Confinement Effects on Water Clusters Inside Carbon Nanotubes

Article pubs.acs.org/JPCC

Confinement Effects on Water Clusters Inside Carbon Nanotubes J. Hernández-Rojas,*,† F. Calvo,‡ J. Bretón,† and J.M. Gomez Llorente† †

Departamento de Física Fundamental II and IUdEA, Universidad de La Laguna, 38205, La Laguna, Tenerife, Spain LASIM, Université de Lyon and CNRS UMR 5579, Bât. A. Kastler, 43 Bd du 11 Novembre 1918, F69622 Villeurbanne Cedex, France

‡

S Supporting Information *

ABSTRACT: The effects of confinement on water clusters inside nonmetallic carbon nanotubes with radii ranging between 4 and 7.5 Å have been computationally investigated by means of global optimization and finite temperature simulations. The water−water interaction is described by the TIP4P rigid body potential, and a Lennard-Jones potential is used for the water− carbon interaction. Water clusters containing up to 20 molecules are found to form 1D chainlike configurations for the narrow (7, 5) nanotube and 2D ladderlike structures in the (7, 6) tube. In wider tubes, 3D configurations are then formed showing helical motifs, ringlike or closed cage structures, before the most stable structure on flat graphene is eventually found. The same results are obtained by replacing the fully atomistic water−nanotube potential by its continuous approximation [Bretón, J.; González-Platas, J.; Giradet, C. J. Chem. Phys. 1994, 101, 3334], indicating a negligible effect of corrugation. The effects of additional nanotubes were also considered with the adsorption energies being found to converge rather quickly already for the triple-wall tube. Parallel tempering Monte Carlo simulations of the water octamer reveal a counterintuitive decrease in the melting point relative to the free-standing case. Molecular dynamics simulations show that melting is concomitant with some axial diffusion of the water molecules, and with radial diffusion perpendicular to the tube axis remaining limited. In accordance with previous studies concerned with bulk water, the weakening of the cluster thermal stability is interpreted as being caused by the hydrophobic character of the carbon−water interaction.

1. INTRODUCTION Water confined in nanoscale cavities presents unusual properties that differ from both the bulk and gas phases.1 Confined water molecules can bind and form small clusters, whose hydrogen-bond network is strongly modified by the confining conditions.2 Nanoscale pores that can be filled with water include biomolecular species3 such as aquaporins,4 geological compounds such as zeolites5,6 or silica pores,7 or hollow carbon-based materials. Water in contact with carbon substrates is relevant in various areas of science and technology. The adhesion and wetting of water on graphite surfaces has shown interesting lubricant properties.8 The universal nature of water as a combustion product, together with the chemical inertness of graphite under extreme conditions, also makes the water− carbon interaction involved in the design of corrosion-free combustion chambers and rocket nozzles.9 Monolayer graphite (graphene) and carbon nanotubes (CNTs) were found as highly sensitive gas sensors.10,11 At the nanoscale, water could be encapsulated into fullerenes owing to a series of cage-opening chemical reactions.12 Several theoretical works have since investigated the structural and finite temperature behavior of small water clusters encapsulated into fullerenes using either empirical potentials13,14 or a more explicit description of electronic structure.15,16 These works generally found a strong influence of confinement on the structure of water clusters. © 2012 American Chemical Society

Carbon nanotubes are probably more promising candidates for technological applications. One of their very remarkable features that was found was their ability to wet liquids having a surface tension below about 200 mN m−1 (ref 17) enabling filling of the nanotubes through suction mechanisms.18 The first successful introduction of water into carbon nanotubes was reported in 2001 for single-wall tubes19 and in 2004 for multiwall tubes.20 CNTs have shown great potential for achieving ultraprecise fluid conduction,21,22 which may allow selective drug delivery in living cells.23 Fluid flow in CNTs was found both experimentally24−26 and computationally27,28 to be enhanced dramatically relative to normal (unconfined) fluids and also to proceed pulselike.29 In the case of water, the molecular origin of fast transport was recently identified as caused by the curvature dependence of the friction coefficient,30 and thermodynamical arguments have been invoked as well.31 Starting with the pioneering work of Gordillo and Marti,́ 32 a wealth of computational studies have been devoted to modeling water in CNTs.33 The computational approach is further motivated by the difficulty of tailoring carbon nanotubes with desired morphology and by their possible toxicity34 that both hinder experimental studies. An important result of these Received: April 27, 2012 Revised: July 19, 2012 Published: July 19, 2012 17019

