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transport properties of water confined within long narrow carbon nanotubes. ... of confined water are studied at time scales in excess of 500 ps, a Fi...
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NANO LETTERS

The Mechanism of Water Diffusion in Narrow Carbon Nanotubes

2006 Vol. 6, No. 4 633-639

Alberto Striolo* School of Chemical Biological and Materials Engineering, The UniVersity of Oklahoma, Norman, Oklahoma 73019 Received November 14, 2005; Revised Manuscript Received January 30, 2006

ABSTRACT Carbon nanotubes show exceptional physical properties that render them promising candidates as building blocks for nanostructured materials. Many ambitious applications, ranging from gene therapy to membrane separations, require the delivery of fluids, in particular aqueous solutions, through the interior of carbon nanotubes. To foster these and other applications, it is necessary to understand the thermodynamic and transport properties of water confined within long narrow carbon nanotubes. Previous theoretical work considered either short carbon nanotubes or short periods of time. By conducting molecular dynamics simulations in the microcanonical ensemble for water confined in infinitely long carbon nanotubes of diameter 1.08 nm, we show here that confined water molecules diffuse through a fast ballistic motion mechanism for up to 500 ps at room temperature. By comparing the results obtained for the diffusion of water to those obtained for the diffusion of a reference Lennard-Jones fluid, we prove here that long-lasting hydrogen bonds are responsible for the ballistic diffusion of water clusters in narrow carbon nanotubes, as opposed to spatial mismatches between pore−fluid and fluid−fluid attractive interactions which, as shown previously by others, are responsible for the concerted motion of simple fluids in molecular sieves. Additionally we prove here for the first time that, despite the narrow diameter of the carbon nanotubes considered which may suggest the existence of single-file diffusion, when the trajectories of confined water are studied at time scales in excess of 500 ps, a Fickian-type diffusion mechanism prevails. Our results are important for designing nano fluidic apparatuses to develop, for example, novel drug-delivery devices.

Because of exceptional mechanical and electronic properties, carbon nanotubes1 are important building blocks for nanocomposite materials and nanomachinery. Exotic carbonnanotube-based apparatuses such as nanosyringes,2 nanotubeembedded membranes,3 and gating mechanisms4 have been envisioned. The hydrophobic interior of carbon nanotubes is also considered, at a first approximation, as model for ion channels,5 even though it is recognized that only by accounting for the complex topology and chemical structure of membrane proteins it is possible to gain physical insights about the ion-transfer mechanism.6-8 Whether our interest is stimulated by materials science or by biological applications, it is crucial to understand the thermodynamic and transport properties of aqueous solutions within carbon nanotubes. Because of the reduced diameter of carbon nanotubes (in the order of nanometers), thermodynamic and transport properties of confined water differ substantially from those observed in the bulk.9 Several groups, including ours, have studied the adsorption of water in carbon nanotubes.10-15 Koga et al.16,17 investigated the structure of water within single-walled carbon nanotubes (SWNTs) of diameter 1.4 nm and found that water forms ice nanotubes composed by a rolled sheet of cubic ice. Recent neutron scattering studies, combined with molecular simula* Phone: 1 405 325 5716. Fax: 1 405 325 5813. E-mail: [email protected]. 10.1021/nl052254u CCC: $33.50 Published on Web 03/02/2006

© 2006 American Chemical Society

tions, reveal that water in SWNTs of diameter 1.4 nm forms a core-shell structure in which the cubic ice sheet described by Koga et al.16 is coupled with a chainlike configuration at the center of the shell.18 Predictions from grand canonical Monte Carlo (GCMC) simulations19 agree with the experimental observations. When simple fluids are simulated within narrow molecular sieves, it has been shown that their diffusion is dominated by concerted events in which multiple molecules move simultaneously.20,21 These concerted events are due to strong mismatches between the distance between binding sites along the pore axis and that between adsorbed molecules that minimizes adsorbate-adsorbate interactions.22 On the contrary, the inherent smoothness of the interior of carbon nanotubes generates exceptionally fast diffusion coefficients for simple gases,23-25 as well as for water.10,26,27 Concerning water, the scientific interest was centered initially on its behavior within short, narrow tubes. Hummer et al.28 studied one SWNT with diameter of 0.8 nm and length of 1.35 nm immersed in bulk water. They found that water molecules move occasionally along the nanotube axis via bursts of hydrogen-bonded clusters of molecules. Experimental observations of water diffusion through bacterial potassium channels seem to corroborate the theoretical predictions.29 Beckstein and Sansom30 extended those pioneering simula-

