemistry, Baker Laboratory, Cornell University, Ithaca, NY 14853-1301, USAb Department of Chemistry, US Naval Academy, Annapolis, MD 21402, USAc Department of Chemistry, Okayama University, 3-3-1, Tsushima, Okayama 700-8530, Japand Department of Chemistry and Center for Materials Research and Analysis, University of Nebraska,

Lincoln, NE 68588, USA

Abstract
Phase behavior of quasi-one-dimensional water con1ned inside a carbon nanotube is studied
in the thermodynamic space of temperature, pressure, and diameter of the cylindrical container.
Four kinds of solid-like ordered structures—ice nanotubes—form spontaneously from liquid-like
disordered phases at low temperatures. In the model system that comprises of TIP4P watermolecules interacting with each other via short-range Lennard–Jones and long-range Coulomb
site–site potentials under a periodic boundary condition in the axial direction, the phase change
occurs either discontinuously or continuously depending on the path in the thermodynamic space.
That the isotherms for a given diameter are found to be similar to those around the liquid–gas
critical point of 8uids suggests existence of a phase boundary terminated by a critical point.
The apparently-complex phase behavior is accounted for by noting that the phase boundaries
are layered surfaces in the three-dimensional thermodynamic space and some of the surfaces are
terminated by critical lines.
c 2002 Elsevier Science B.V. All rights reserved.

PACS: 61.46.+w; 64.70.Dv; 68.08.De
Keywords: Con1ned water; Carbon nanotube; Phase behavior

1. Introduction
Phase behavior of water is very rich; there are 13 polymorphic phases of bulk ice
identi1ed experimentally thus far [1], the high- and low-density amorphous phases observed experimentally [2,3], and the high- and low-density liquid phases suggested by∗ Corresponding author. Permanent address: Department of Chemistry, Fukuoka University of Education,

Munakata, Fukuoka 811-4192, Japan. Tel.: +1-607-255-3845; fax: +1-607-255-4137.
E-mail address: kk275@cornell.edu (K. Koga).
c 2002 Elsevier Science B.V. All rights reserved.
0378-4371/02/$ - see front matter 
PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 1 0 7 4 - 9

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simulations in the experimentally inaccessible region [4,5]. Recent studies of phase behavior of water con1ned in narrow spaces have suggested the possibility of various condensed phases of water, and phase transitions between them in quasi-two-dimensional(Q2D) water [6,7]. Here, we present some recent results on water con1ned in very
narrow cylindrical spaces—quasi-one-dimensional (Q1D) water—under various thermodynamic conditions. Speci1cally, we examine water con1ned inside carbon nanotubes
since they provide well-de1ned cylindrical spaces and can be wet by low-surfacetension liquids including water [8].
Encapsulation of a second phase inside carbon nanotubes oLers a new avenue toinvestigate dimensionally con1ned phase transitions [9,10]. When pure liquid water
is encapsulated inside narrow carbon nanotubes, water molecules would be expected
to line up into some quasi-one-dimensional structures, and on freezing, may exhibit
quite diLerent crystalline structures from bulk ice. Crystallization of water is known tobe highly non-universal [11], i.e., the stable (or metastable) ice phase selected upon
crystallization depends sensitively on the external environment. Thus, the 1rst question
we try to address is what type of crystalline lattice could be selected on coolingthe con1ned water in narrow carbon nanotubes. Con1nement may change not only
resulting crystalline structures but also the way liquids freeze. For example, it has been
conjectured [12] that in carbon nanotubes con1ned matter may exhibit an uncommon
solid–liquid critical point [13] beyond which distinction between solid and liquid nolonger exists. Thus the second question to be raised here is how water freezes andmelts inside carbon nanotubes.
2. Model system
The model system consists of N molecules con1ned inside a cylindrical pore of diameter D. The intermolecular interaction is taken to be the TIP4P water whose potential
energy is given by the sum of the long-range Coulomb potential and the short-range
Lennard–Jones (LJ) potential between the interaction sites. The potential function is
O by a switching function. The force 1eld of the model carbon nantruncated at 8:75 A
otube is taken to be a LJ potential integrated over the cylindrical area of the tube. A
periodic boundary condition is applied in the axial direction (the z-direction) of thecylindrical unit cell of length l. In the molecular dynamics simulation the temperature T
and the axial pressure Pzz are controlled, thus the total energy and the length l 8uctuate.
The ‘zigzag’ (R; 0) carbon nanotube considered here is speci1ed by the index R, whichis the number of ‘benzene’ rings along the circumference and, thus, proportional to D.
O and R = 18 to D = 14:2 A.
O Other details
For example, R = 14 corresponds to D = 11:1 A
for the MD simulations and free-energy calculations are given elsewhere [14,15].
3. Structures of Q1D water at low temperatures
Shown in Fig. 1 are structures of the TIP4P water in the cylindrical hollow spaceof the model carbon nanotubes at low- and high-temperatures. The low-T phases arethose reached by cooling stepwise the high-T phases under Pzz =50 MPa. It is clear thatthe high-T phases have disordered structures with incomplete hydrogen-bond networks