dx.doi.org/10.1021/jp304079b | J. Phys. Chem. C 2012, 116, 17019−17028

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Although recent calculations indicate that the filling of metallic and nonmetallic CNT with water shares the same features,38 to avoid more likely complex electrostatic effects in their interaction with small water clusters, we have chosen to limit this study to nonmetallic (chiral) carbon nanotubes varying the radius in the approximate range 4−7.5 Å. 2.1. Potential Energy Surfaces. The closed-shell electronic structure of the water molecules and the CNTs makes empirical methods particularly appropriate for modeling the potential energy surface (PES). We write the total energy as a sum of two contributions

studies was the finding that water in CNTs can exhibit specific ice phases (water nanotubes) that are absent from bulk ice,35,36 very narrow tubes only supporting one-dimensional chains.37,38 Such n-gonal ice rings have since been confirmed by X-ray diffraction analyses. Although challenging, the phase diagram of water in CNTs extended to the diameter variable39 has also received the attention of theoretical40 and experimental groups. In this work, we study relatively small water clusters inside carbon nanotubes. There is an important difference between this confinement and the encapsulation of such clusters by fullerene cavities because the molecules can freely spread along one dimension in the nanotube. In the fullerene case, the number of encapsulated water molecules cannot be infinite, and therefore, the behavior of confined small water clusters may differ significantly from that known experimentally for waterfilled nanotubes.42 However, and as will be shown below, the structural properties obtained for the small water clusters extrapolate to those of the confined infinite system. With respect to bulk water, finite water clusters inside nanotubes present one additional tunable parameter, namely, the number of encapsulated molecules. Although size is usually one primary parameter driving the properties of an isolated finite system on its way toward the bulk limit,43 confinement plays another likely important role in the case of confined clusters. In addition, the free boundaries at the two ends of the cluster are an essential difference with the case of bulk confined water with translational invariance along the tube axis. As precursors of mesoscopic water droplets, water clusters inside CNTs could give useful information on the structural, thermodynamical, and dynamical properties of low-density water in a confined system. The first purpose of the present paper is to investigate computationally the interplay between the number of water molecules and the nanotube diameter on the stable structures of clusters. For this, we use basin-hopping global optimization with a continuous representation of the water−nanotube interaction, which we carefully check against the full atomistic description. Varying the radius of the CNT strongly impacts the stable structure of the clusters, which can be rationalized into a local (size, radius) phase diagram. In our second objective, we consider the finite temperature thermodynamics by extracting the caloric curve of the representative water octamer from Monte Carlo simulations. Our results indicate a depletion in the melting point of the confined cluster below the nominal melting point, in agreement with experimental 44 and theoretical35 studies of bulk water confined in CNTs. Additional molecular dynamics (MD) simulations reveal that melting is concomitant with some significant radial diffusion, whereas diffusion perpendicular to the tube axis remains limited. Comparing the results obtained with the continuous and atomistic interaction potentials between water and the nanotubes, we finally find that corrugation effects do not appreciably influence the structural, thermodynamical, and even dynamical properties of the clusters. This paper is organized as follows. In section 2, we briefly describe the computational methods, including the interaction potentials and some details of the global optimization and simulation methods. The results are presented and discussed in section 3, and section 4 summarizes our conclusions.

Vtot = Vww + Vwc

(1)

where the first and second terms refer to water−water and water−carbon interactions, respectively. For Vww, we chose the popular TIP4P potential,45 which treats water molecules as rigid bodies with two positive charges on the hydrogen atoms and a balancing negative charge close to the oxygen atom. Besides Coulomb forces, the oxygen atoms interact by a Lennard-Jones (LJ) potential. The deficiencies of the TIP4P potential to describe physical properties of bulk water are wellknown. However, this model has proved to be rather accurate in describing the physics of small water clusters.46 For a nonmetallic tube, the water−carbon potential Vwc can first be safely written by a sum over water molecules of pairwise dispersion−repulsion terms between oxygen and carbon atoms. In this so-called atomistic model, these terms are expressed also by an LJ form with the parameters εOC = 0.0930 kcal/mol and σOC = 3.269 Å being calculated as in ref 35 using the Lorentz− Berthelot combination rules from the C−C parameters in ref 47 and the O−O parameters for TIP4P.45 The parameters of the water−carbon interaction have not been as thoroughly tested as those for the water−water interaction.33 For this reason, we have repeated some calculations with an alternative set of C−O parameters that were adjusted by Werder et al.48 in order to reproduce the contact angle of a water droplet on graphite. These parameters are slightly more binding; they read εOC = 0.0937 kcal/mol and σOC = 3.190 Å, and the corresponding results are given as Supporting Information. Calculations employing this explicit potential used periodic boundary conditions in the direction of the tube axis with the nanotubes being represented by a short multiple of their unit cell (still containing between 436 atoms for the (7, 5) tube and 1486 atoms for the (13, 9) tube). This weak interaction is not expected to deform the tube significantly, and assuming a fully rigid tube, a coarse-grained model can be obtained by integrating the LJ interaction over the rolled up carbon sheet with surface density Θ = 0.38 Å−2 (refs 49−52). Within this continuous description, the interaction between the oxygen atom of water molecule i and the entire CNT reads ⎡ ⎛ σ ⎞10 ⎛ σ ⎞4 ⎤ (i) 2 ⎢G12(si)⎜ OC ⎟ − G6(si)⎜ OC ⎟ ⎥ = 2πΘεOCσOC V wc ⎝ R ⎠ ⎝ R ⎠⎦ ⎣ (2)