tions to beyond 50 ns and were able to study the free energy difference between filled and empty carbon nanotubes. The study of water diffusion through membranes composed of an array of aligned carbon nanotubes is now of interest. Because of frictionless water-carbon nanotube interface, extremely high flow velocities have been experimentally observed when the carbon nanotubes are of approximately 7 nm in diameter.31 These studies find theoretical corroboration in earlier molecular simulations for the osmotic water transport through carbon nanotube membranes,32 in which the fast diffusion of water through the nanotubes of diameter 0.81 nm and length 1.34 nm was ascribed to the single-file diffusion mechanism. Recent molecular-simulation studies for the transport of water-methanol mixtures through nanotubes (hydrophobic and hydrophilic) under a chemical potential gradient indicate that the fluid transport across the pore is controlled by pore-entrance effects,33 in qualitative agreement with previous theoretical studies for carbon nanotubes immersed in bulk water.28,34,35 In diffusion studies it is common practice to integrate the equation of motion for water molecules confined within infinitely long nanotubes. The self-diffusion coefficient is then obtained from the average mean square displacement of water molecules as a function of time. It is often assumed that the motion of confined water follows the Fickian mechanism, even in the narrowest tubes considered. Using this procedure, Liu and Wang studied the transport properties (self-diffusion coefficient, thermal conductivity, and viscosity) for water in SWNTs with diameters from 1.1 to 2.1 nm.36 They found that the axial diffusion coefficient is lower than that characteristic of bulk water and that it decreases as the pore diameter narrows. For the largest nanotube considered (2.1 nm in diameter), the calculated diffusion coefficient equals 0.9423 × 10-9 m2/s. It should however be pointed out that Fickian motion only occurs when the molecules pass each other chaotically in the direction of flow. This may not be true when fluid molecules are confined in long and narrow nanotubes. Understanding the mechanism of water diffusion under these circumstances, the goal of this contribution, is crucial for designing nanomachinery and synthetic membranes whose thickness reaches 5-10 µm3 as well as for the development of novel drug-delivery devices. The availability of carbon-nanotube-based membranes37 will permit the experimental verification of our theoretical predictions. The diffusion of water in bulk solutions occurs through a Fickian-type mechanism according to which the mean square displacement (msqd) scales linearly with time. The diffusion of confined water is often expected to obey the Fickian dependence, provided that the molecular motion is chaotic and that water molecules can pass each other in the direction of motion. When water molecules are confined in a narrow one-dimensional channel, they are prevented from passing each other, and the diffusion could become of the singlefile type in which the msqd scales with the square root of time.38 When confined water molecules move in a highly coordinated fashion, it is possible that the diffusion occurs through a ballistic mechanism, in which the msqd scales with 634

Figure 1. Representative simulation snapshots for water confined in (8:8) SWNTs. We show the view parallel (left) and that perpendicular (right) to the pore axis. The carbon atoms are not shown for clarity. The yellow lines connect the centers of nearestneighbor carbon atoms. Oxygen atoms are shown in red; hydrogen atoms are shown in light blue.

the square of time. The three motion mechanisms are mathematically described as dr2 ∝ D dt dr2 ∝ F dt1/2