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K. Koga et al. / Physica A 314 (2002) 462 – 469

Fig. 1. (a) Square, (b) pentagonal, (c) hexagonal, (d) heptagonal ice nanotubes at 240 K (except d at 230 K)
in the carbon nanotubes of R=14, 15, 16, and 17, and (e) – (h) the corresponding liquid phases at 320 K(except h at 300 K). See text for index R. The ice nanotubes form upon cooling under the axial pressure of
50 MPa in MD simulations. Shown here are potential-energy local minimum structures.

whereas the low-T phases have ordered structures with perfect networks. Speci1cally,
the low-T phases are Q1D n-gonal ‘ice nanotubes’ [16] composed of n-membered rings.
The square (n = 4) ice nanotube forms in the R = 13 and 14 systems, the pentagonal,
hexagonal, and heptagonal ice nanotubes are found in the R = 15, 16 and 17 systems,
respectively. All the Q1D ice nanotubes satisfy the ice rule as bulk ice does: every
molecule is hydrogen-bonded to exactly four nearest neighbor molecules. From the icerule it follows that the OH ‘arms’ of molecules along each edge of n-gonal prism areeither all up or all down and that those along each n-membered ring line up either
clockwise or counterclockwise. Another rule special for the Q1D ice is that the all-up
and all-down OH arms are found alternatively. (In the cases of n odd, this rule is
not fully satis1ed.) Besides their regular structures the low-T phases have diLusivity
as small as that of bulk ice [14], which indicates that the ‘ice’ nanotubes are indeed
solids or solid-like.
4. Phase behavior of Q1D water
Now we consider how the liquid-like disordered phases turn into the solid-like
phases. Fig. 2 shows molar volume changes against temperature for six systems (R =13; : : : ; 18) under 50 MPa. For R = 16 and 17, there are discontinuous changes involume and hysteresis loops. It is meant by ‘discontinuous’ that there is a suddenchange of the quantity when measured in units of 5 K. These discontinuous changes

K. Koga et al. / Physica A 314 (2002) 462 – 469
1

465

0.9

(a)

(b)

R=13

R=14

0.9
0.8
0.8

0.7
240

260

280

300

320

0.7

0.7
200 220 240 260 280 300 320 340 360
0.56

(c)

(d)

R=15

R=16

Volume/(Nr 2 ) (Å)

0.54
0.65
0.52

0.5
0.6
0.48

0.55
240

260

280

300

320

0.44

0.46
240

260

280

300

320

0.38

(e)

(f)

R=17

R=18

0.36
0.42

0.34
0.4

200

220

240

260

280

300

0.32
200

220

240

260

280

300

Temperature(K)Fig. 2. (a–f) Volume per molecule vs. temperature for water con1ned in the carbon nanotubes of R = 13–18.
The applied axial pressure is 50 MPa. Filled and open marks indicate the cooling and heating process,respectively.

correspond to structural changes between liquid-like and solid-like phases (hexagonal
and heptagonal ice nanotubes). For R = 13, 14, and 15, the volume changes without
any abrupt drop or jump and no hysteresis appears—an indication that the liquid-like

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K. Koga et al. / Physica A 314 (2002) 462 – 469
0.9

Volume/(Nr 2) / Å

280K
300K
330K
360K
0.8

0.7

0.6

200

400
600
800
Pressure (axial) / MPa

1000

Fig. 3. Isotherms in the Pzz –V plane for con1ned water in the (14,14) carbon nanotube at temperature
280 K (circles), 300 K (squares), 330 K (triangles), and 360 K (diamonds). Filled and open marks indicate
the compressing and decompressing process, respectively.