in which R is the tube radius and si = ri/R with ri the distance of the oxygen atom to the tube axis. The functions G12 and G6 result from the integration of the repulsive and attractive contributions to the LJ potential over the cylinder and are expressed as51

2. METHODS The carbon nanotubes investigated in this work are referred to by their chiral indices (n, m) from which the tube radius R is obtained straightforwardly (see Supporting Information). 17020


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⎧ ⎡ 128 1 ⎨E(qi)⎢ 2 160(1 − si) (1 + si) ⎩ ⎣ (1 + si)8

energy. Molecular motion was analyzed in terms of the mean square displacement with the components along and perpendicular to the tube axis providing in turn the data needed for calculating the radial and axial diffusion constants. As was the case for global optimization, the finite temperature simulations were carried out using both the atomistic and continuous representations of the water− nanotube interaction.

⎪

⎪

104 99 + 2 6 (1 − si)4 (1 + si)4 (1 − si) (1 + si) 104 128 ⎤ ⎥ + + (1 − si)8 ⎦ (1 − si)6 (1 + si)2 +

−

4K (qi) ⎡ 16 15 ⎢ + (1 + si)2 ⎣ (1 + si)6 (1 − si)2 (1 + si)4

+

15 16 ⎤⎫ ⎥⎬ + ⎪ (1 − si)4 (1 + si)2 (1 − si)6 ⎦⎭

3. RESULTS AND DISCUSSION The properties of gas-phase water clusters modeled by the TIP4P model are rather well documented;60,61,66−72 the popularity of this potential is justified by its relatively good energetics with respect to more sophisticated approaches.46 In addition to isolated species, it is important to compare the properties of clusters confined in nanotubes to those obtained for clusters on flat carbon surfaces.54 3.1. Structural Properties. The global minima were determined for all clusters containing between 2 and 20 molecules and for a broad range of nanotube radii corresponding to various chiral indices. Rather than providing detailed information about all these different clusters, we have selected some representative sizes to illustrate the most salient structural features of confined clusters. The binding energies of the most stable structures for the water octamer are shown in Figure 1 as a function of nanotube

⎪

(3)

and G6(si) =

⎡ ⎤ 1 + si2 1 ⎢ ⎥ 4 E ( q ) − K ( q ) i i (1 − si)2 (1 + si)3 ⎣ (1 − si)2 ⎦ (4) 1/2

where qi = 2si /(1 + si) and with E(qi) and K(qi) the elliptic integrals of first and second kind, respectively. These integrals were evaluated using a simple but accurate polynomial approximation.53 In the limit of R → ∞, eq 2 converges to the interaction between a water molecule and the planar graphene sheet50,54 (see Supporting Information). 2.2. Global Optimization. Putative global energy minima of endohedral (H2O)N@CNT clusters with N ≤ 20 were located using the basin-hopping (BH) method55 also known as the Monte Carlo plus energy minimization approach of Li and Scheraga.56 This unbiased technique has been particularly successful for the global optimization of various molecular systems57−59 and, notably, water clusters.60−64 Suitable parameters for the present BH simulations were determined for all cluster sizes on the basis of preliminary tests on (H2O)12@CNT at various CNT radii. These benchmarks consisted of 104 minimization steps and were initiated from independent random geometries inside the CNT, varying the temperature and the target acceptance ratios of the Monte Carlo simulation. The results below were obtained at a constant temperature of kBT = 1.5 kcal/mol. A total of eight runs of 5 × 103 basin-hopping steps each were performed for N ≤ 14, and eight runs of 105 steps each for each size 15 ≤ N ≤ 20 were performed. Similar parameters were used for calculating the global minima in the graphene limit. This should ensure a reasonably high degree of confidence. 2.3. Thermodynamics and Dynamics. Parallel tempering Monte Carlo simulations were employed to simulate the statistical behavior of the water octamer at thermal equilibrium for different CNT radii. Starting from the global minima determined by basin-hopping, the simulations were carried out using 40 replicas geometrically allocated in the temperature range 10−300 K, and for each replica several series of 107 Monte Carlo cycles were performed. As a complement, constant-energy molecular dynamics simulations were performed to evaluate the extent of diffusion of the water molecules inside the nanotubes. Microcanonical trajectories were integrated using the leapfrog algorithm with quaternion coordinates and a time step of 2 fs for total integration times of 1 ns per trajectory. The results were averaged over 400 independent trajectories for each total