(1)

dr2 ∝ B dt2 From top to bottom, the three equations represent the Fickian, single-file, and ballistic diffusion mechanism, respectively. dr2 is the msqd, dt is time, and D, F, and B are proportionality coefficients. D is the self-diffusion coefficient, and F is the single-file mobility. Deviations from the time dependencies expressed in eq 1 reflect changes in the diffusion mechanism, which could be due to pore-fluid interactions, pore size, pore connectivity, etc. The goal of the present study is to understand the mechanism of water diffusion through infinitely long narrow carbon nanotubes. We are also interested in understanding whether the transport properties of confined water differ from those of simple fluids. To tackle this issue we use molecular dynamics simulations in the microcanonical ensemble. We integrate the equation of motion for confined water molecules over several nanoseconds (up to 18 ns). The simulation details are reported in the Appendix. We are interested in understanding if and when singlefile diffusion can occur for confined water molecules. We restrict our analysis to water confined within (8:8) SWNTs. The diameter of these nanotubes, defined as the distance between opposite carbon atoms, is 1.08 nm. Water adsorbed in (8:8) SWNTs forms cubic water nanotubes.19 In Figure 1 we report one simulation snapshot for (8:8) SWNTs filled with water molecules. The snapshot shown in Figure 1 corresponds to the final configuration of our GCMC simulations.19 The (8:8) SWNT in Figure 1 is 3.7 nm in length. We conduct here molecular dynamics simulations in SWNTs that are obtained by replicating four times along the axis the tubes shown in Figure 1. Two hundred eight water molecules are necessary to fill the resulting (8:8) nanotube, 14.8 nm long, at 298 K. The mechanism of diffusion is reflected on the scaling behavior of the msqd as a function of time. We report in Figure 2 the msqd along the pore axis for water in (8:8) SWNTs for up to 500 ps of trajectory. The results seem to indicate that the diffusion of water within the nanotubes follows a ballistic-type mechanism for several hundreds of Nano Lett., Vol. 6, No. 4, 2006

Figure 2. Mean square displacement along the pore axis, dz2, as a function of time for water molecules confined in (8:8) SWNTs (red). Simulation temperature is T ) 298 ( 20 K. The nanotubes are completely filled with water molecules (208 molecules). The three black lines represent the scaling behavior expected for diffusion mechanisms of the ballistic (dotted), Fickian (continuous), and single-file types (dashed).

picoseconds, and it becomes even faster toward the end of the 500 ps of trajectory considered. This behavior is representative of a collective motion for the confined water molecules, which is corroborated by the visual observation of the animations of sequences of simulation snapshots available as Supporting Information.39 The concerted motion of confined water molecules has been observed by others,4,28,32 but single-file diffusion was proposed to explain the anomalous transport of water molecules confined in short nanotubes. Interestingly, we find no evidence of single-file diffusion even though the animations and the simulation snapshots show that water molecules do not pass each other during the whole duration of the simulation when the (8:8) SWNTs are filled with water. This unexpected result is probably a consequence of the length of the nanotubes considered in our calculations. The transport mechanism of water through short carbon nanotubes as those considered in the literature28,30,33 is dictated by pore-entrance effects, whereas pore-entrance effects are not accounted for in our calculations. It is worth noting that when a real membrane is considered in which the pores are obtained by arrays of aligned SWNTs,3,31 both pore entrance/exit effects and morphological defects in the tubes will influence the transmembrane flux. To evaluate the importance of pore entrance/ exit effects by molecular simulation, it is possible to employ nonequilibrium methods, provided that thin membranes are considered (up to 10-100 nm in thickness),40-42 or the local equilibrium flux method, when thicker membranes are of interest.43 With this latter method it has been shown that pore entrance/exit effects become negligible when methane diffuses through zeolites thicker than 100 nm. The experimental conditions that we intend to mimic with our simulations are characterized by membranes of thickness of the order of 5-10 µm; thus we expect pore entrance/exit effects to be minimal.44,45 The verification of this hypothesis is deferred to future work. Another possible source of discrepancy between our results and the interpretations proposed in the literature consists of the number of water molecules adsorbed within the SWNTs. Nano Lett., Vol. 6, No. 4, 2006

Figure 3. Mean square displacement along the pore axis for water in (8:8) SWNTs. Simulation temperature is T ) 298 ( 20 K. The blue, red, green, turquoise, and violet lines are for 10, 28, 58, 108, and 208 adsorbed water molecules, respectively. The three black lines represent the scaling behavior expected for diffusion mechanisms of the ballistic (dotted), Fickian (continuous), and singlefile types (dashed).