phase turns continuously into the solid-like phase (square for R = 13 and 14 and pentagonal for R=15). In the R=18 system the liquid-like disordered phase remains at thelowest temperature. These results indicate that along the isobar Pzz = 50 MPa con1ned
water undergoes continuous phase changes to square (n = 4) and pentagonal (n = 5)
ice nanotubes but discontinuous phase changes to hexagonal (n = 6) and heptagonal(n = 8) ice nanotubes. This does not mean, however, that the phase change is always
continuous in the narrower carbon nanotubes; indeed richer phase behavior is found in
the pressure-temperature plane at 1xed R. For R = 15 under Pzz = 500 MPa the con1ned
water exhibits on cooling a discontinuous phase change to the hexagonal ice nanotube,
instead of the continuous change to pentagonal solid-like phase at 50 MPa. For R = 14
under Pzz = 200 MPa the water undergoes a discontinuous phase change to pentagonal ice nanotube, instead of the continuous change to the square solid-like phase at
50 MPa, and under 500 MPa the liquid phase continuously turns into the pentagonal
ice nanotube. These results illustrate complex phase behavior of water con1ned in the
Q1D systems.
To better understand the phase behavior isotherms in the Pzz –T plane were obtainedat R = 14 and 15. Fig. 3 shows the result for the R = 14 system. At 280 K, volume
shows a sudden drop at 200 MPa upon compressing and a sudden increase at 170 MPa
upon decompressing; i.e., a hysteresis loop is observed. Structural analysis shows thatthe low-density phase is a liquid-like disordered phase while the high-density phase isa solid-like pentagonal ice nanotube. The diLusion constant of the pentagonal nanotube(at 200 MPa) and the disordered phase (at 190 MPa) on compressing-path is 1:0×10−10
and 9 × 10−7 cm2 s−1 , respectively. The diLusivity data reinforce that the high-density
phase is solid-like while the low-density phase is liquid-like. At 300 K, however, thetransformation appears to be continuous since no abrupt jumps in volume are found.
Long-time (200 ns) simulations were used in the pressure range between 250 andK. Koga et al. / Physica A 314 (2002) 462 – 469

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330 MPa, in which the isotherm exhibits a marked decline. At 330 K, the changes
in volume become much less abrupt, and 1nally, at 350 K, the changes in volume
are essentially smooth. At the two higher temperatures, structural analysis shows thatthe high-density phase is a solid-like pentagonal-nanotube phase and the low-density
phase is a liquid-like phase. In summary, the isotherms at R = 14 seem analogous
to those of 8uids near the critical point. For the R = 15 system, along the isotherms
examined (T = 280, 300, 320 K) the volume drops discontinuously at certain pressureson compressing. The low-density phases are liquid-like at these temperatures. (Thepentagonal feature appears at 280 K though.) The high-density phases are the hexagonal
ice nanotube (even at 320 K). Unlike in the R=14 system, no continuous phase change
from liquid-like to solid-like phases is found.
5. Phase diagram
Before we discuss the phase diagrams of con1ned water, some remarks are given
on what we mean by ‘phase boundary’ in the Q1D systems. We note that althoughcomputer simulation cannot prove or disprove the existence of a true phase transition, or
singularity in the partition function, in the thermodynamic limit N → ∞, extrapolationof its results at 1nite N to the thermodynamic limit is commonly accepted in 2D and 3D
systems because we know there exist true phase transitions. A Q1D system, however,
may or may not exhibit a true phase transition in the thermodynamic limit (dependingon the range of the intermolecular interaction [17,18]), and we do not discuss here theexistence of a true phase transition in the thermodynamic limit. On the other hand,
our computer experiments demonstrate that water in carbon nanotubes exhibits abruptchanges in density, energy, entropy, and structure and corresponding hysteresis loops
when certain paths are taken in the thermodynamic space. Furthermore, each single
phase is well de1ned throughout the unit cell with length of the order of 10 nm (orO It is in this sense that we use the terms ‘phase transition’ and ‘phase boundary’;
100 A).
the concepts of phase boundary, phase transition, and phase diagram in Q1D systems
are as useful as in 2D and 3D systems.
A schematic phase diagram for the R = 14 system is given in Fig. 4. The phaseboundary separates the high-density and low-density phases, which are referred to as
pentagonal and square phases for convenience. At low temperatures (e.g., 200 K) boththe pentagonal and square phases are solid-like; at medium temperatures (e.g., 270 K)
the former is solid-like while the latter is liquid-like. Since the phase change becomescontinuous above 280 K, the phase boundary should terminate at some point around thattemperature. The point is referred to as a ‘critical point’ in the same restricted sense
discussed above. This phase diagram explains the complex phase behavior observedin the R = 14 system. The continuous change from liquid-like structure to solid-likesquare ice nanotube at 50 MPa on cooling (Fig. 2b) does not involve any crossing of
the phase boundary; the path of 200 MPa isobar crosses the phase boundary, thus thediscontinuous change to the pentagonal phase; and the path of 500 MPa isobar does
not cross the boundary (thus the continuous change to the pentagonal phase) but does
cross the extension of the phase boundary where a marked change is found.