Figure 1. Total binding energy of the global minimum of the water octamer encapsulated in a CNT with increasing radius as obtained from the continuous and atomistic interaction models. The global minima are identified by arrows, and the energies in the graphene limit are highlighted.

radius and in the graphene limit of vanishing curvature. This figure also compares the predictions of the continuous and atomistic water−nanotube interaction models. In tubes as narrow as (7, 5), and in agreement with earlier studies,37,38 clusters consist of a one-dimensional hydrogen-bonded chain. As the tube diameter increases, the molecules can fill the opening space and the clusters become progressively more compact. Two, three, and four molecules can thus accommodate planes perpendicular to the axis in the (7, 6), (8, 6), and (8, 7) tubes, respectively. From (8, 7) and in larger tubes, the cubic structure of the octamer known in the unconfined case66 is recovered. The same minimum is obtained in the graphene limit. From the energetic point of view, both models predict an optimal binding energy for the octamer in the (10, 5) tube, which corresponds to a threshold beyond which the water− 17021



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exhibits a global minimum at variance with the less compact structure obtained in both the graphene limit and the isolated case. Clearly, an even larger tube radius is required to support this endohedral configuration. As R exceeds 5.9 Å, additional types of structures are sometimes found, such as hexagonal tubes or clathrate-like configurations with an atom surrounded by a closed cage. Such structures are depicted in Figure 3 for

nanotube interaction decreases while the water−water interaction remains practically constant. The atomistic and continuous models are also in quantitative agreement with each other; corrugation leads to a very minor decrease in binding energy. This result supports the use of the continuous model, which is computationally much cheaper, for carrying out global optimization in broad ranges of cluster size and tube diameters. Results with the C−O parameters provided by Werder et al.48 are very similar to those depicted in Figure 1 (see Supporting Information). The most noticeable variation is the shift of the optimal binding energy for the octamer from the (10, 5) to the (8, 7) tube, but no qualitative change is observed. Only for the (7, 6) tube, the binding energy with these alternate parameters is higher in magnitude than the reference value obtained with the present set of parameters. On the basis of these limited observations, the constraints experienced by the cluster confined in the nanotube can be seen as imposing the wetting of the inner wall of the tube through distortions in the hydrogen-bond network. This picture is confirmed by looking at the global minima for other cluster sizes, and we show in Figure 2 the variations of the

Figure 3. Examples of cage structures found for specific cluster sizes and nanotube diameters. Left: hexagonal water tube obtained for N = 18 in the (12, 5) CNT; right: clathrate-like structure obtained for N = 20 in the (12, 7) CNT.

N = 18 in the (12, 5) tube and for N = 20 in the (12, 7) tube, respectively. Interestingly, these structures bear even more similarity with the ice phases reported in bulk studies.2,35,36 The hexagonal ring structure is also found for the 12-mer cluster in CNTs with similar radii close to 6 Å, and we anticipate that the hexagonal ring motif could be found for larger clusters having several molecules with a multiple of 6. The clathrate-like configuration with one molecule enclosed in a 19-molecule cage is not uncommon in TIP4P water clusters as it was also reported as the global minima in free clusters.60 Again, we expect such conformations to be stabilized by the nanotube but for slightly larger radii in the approximate range of 6.5−7 Å. These clathrate-like structures also look as cluster equivalent to tubular structures with a water chain along the axis previously identified by Koga et al.35 It is a remarkable result that the structures found for these relatively small clusters are in close correspondence with the phases reported in studies of water-filled nanotubes. Apart from the aforementioned similarities, our global minimum structures show full agreement with the low-temperature region of the bulk temperature−radius phase diagram derived theoretically in ref 40. Therefore, the structural properties of CNTencapsulated water converge rapidly as the number of water molecules increases. Our results are also consistent with the experimental phase diagram reported in ref 41 in the small region where both studies overlap for the larger nanotube radii considered here. In absence of confinement, other minima are favored over these structures. However, the substrate may still play some role, and in the case of graphene, the global minima sometimes differ from the ones of free clusters because of a favorable contact energy for specific isomers. This finding was also reported for multilayer graphite54 and concerns the same oddsized clusters of N = 7, 11, 13, 15, 17, and 19 as well as the hexamer. Such structural transitions are made possible by the extremely rough energy landscapes of water clusters75 and are not systematic. In particular, we do not find evidence for deformations that strongly increase the contact area between the cluster and the substrate as expected from a hydrophobic substrate, that is, the water−water interaction dominates the energetics and the flat substrate plays a minor role. In large carbon nanotubes, convergence of the global minimum to the