Several previous studies reported fast diffusion of clusters of water molecules through short empty nanotubes.4,28 In the results shown in Figure 2, the nanotubes are filled, and it is expected that the diffusion is slower in filled nanotubes compared to that observed in carbon nanotubes partially filled with water. In what follows we study the effect of changing the number of water molecules adsorbed within the SWNTs on the collective transport properties. Another source of disagreement could be ascribed to the correct interpretation of the simulation results. It is known that that when the diffusion mechanism is interpreted by fitting theoretical results to eq 1, it is of fundamental importance to consider the limit at infinite time for the mean square displacements. We report below the msqd obtained by analyzing trajectories for at least 2 ns. To understand how the transport properties of confined water vary as the degree of filling of the nanotubes changes, we reduce the number of water molecules in the nanotubes and we repeat our simulations for up to 18 ns of production time. These long simulations allow us to compute mean square displacements for trajectories of at least 2 ns, and in some cases up to 10 ns. In Figure 3 we report the msqd obtained for water in partially filled (8:8) SWNTs. For clarity we only report data for 10, 28, 58, 108, and 208 adsorbed water molecules, but we conducted simulations for several other degrees of filling. The general conclusions discussed here are valid in all cases considered. Several observations arise from the analysis of the results shown in Figure 3. As discussed in Figure 2, at short times we observe a ballistic-type diffusion mechanism for water adsorbed within (8:8) SWNTs at 298 K. This coordinated motion mechanism is observed for all degrees of water filling considered and persists for several hundreds of picoseconds. At long times (above 500 ps) the diffusion becomes slower rather than faster as erroneously suggested when the msqd is monitored for only 500 ps, as shown in Figure 2. The mechanism appears to become of the Fickian type, even though some deviations are observed, possibly due to pore635

water interactions. Similar results for the msqd as a function of time were reported for the diffusion of polymers through SWNTs,46 although the ballistic-type mechanism lasted for up to only 40 ps in that study. To emphasize the relevance of our findings, we point out that the msqd for bulk liquid water is ballistic for a few picoseconds (generally less than 10), after which the diffusion follows a Fickian mechanism. Molecular dynamics simulations of up to 100 ps are sufficient to reliably estimate the self-diffusion coefficient for bulk water.47 On stark contrast, we note that the ballistic-type diffusion mechanism observed for water confined in (8:8) SWNTs persists for up to 500 ps. It is also interesting to note that as the number of water molecules adsorbed within the (8:8) SWNTs decreases, the ballistic diffusion persists for longer times. Additionally, we note that the msqd curves in the logarithmic plot shown in Figure 3 are shifted to larger dz2 as the number of confined water molecules decreases. This indicates that the coordinated motion of confined water molecules, organized in clusters as shown later, is faster as the size of the cluster decreases. Consequently, the “ballistic diffusion coefficient” increases as the number of water molecules decreases. The last two observations suggest that as the number of water molecules decreases the concerted motion of the clusters of water molecules persists for longer times and it is faster, in agreement with visual observation of animations obtained from sequential sequences of simulation snapshots.39 For simple fluids confined in molecular sieves, for which concerted motion events are also observed,20 the diffusion coefficient for single adsorbed molecules can be slower than that for clusters of adsorbed molecules. Our results show the opposite dependence on density. The difference between the results reported here and those obtained for simple fluids in molecular sieves20 is due to the different molecular mechanisms responsible for diffusion in the two cases. The hopping events observed for simple fluids in molecular sieves are due to mismatches between pore-fluid and fluid-fluid attractive interactions, while the collective motion observed for water in SWNTs is due to the formation of hydrogen bonds between adsorbed water molecules and to the smooth and weakly attractive SWNTs interior surface, as discussed later. When simple fluids are simulated within SWNTs, concerted motion is observed for long times when a concentration gradient is applied across the pores, but not otherwise.24,48 The transport diffusion coefficient observed when a concentration gradient is applied does not depend significantly on the density of the adsorbed molecules, suggesting that the inherent SWNTs smoothness, which results in near-specular reflections of molecules when they contact the nanotube wall,25 is responsible for the fast diffusion. The results reported in Figure 3, compared to those in Figure 2, indicate that it is necessary to conduct molecular dynamics simulations for more than 2 ns when it is desired to investigate the correct diffusion mechanism for water confined within narrow SWNTs. As discussed elsewhere,19 when water fills (8:8) SWNTs it shows an ordered structure. As shown by the snapshots reported later, the order is somewhat lost when water does not completely fill the 636