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K. Koga et al. / Physica A 314 (2002) 462 – 469

Pressure (axial) / MPa

400

R’ < R
200
Pentagonal

R’ > R

Square
0
200

250

300

Temperature / K

Fig. 4. A schematic T –Pzz phase diagram of con1ned water in the R = 14 system. The phase boundary(solid line) divides the low-density (square) and high-density (pentagonal) phases. At low temperatures bothphases are solid-like, at medium temperatures the former is liquid-like while the latter is solid-like, and at
the high temperatures both are liquid-like. The phase boundary terminates at a critical point around 280 K.
The arrows indicate the direction to which the phase boundary would be shifted if the diameter of the poreis slightly increased (R ¿ R) or decreased (R ¡ R).

For the R=15 system the low-density and high-density phases separated by the phaseboundary are referred to as pentagonal and hexagonal phases because the former is apentagonal ice nanotube at low temperatures while the latter is a solid-like hexagonal
ice nanotube. Unlike in the R = 14 system, no continuous change from one to theother phases is found in this system. Thus it seems that the phase boundary does
not terminate at a critical point. For the R = 16 systems, the low-T and high-density
phase is a solid-like hexagonal ice nanotube while the high-T and low-density phaseis liquid-like. No continuous path is found in this system. For the R = 17 systems, thelow-T phase is a solid-like pentagonal ice nanotube and the high-T phase is liquid-like.
Note that the low-T phase is a low-density phase unlike in the other systems (SeeFig. 2). Thus the phase boundary should have a negative slope in the T –Pzz plane.
Now we examine how the phase diagrams at discrete values of R relate with eachother. First we consider a system slightly wider than R = 14. Then one would 1nd thephase boundary shifted slightly downward with respect to that in the R=14 system. Thatis, the equilibrium pressure at given T decreases with increasing the radius of the carbon
nanotube. Le Chatelier’s principle explains this qualitatively: increasing the radius has
the eLect of lowering the pressure PN exerted on the cylindrical wall, which in turninduces a process that increases PN . The process is to drive the system to the pentagonalphase, because at the phase equilibrium PN exerted by the pentagonal portion is higher
than that by the square portion. A more precise explanation is given by the Clapeyron
equations for the quasi-one-dimensional system [15] by which one can obtain the rates
eqof the boundary shift, (@Pzz=@R)T and (@T eq =@R)Pzz . Indeed MD simulations at R = 14:2
have shown the shift of the phase boundaries that is consistent with prediction from

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469

the Clapeyron equations. With increasing R the square-pentagonal phase boundary witha critical point moves downward to low Pzz and eventually disappears. At R = 15
the square-pentagonal boundary no longer exists, and thus we observe the continuouschange from a liquid-like state to a solid-like state of the pentagonal ice nanotube under
Pzz =50 MPa. However there is a pentagonal–hexagonal phase boundary at R=15. Thisboundary would moves upward to high-Pzz with decreasing R and is no longer foundat R = 14. At R = 13, no discontinuity is found when the liquid-like state turns into
the solid-like square ice nanotube. However, simulations at smaller R show that there
is a phase boundary that separates a low-density liquid-like phase and the solid-likesquare ice nanotube phase. Thus no discontinuity at R = 13 is not because the system
is too narrow to exhibit any phase transition but because it is too narrow for thesquare-pentagonal phase boundary and too wide for the liquid-square phase boundary.
In summary the global phase diagram in the Q1D system seems to be the following:
the phase boundaries in the TPzz R space are found to be surfaces that have very
steep slopes with respect to the R-axis, and two of them (the square-pentagonal andliquid-square phase boundary) appear to terminate at a ‘critical line’.
Acknowledgements
KK wishes to thank Professor B. Widom for valuable discussions on the phasetransition. KK and HT are supported by Japan Society for the Promotion of Science(JSPS), the Japan Ministry of Education, and IMS. XCZ is supported by US National
Science Foundation and ONR. KK is visiting Cornell University under JSPS Fellowshipfor Research Abroad 2001.
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