Figure 2. Total binding energies of global minima of (H2O)N clusters encapsulated in CNTs with increasing radius, for N = 4, 8, 12, 16, and 20, as obtained from the continuous interaction model. The structures obtained for N = 20 are indicated by arrows, and the energies in the graphene limit are highlighted.

total binding energies for clusters containing 4, 8, 12, 16, and 20 molecules and for increasing tube radius. Here, the continuous interaction model was used. As was the case for the octamer, the binding energies exhibit nonmonotonic variations with tube radius, and as expected, the optimal radius increases with cluster size. For clarity, the global minima are depicted only for the 20-mer, but some general trends that are valid for all the clusters considered can still be inferred. All clusters show the one-dimensional chain as the global minimum for the narrowest (7, 5) tube and the twodimensional, ladderlike structure for the subsequent (7, 6) tube. The distorted helix found for the (8, 6) tube is reminiscent of the spiral-like structure of bulk water found in simulations by several authors.73,74 In larger tubes, more regular n-gonal tubes are favored in a fashion also observed by Koga et al. for bulk water under pressure.35 Even for the largest (11, 6) tube considered in this figure (R = 5.8 Å), the 20-mer cluster 17022



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graphene limit is then mainly driven by geometric arguments. However, geometry alone cannot explain the existence of optimal radii as those shown in Figure 2 and Figure 1. A hollow cylinder with hard wall interaction with the water molecules gives a binding energy that decreases monotonically with the cylinder radius and tends to a constant asymptotic value. We show below that the CNT−water interaction has a significant effect on the finite temperature properties of the encapsulated clusters. Therefore, this interaction plays a noticeable role in the physical properties of water clusters. The number NH of hydrogen bonds in the cluster is in direct correspondence with the strength of the water−water bond, and this quantity is useful to disentangle the role of the two interactions on the stability of the global minima. By definition, such bonds occur if the O−H distance remains below 2.5 Å and the H···O−H angle is greater than 130°. The variations of the number of hydrogen bonds with increasing tube radius are shown in Figure 4 for the same selection of cluster sizes as

Figure 5. Structural map of water clusters inside carbon nanotubes with increasing radius (ordinate) as a function of the number of water molecules (abscissa). Various types of configurations are denoted by specific symbols given in the legend with circles in case the structure is the same as in the isolated case. Symbols for the global minima obtained on the flat graphene surface are also indicated in the upper part.

minimum of the confined cluster has the same structure as in the isolated case. In about half of the case studies, the global minimum of the water cluster on the graphene surface coincides with that of the corresponding isolated water molecule. Independently of cluster size, the one-dimensional chain and the two-dimensional square ladder are the only structures available in the smallest (7, 5) and (7, 6) tubes, respectively. Helices with three molecules in contact with the tube are found systematically for the (8, 6) tubes, but the spiral motif is also occasionally encountered for larger clusters in larger tubes. The square ring (fused cubes) structure is preferred for nanotube radii higher than 5 Å, and pentagonal rings are found above 5.7 Å. The global minima of free clusters are usually recovered for tubes with sufficiently large radius, which is in keeping with the stronger magnitude of the hydrogen bond relative to the water−tube interaction. However, in some rare cases (denoted by an asterisk in Figure 5), the interaction with the tube leads to a structural transition favoring higher-lying isomers. These isomers usually expose a slightly larger area to the tube, improving the water−tube interaction to the expense of water−water binding. In the hexamer, for instance, the most stable structure for all radii larger than 5.5 Å is the so-called book isomer (a bent double square) instead of the cage global minimum.60 In clusters containing at least 10 molecules, confinement impacts their optimal structures even for large CNT diameters; the fused cubes or pentatonal rings are the global minima in the isolated clusters only for sizes that are multiples of 4 or 5, respectively. The overall trends exhibited by the structural diagram suggest that the n-gonal motif should be generally observed for clusters for which the global minimum in the graphene limit exceeds the tube dimensions. The formation of an additional water chain at the center of the hexagonal ring reported by Koga et al.35 for bulk water under pressure could have an analogue here for appropriate tube diameters, but probably rings larger than the hexagon would be needed to insert axial molecules in the absence of pressure or temperature. Apart from the aforementioned one, there is no noticeable difference

Figure 4. Number of hydrogen bonds in the global minima of (H2O)N clusters encapsulated in CNT with increasing radius for N = 4, 8, 12, 16, and 20. The values obtained for the global minima on the flat graphene surface are indicated on the right.