Figure 4. Sequence of simulation snapshots obtained for 68 water molecules confined within (8:8) SWNTs at 298 K. The sequence illustrates the formation of one small cluster of water molecules that breaks free from the large hydrogen-bonded cluster, flows through the tube, and is adsorbed again on the larger cluster. The black arrows indicate the direction of concerted motion of the clusters of water molecules. The length of the arrows is qualitatively indicative of the speed of the clusters. The smaller cluster diffuses faster through the SWNTs than the larger cluster does.

SWNTs. The results shown in Figure 3 suggest that the molecular mechanism responsible for water diffusion does not depend substantially on the density of confined water molecules, thus we conclude that the organization of water molecules within a cluster (whether ordered or not) does not influence the mechanism of motion. To corroborate our conclusion, we analyzed the trajectories of single water molecules (results not reported for brevity). Those data always indicate a collective motion for all the water molecules that form a cluster, whether ordered or not. To appreciate the molecular mechanism of water diffusion, we report in Figure 4 a sequence of three consecutive simulation snapshots obtained when 68 water molecules are adsorbed in (8:8) SWNTs at 298 K. The sequence of simulation snapshots shows that initially one cluster of water molecules is moving along the pore axis in the direction of the black arrow. After some time, a small cluster composed of 4 molecules manages to break free from the larger cluster and moves rapidly along the pore axis in the direction opposite to that along which the larger cluster moves. Because of periodic-boundary conditions, the two clusters eventually meet and coalesce. Frequent events such as that represented in Figure 4 are responsible for burst recorded in the msqd as a function of time and render ragged the curves shown in Figure 3. Detailed analysis of the motion of each single cluster of water molecules in SWNTs is available in the literature30 and is not repeated here. Next, we investigate the physical origin of the unexpected behavior observed at time scales as long as 500 ps and discussed earlier in this paper. Hydrogen-bonded interactions are considered responsible for many peculiar properties of water, in the bulk as well as under confinement.49 Thus we ascribe the persistent ballistic diffusion mechanism observed for confined water to the establishment of long-lasting hydrogen bonds between different water molecules. According to our hypothesis, the diffusion mechanism should not be a persistent ballistic diffusion when hydrogen bonds are prevented from forming between adsorbed water molecules. To test our hypothesis, we suppress the electrostatic interactions between water molecules, which are responsible for the establishment of hydrogen bonds. By doing so we are studying a reference system of Lennard-Jones (LJ) fluid in SWNTs. Under ambient conditions for liquid water (T ) Nano Lett., Vol. 6, No. 4, 2006

Figure 6. Representative equilibrium simulation snapshots for water confined in (8:8) SWNTs at 298 K. From top to bottom the snapshots are for 148, 108, 68, and 28 water molecules confined within the tube. The bottom panel is for the configuration for 10 reference LJ molecules in (8:8) SWNTs at 298 K.

Figure 5. Mean square displacement along the pore axis as a function of time for the reference LJ fluid in (8:8) SWNTs. Simulation temperature is T ) 73 ( 5 K (top) and 43 ( 5 K (bottom). The blue, red, green, turquoise, and violet lines are for 10, 18, 38, 108, and 208 adsorbed LJ molecules, respectively. The three black lines represent the scaling behavior expected for diffusion mechanisms of the ballistic (dotted), Fickian (continuous), and single-file types (dashed).

298 K), the reference LJ fluid is at supercritical conditions (T* ) kBT/ ) 3.84).50 To compare the behavior of confined water to that of the reference LJ fluid, it is necessary to lower the temperature of the system. We conduct molecular dynamics simulations for the reference LJ fluid at 78 and at 43 K. Additional simulations were conducted at 298 K but are not discussed here for brevity. In Figure 5 we contrast the msqd obtained for the LJ fluid at the two temperatures. As in the case of water (see Figure 3), we consider different degrees of filling for the reference LJ fluid within the (8:8) SWNTs. Our results indicate that the diffusion mechanism for the reference LJ fluid is significantly different compared to that of confined water. As expected, both at 78 and at 43 K the msqd at short times is indicative of ballistic-type diffusion, in which the confined reference LJ fluid molecules move in a coordinated fashion. This indicates that the high degree of confinement provided by (8:8) SWNTs forces the reference LJ molecules to move in a highly coordinated fashion. However, we note that the ballistic-type diffusion mechanism persists for shorter times compared to the results obtained for water (see Figure 3). In particular, when T ) 43 K, the ballistic diffusion mechanism persists for less than 200 ps. As the temperature increases to 78 K, the ballistic diffusion persists for less than 100 ps because the thermal motion becomes more important and the clusters of LJ molecules form and break more often compared to the results obtained at 43 K. After the short lag time during which the Nano Lett., Vol. 6, No. 4, 2006

msqd scales linearly with the square of time, the analysis of the trajectories for the confined reference LJ fluid indicates that the msqd scales linearly with time, indicating that the Fickian-type diffusion is established. As observed for confined water, we note that the diffusion becomes faster as the number of LJ adsorbed molecules decreases and also that the diffusion (both the ballistic-type and the Fickian-type) is faster at larger temperatures, as expected. This result differs from data obtained for the diffusion of simple fluids through molecular sieves in which the self-diffusion of a cluster of adsorbed molecules can be faster than that of single adsorbed fluid molecules20 but is in qualitative agreement with simulation results obtained for simple fluids in SWNTs.51,52 As discussed earlier, the ragged surface of molecular sieves is responsible for the peculiar behavior observed for confined simple fluids, while SWNTs are characterized by smooth surfaces. The results in Figure 5 also provide no evidence for the single-file diffusion observed for LJ fluids in reconstructed models of porous carbons with diameters of approximately 0.4 nm.53 As indicated by visual analysis of sequences of simulation snapshots, (8:8) SWNTs are large enough to allow the reference LJ molecules to pass each other chaotically in the direction of motion, thus preventing the establishment of single-file diffusion. To visualize the reasons beyond the different behavior just discussed, we show in Figure 6 equilibrium simulation snapshots of (8:8) SWNTs partially filled with 148, 108, 68, and 28 water molecules or with 10 reference LJ molecules at 298 K. The only difference between water and reference LJ molecules lays in the electrostatic interactions that are turned off to simulate the reference LJ fluid. It is clear from visual observation of the snapshots reported in Figure 6 that electrostatic interactions are responsible for the formation of clusters of water molecules within the SWNTs. The concerted motion of these clusters determines the ballistictype diffusion mechanism evident from the analysis of msqd for up to 500 ps (see Figures 2 and 3). When we suppress the electrostatic interactions, hydrogen bonds cannot form between different molecules and the reference LJ fluid occupies all the available volume, although few clusters of LJ fluids are observed at 43 and 78 K. Consequently, the diffusion mechanism for the reference LJ fluid is drastically different compared to that of water, as discussed in Figure 5. 637