previously discussed. In addition to cluster size, NH increases with dimensionality of the global minimum, reaching a maximum when the helical structure is formed for the (8, 6) tube. From (n, m) = (8, 6) and in larger tubes, and for 4-, 8-, and 12-mer clusters, NH takes the same values as those obtained in the graphene limit. Less markedly, NH also depends on the ring structure of the cluster because larger rings are associated with fewer hydrogen bonds. This is simply the consequence of the water molecules in n-gonal rings to participate in four hydrogen bonds and only three for the outer rings. Therefore, the average NH will always lie between 3 and 4, and structures with smaller but more numerous rings will have a higher NH. Such an effect is visible for the 16- and 20-mer clusters and for tube radii exceeding 5.5 Å. In these situations, the global minima change from fused cubes to pentagonal rings (see Figure 2) for increasing tube diameter before changing again in the graphene limit. The various types of structures predicted by the continuous model can be categorized into several families from which the (radius, size) structural phase diagram presented in Figure 5 may be inferred. This diagram provides a simple view of many of the structural results obtained in this work, which may be useful in future theoretical and experimental research. For this reason, diagrams of this kind have been used often in other studies.76−78 Figure 5 also highlights situations where the global 17023



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water−CNT interaction are represented in Figure 7 as a function of temperature for three very distinct tubes in which

between the results obtained with the two sets of C−O parameters (see the Supporting Information). The continuous interaction model can be straightforwardly extended to account for coaxial, multiple-wall nanotubes (MWCNTs). We have repeated some global optimizations for the water octamer encapsulated into double-, triple, and quadruple-wall tubes with fixed distance between successive tubes equal to the interplane distance in graphite (3.4 Å). No water molecule is allowed to reside between nanotube walls as strong energetic penalties would disfavor such situations. The binding energies of this cluster are shown in Figure 6 as a

Figure 7. Heat capacity of the water octamer inside (7, 5), (7, 6), and (13, 9) carbon nanotubes as obtained from Monte Carlo simulations using the continuous and atomistic interaction models. The extent of the cluster in the nanotube is visualized in each panel.

Figure 6. Total binding energy of the water octamer confined in single-, double-, triple-, and quadruple-wall carbon nanotubes with intertube distance equal to 3.4 Å for increasing internal tube radius. The limits at zero curvature are highlighted for the corresponding number of graphene sheets as well as for graphite.

the global minimum is a 1D chain, a 2D ladder, or the 3D cube. The thermodynamical analysis can be assisted by considering the average radial distance ⟨ρ⟩ of oxygen atoms in the cluster, whose variations with temperature are represented in Figure 8. This quantity itself is better understood from the knowledge of the spatial extent of the cluster inside the tube, which can be

function of increasing inner radius. Encapsulating the inner CNT into additional tubes shifts the binding energy of the cluster toward lower values, and there is no change in the global minimum. The energies obtained for three- and four-wall nanotubes are within 0.5% of each other, and the same convergence holds for multilayer graphene (see right part of Figure 6). In particular, we recover a previous observation54 that the interaction between a water molecule and a semiinfinite graphite substrate is well accounted for by the first four graphene layers only. The energy difference between the double- and single-tube encapsulations steadily decreases in magnitude with increasing inner tube radius, reaching its lowest value in the flat graphene limit. This loss in energy reflects the stronger and more isotropic bonding of the tightly confined clusters having their center of mass near the tube axis, whereas it is favorable for the clusters to move toward the inner wall in large tubes. Considering the very complex energy landscapes of water clusters,75 structural transitions could well take place in large encapsulated clusters simply by adding a second nanotube. In the Supporting Information, the results obtained with the alternate set of C−O parameters from Werder et al.48 are presented. Apart from the slight shift in the optimal binding energy to a lower tube radius, the general behavior is very similar. 3.2. Finite-Temperature Properties. We now discuss the possible role of temperature on clusters encapsulated in singlewall carbon nanotubes as analyzed from the results of parallel tempering Monte Carlo simulations. We again focus on the water octamer for which accurate reference data are available in the isolated case.66,79−81 The canonical heat capacities obtained from both the atomistic and the continuous models for the

Figure 8. Average radial distance of water molecules from the octamer inside (7, 5), (7, 6), and (13, 9) carbon nanotubes as obtained from Monte Carlo simulations using the continuous and atomistic interaction models. 17024