In conclusion, we have demonstrated that the transport of water molecules through an infinitely long narrow carbon nanotube occurs via a coordinated motion, independently on the degree of filling of the nanotubes, for up to 500 ps. As a consequence, the mean square displacement follows a ballistic-type dependence on time for up to several hundreds of picoseconds. Concerted motion for highly confined fluids were observed previously for LJ fluids in molecular sieves, for polymers in SWNTs, and for LJ fluids in SWNTs provided that a concentration gradient is applied across the pore. As opposed to the results reported in those situations, the ballistic motion reported here lasts for exceptionally long times (up to 500 ps) and it is due to a molecular mechanism that results from long-lasting hydrogen bonds between adsorbed water molecules, weak carbon-water attractive interactions, and smooth SWNTs surfaces. When the mean square displacement is analyzed beyond 1 ns, the simulation results indicate that the diffusion mechanism changes and becomes Fickian in all cases considered here. Consequently production times of tens of nanoseconds are required to reliably estimate the selfdiffusion coefficient for water confined in narrow carbon nanotubes. Surprisingly we find no evidence of single-file diffusion in any of the situations considered, although we expect that the transport properties will change when water-substrate interactions become more attractive, as for example could happen for water confined in metal oxide nanotubes. The results reported here are crucial for achieving the complete understanding of the properties of confined water that is undoubtedly necessary for the design of novel nanomachinery such as nanosyringes or carbon-nanotubeembedded membranes for the controlled delivery of nanometer quantities of aqueous solutions. Acknowledgment. This work was partially supported by the Vice President for Research at the University of Oklahoma, Norman, and by the U.S. Department of Energy (Grant Number DE-FG0298ER14847). We acknowledge generous allocations of computer time at the supercomputer facilities of Oak Ridge National Laboratory, those of NERSC, Berkeley, and those of NPACI, San Diego. The author is indebted to Keith E. Gubbins, Peter T. Cummings, Ariel A. Chialvo, and Jorge Pikunic who inspired the work presented here with interesting discussions and continuous encouragement. Appendix. Simulation Details. The nanotubes considered here are of the armchair type and they are described by Lennard-Jones carbon atoms maintained fixed during the simulations. The self-diffusion coefficients predicted for simple fluids through rigid SWNTs can be faster than those predicted through flexible SWNTs.54 This effect is important at low loadings, and it becomes less important as the density of confined fluids increases.48 For the scope of the present work, which is to investigate the mechanism of water diffusion within narrow SWNTs, it is reasonable to consider rigid SWNTs for economy of computing time because carbon atom vibrations in SWNTs are not expected to affect the 638

tendency of adsorbed water molecules to form hydrogenbonded clusters. The equations of motion are integrated within the microcanonical (NVE) ensemble in which the number of particles in the system (N), the volume of the simulation box (V), and the total energy of the system (E) are maintained constant. The software package DL-POLY,55 version 2.14, is used to carry out the simulations. The integration time is 0.5 fs to ensure constant total energy and temperature throughout the simulations. We never observe temperature fluctuations in excess of 20 K from the set point for the simulations at 298 K and in excess of 5 K from the set point for the simulations at 78 and 43 K. The simulations are conducted for up to 18 ns. To ensure equilibration, the first 200-500 ps of simulation are discarded. During the equilibration we implement a temperature rescaling algorithm every 1000 integration steps. During the production run the temperature is allowed to fluctuate and the configuration of the system is stored every 5.0 ps. The sequence of configurations is used subsequently to calculate the mean square displacement (msqd) as a function of time. We only consider the msqd along the pore axis. To compute the msqd we use 15 origins, separated by 500 ps, and we compute averages over all molecules in the system. The msqd analysis is performed over at least 2 ns of data, but we often extend the analysis to 10 ns. To assess the reliability of our results, we repeat selected simulation runs (e.g., those shown in Figures 2 and 3) with different initial conditions (e.g., positions and velocities of confined water molecules). No significant difference was noticed on the conclusions here reported. Water molecules are described via the SPC/E model.56 The geometry of each water molecule is maintained rigid by using the SHAKE algorithm. Water-carbon interactions are modeled according to Lennard-Jones potentials as described in our earlier publications.13,19 Water-water interactions include one Lennard-Jones term to account for dispersive interactions that are augmented by electrostatic interactions to account for the formation of hydrogen bonds. The electrostatic interactions are treated according to the Ewald summation formalism. To simulate the reference Lennard-Jones fluid confined in the SWNTs, we suppress the electrostatic interactions and fluid-fluid interactions are described only through the dispersive part of the SPC/E model. The vaporliquid critical temperature for the reference LJ fluid is lower than that for SPC/E water. For Lennard-Jones fluids with cutoff at 2.5 times the diameter of the molecule, the reduced vapor-liquid critical temperature (TC* ) kTC/, where TC* is the reduced critical temperature, k is the Boltzmann constant, and  is the usual Lennard-Jones energy parameter) is estimated to be 1.176.57,58 Because the vapor-liquid critical temperature for SPC/E water is approximately 635 K,59 to conduct our simulations for the confined fluid at thermodynamic conditions similar (in terms of reduced temperature) to those at which we conduct the simulation for SPC/E water at 298 K, we need to lower the temperature to approximately 43 K. For the reference LJ fluid we conduct molecular dynamics simulations at 43, 78, and 298 K. The reference Nano Lett., Vol. 6, No. 4, 2006