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gauged from the global minima depicted within the corresponding tubes in Figure 7. In the (7, 5) tube, the heat capacity only shows some very smooth variations with temperature with a small bump near 20 K. This behavior is due to the low structural diversity granted by the onedimensional chain, the bump being caused by some flipping of the molecules, keeping the number of hydrogen bonds constant and equal to 7. This transition proceeds with a slight shift of the molecules toward the tube axis, but confinement keeps the radial distance close to the equilibrium value even at high temperatures. Clear peaks found in the heat capacity for octamers confined in the larger (7, 6) and (13, 9) tubes give strong evidence that proper phase changes take place. In the (7, 6) tube, hydrogen bonds at the two ends of the ladder structure are broken above 180 K and form pieces of 1D chains before eventually dissociating at higher temperatures. This gives rise to a significant latent heat and is attested in Figure 8b by the drop in the average radial distance at this temperature. The global minimum of the octamer encapsulated in the (13, 9) tube is a cube weakly bound next to the curved inner surface. At low temperature, thermal expansion pushes the cluster away from the tube bringing its center of mass closer to the axis. Melting occurs near 170 K with a significant increase in ⟨ρ⟩. For this tube, the cluster has more room to melt and can access a broader range of higher-lying minima, especially those having a larger contact area with the tube, and this is the reason for the increase. In these minima, some of the hydrogen bonds may be broken. It is striking that the atomistic and continuous water interaction models yield essentially identical caloric curves and geometrical properties for the three tubes, and additional calculations for other clusters confirm this finding (results not shown). The negligible role played by corrugation on the global minima is thus confirmed on the thermodynamics. The depletion in the melting point was previously reported in experimental44 and theoretical35 works on bulk water in CNTs, and it has also been found in the case of water clusters in exohedral adsorption on fullerenes.82 This feature reflects the weakening of hydrogen bonds for water molecules in direct contact with the hydrophobic substrate. Thermodynamical results obtained with the alternate set of C−O parameters from Werder et al.48 are given in the Supporting Information. Differences between the two sets of parameters in this case become more significant when the tube radius is comparable to the cluster dimensions, whereas the behavior in narrow (7, 5) and broad (13, 9) tubes remains quantitatively similar. In the (7, 6) tube, the melting point of the octamer shifts by about 30 K to higher temperatures. This higher thermal stability is consistent with the higher binding energy found at this very specific radius. The average radial distance is generally higher with the parameters from Werder et al., which was anticipated because of the shorter C−O bond distance of this model. For the same clusters, we have also examined the selfdiffusion of water molecules into their cylindrical cavity by estimating the diffusion coefficient D from the long-time slope of the mean-square displacement of the molecular centers of mass. Here, we also distinguish displacements parallel and perpendicular to the tube axis, thereby providing axial and radial diffusion coefficients that are represented in Figure 9 as a function of microcanonical temperature. The average rootmean-square (rms) bond length fluctuation δ, also known as

Figure 9. Self-diffusion coefficients for the water octamer inside (7, 5), (7, 6), and (13, 9) carbon nanotubes as obtained from molecular dynamics simulations using the continuous (black symbols) and atomistic (red symbols) interaction models. Axial and radial diffusion coefficients are presented by full and empty symbols, respectively. The insets show the rms bond length fluctuation indices δ as a function of temperature.

the Lindemann index, was also calculated from the molecular dynamics trajectories. Its variations with temperature, shown as insets in Figure 9, display smooth increases but sharper rises near the melting point for the clusters confined in the two larger tubes. In the narrowest tube, the progressive increase of δ is due to the more frequent dissociations of molecules at both ends of the chain. For the three systems, the diffusion coefficients steadily increase with temperature. Radial diffusion does not carry any particular signature of the melting phase change. However, it has a markedly larger value in the larger tube, which indicates some mobility related to the much lesser confinement. Axial diffusion remains marginal at low temperatures but shows some clear jumps near 230, 180, and 170 K for the three tubes with increasing diameter. The two latter temperatures coincide with the melting point of the clusters, and for the narrowest tube, this axial diffusion is related to dissociation of the less bound, outer molecules at the ends of the chain. Clusters encapsulated in nanotubes should thus manifest some spontaneous radial mobility if their dimensions are lower than the nanotube diameter, but thermally activated axial diffusion is otherwise more likely. We also notice that the same diffusion constants are obtained with the atomistic and continuous interaction potentials with the diffusion dynamics being thus poorly influenced by corrugation effects. This is consistent with our previous results on the structure and thermodynamics but could seem at variance with other computational studies reporting fast transport of bulk water in CNTs.25,30 However, the situation considered in such studies was radically different because in addition to a bulk fluid they usually imposed a flow or different pressures between the two ends of the tube. There is no pressure to hold the clusters in the tube, and thermal fluctuations of the tube acting as the thermostat could be 17025



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because of their sensitivity to the interaction potentials. While TIP4P is reasonably accurate for free clusters,46 parameters for the water−carbon interaction are somewhat scattered.33 In this work, we have checked two sets of LJ parameters for the latter interaction. We found that structural and thermodynamical results are practically unchanged except for tubes that are of comparable radius to the cluster itself. In this case of the (7, 6) tube, the stronger binding exerted by the C−O interactions with parameters of Werder et al.48 thermally stabilizes the water octamer by about 30 K. This result in itself shows that the influence of the nanotube on the thermodynamics is not purely geometric (as if it were a hard wall cylinder), but it has a clear energetic component. Future work could be aimed at defining more accurately the water−carbon interaction against more sophisticated descriptions involving explicitly the electronic structure and taking dispersion effects into account. Another perspective could be estimating the dissociation time scales in order to determine the lifetime of such clusters as a function of their size, the tube diameter, and the temperature. This quantity could be important when describing nanotubes filled with water at low density, and more generally, it points out the issue of nucleation under confined geometries. One practical strategy to extend the present work to determine these lifetimes combines molecular dynamics simulations and rate theories.83 Finally, it would also be interesting to consider other types of confinements, such as nanoporous amorphous carbon, which has been suggested for applications in gas separation,84 or hydrophilic environments such as nanoscale silica capillaries. By probing regimes with stronger or more diverse water−substrate bondings, completely different structures and finite temperature properties could arise. This further raises the issue of accurate models for the water−substrate interaction, but the approximation of a rigid substrate would also become questionable.

sufficient to vaporize them over long times. However, the close resemblance between the structural properties of small water clusters confined in CNTs and those of the corresponding bulk systems is not expected for the dynamical and thermodynamical properties because of the stronger size effects involving the contribution from the two free extremities that cannot be neglected. We also note from these calculations that dynamical and thermodynamical features are, unlike structural features, much more dependent on the number of encapsulated water molecules, and we are therefore far from the bulk limit.

4. CONCLUSIONS AND PERSPECTIVES Confinement of bulk atomic and molecular systems can alter their static, dynamical, and thermodynamical properties to a major extent. Such alterations also take place for finite-size clusters for which the intrinsic behavior in the gas phase is expected to be even more sensitive to a constraining environment. In the present work, water clusters encapsulated into nonmetallic carbon nanotubes have been computationally examined by means of global optimization and molecular simulation. Using the TIP4P potential for describing water clusters and an atomistic or a continuous model for the interaction between water and the tube, the stable structures were found to be essentially dominated by geometric confinement. Along the lines of earlier studies,35−40 water clusters preferentially arrange into structures of increasing dimensionality with increasing tube diameter. 1D chains are formed in the (7, 5) tube, 2D ladders are formed in the (7, 6) tube, and 3D helical and then n-gonal ring or clathrate-like structures are formed in larger tubes. The global minima obtained for a substrate with vanishing curvature (flat graphene sheet) are generally recovered once their dimensions become comparable to the tube diameter. Some exceptions occur in which local minima with a larger contact area with the tube are favored. The structures obtained in the graphene limit often differ from the global minima for the free clusters, which is largely due to their very rugged energy landscape with huge numbers of close-by minima. Taking the octamer as an example, the thermodynamical behavior was also found to be strongly affected by confinement at least through its consequences on available structures. Solidlike and liquidlike phases can be clearly identified as soon as two-dimensional clusters become allowed by the tube diameter with the disordered phase having floppy end chains. When the tube is large enough to support the 3D cubic structure, the melting point is depleted with respect to the freecluster reference. This behavior consistent with previous studies on bulk35,44 and clusters82 has its origins in the weakening of hydrogen bonds near the hydrophobic carbon substrate. Melting was also found to occur with some rise in the axial self-diffusion with radial diffusion being only significant for the least confined clusters. One major result of the present work is the nearly identical results obtained for the atomistic and coarse-grained representations of the water−nanotube interaction. Although expected at high temperatures, this finding was more surprising in the 0 K limit of static structures; because of the ruggedness of the energy landscape, changes in energy as small as corrugation barriers could have led to structural transitions. We also do not find evidence for any marked role of corrugation on the self-diffusion coefficients. As pointed out by Alexiadis and Kassinos,33 the conclusions drawn from molecular simulation should be taken cautiously

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ASSOCIATED CONTENT

S Supporting Information *

Formulas for the tube radius as a function of chiral indices, interaction with the flat graphene surface, and energetic and thermodynamical simulation results obtained with the alternate set of C−O Lennard-Jones parameters of Weder et al.48 This material is available free of charge via the Internet at http:// pubs.acs.org/.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS F.C. would like to thank the Pôle Scientifique de Modélisation Numérique (PSMN) in Lyon, France, for generous allocations of computer time. J.H.-R. acknowledges a fellowship (PR20110337) from Ministerio de Educación, Cultura y Deportes (Spain) and thanks LASIM for its hospitality. J.H.-R., J.B., and J.M.G.Ll. also acknowledge financial support from Ministerio de Ciencia e Innovación (Spain) under Contract No. FIS200907890. 17026



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Confinement Effects on Water Clusters Inside Carbon Nanotubes

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