LJ fluid-carbon interactions are described in the same manner as we describe water-carbon interactions. The initial configurations for the carbon nanotubes filled with water molecules are the final configurations obtained in our previous GCMC simulations.19 Because of the tendency of confined water molecules to form hydrogenbonded clusters, the results for the msqd at low loadings may be affected by the length of the simulation box. To ensure that finite-size effects are not affecting our results, we replicate the GCMC configurations four times along the axis of the nanotubes. Consequently the length of the (8:8) SWNTs is 14.8 nm. The diameter of the (8:8) SWNTs, defined as the distance between the centers of the carbon atoms across the diameter, is 1.08 nm. To apply the threedimensional version of the Ewald summation algorithm, the nanotubes are placed along the Z axis in orthorhombic simulation boxes of size 4.0 nm in the X and Y directions. The size of the simulation box in the Z direction is equal to the length of the SWNTs. Periodic boundary conditions are implemented in the three directions. Supporting Information Available: Video clips of collective motion of confined water particles. This material is available free of charge via the Internet at http:// pubs.acs.org. References (1) Iijima, S. Nature 1991, 354, 56. (2) Lopez, C. F.; Nielsen, S. O.; Moore, P. B.; Klein, M. L. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 4431. (3) Holt, J. K.; Noy, A.; Huser, T.; Eaglesham, D.; Bakajin, O. Nano Lett. 2004, 4, 2245. (4) Beckstein, O.; Biggin, P. C.; Sansom, M. S. P. J. Phys. Chem. B 2001, 105, 12902. (5) Yang, L.; Harroun, T. A.; Weiss, T. M.; Ding, L.; Huang, H. W. Biophys. J. 2001, 81, 1475. (6) Berne`che, S.; Morais-Cabrl, J. H.; Kaufman, A.; MacKinnon, R. Nature 2001, 414, 43. (7) de Groot, B. L.; Grubmu¨ller, H. Science 2001, 294, 2353. (8) Tajkhorshid, E.; Nollert, P.; Jensen, M. O.; Miercke, L. J.; O’Connell, J.; Stroud, R. M.; Schulten, K. Science 2002, 296, 525. (9) Yarin, A. L.; Yazicioglu, A. G.; Megaridis, C. M.; Rossi, M. P.; Gogotsi, Y. J. Appl. Phys. 2005, 97, 124309. (10) Brovchenko, I.; Geiger, A.; Oleinikova, A. Phys. Chem. Chem. Phys. 2001, 3, 1567. (11) Giaya, A.; Thompson, R. W. J. Chem. Phys. 2002, 117, 3464. (12) Noon, W. H.; Ausman, K. D.; Smalley, R. E.; Na, J. Chem. Phys. Lett. 2002, 355, 445. (13) Striolo, A.; Gubbins, K. E.; Chialvo, A. A.; Cummings, P. T. Mol. Phys. 2004, 102, 243. (14) Striolo, A.; Gubbins, K. E.; Burchell, T. D.; Simonson, J. M.; Cole, D. R.; Gruszkiewicz, M. S.; Chialvo, A. A.; Cummings, P. T. Langmuir 2005, 21, 9457. (15) Striolo, A.; Chialvo, A. A.; Gubbins, K. E.; Cummings, P. T. J. Chem. Phys. 2006, 124, 074710. (16) Koga, K.; Gao, G. T.; Tanaka, H.; Zeng, X. C. Nature 2001, 412, 802. (17) Koga, K.; Parra, R. D.; Tanaka, H.; Zeng, X. C. J. Chem. Phys. 2000, 113, 5037.

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