Report

# Fast Mass Transport Through Sub-2-Nanometer Carbon Nanotubes

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Science  19 May 2006:
Vol. 312, Issue 5776, pp. 1034-1037
DOI: 10.1126/science.1126298

## Abstract

We report gas and water flow measurements through microfabricated membranes in which aligned carbon nanotubes with diameters of less than 2 nanometers serve as pores. The measured gas flow exceeds predictions of the Knudsen diffusion model by more than an order of magnitude. The measured water flow exceeds values calculated from continuum hydrodynamics models by more than three orders of magnitude and is comparable to flow rates extrapolated from molecular dynamics simulations. The gas and water permeabilities of these nanotube-based membranes are several orders of magnitude higher than those of commercial polycarbonate membranes, despite having pore sizes an order of magnitude smaller. These membranes enable fundamental studies of mass transport in confined environments, as well as more energy-efficient nanoscale filtration.

Carbon nanotubes, with diameters in the nanometer range and atomically smooth surfaces, offer a unique system for studying molecular transport and nanofluidics. Although the idea that water can occupy such confined hydrophobic channels is somewhat counter-intuitive, experimental evidence has confirmed that water can indeed occupy these channels (1, 2). Water transport through molecular-scale hydrophobic channels is also important because of the similarity of this system to transmembrane protein pores such as aquaporins (3). In recent years, numerous simulations (4, 5) of water transport through single-walled carbon nanotubes (SWNTs) have suggested not only that water occupies these channels, but also that fast molecular transport takes place, far in excess of what continuum hydrodynamic theories would predict if applied on this length scale. Molecular dynamics (MD) simulations attribute this enhancement to the atomic smoothness of the nanotube surface and to molecular ordering phenomena that may occur on confined length scales in the 1- to 2-nm range (4, 5). For similar reasons, simulations of gas transport through SWNTs (6) predict flux enhancements of several orders of magnitude relative to other similarly sized nanoporous materials. Membrane-based gas separations, such as those using zeolites (7), provide precise separation and size exclusion, although often at the expense of throughput or flux. It may be possible to use SWNTs to create a membrane that offers both high selectivity and high flux.

To investigate molecular transport on this length scale, we need to fabricate a carbon nanotube membrane that has a pore size of 1 to 2 nm. Researchers have recently fabricated multiwalled carbon nanotube (MWNT) membranes with larger pore diameters (6 to 7 nm) by encapsulation of vertically aligned arrays of MWNTs (8, 9) and by templated growth within nanochannel alumina (10). Enhanced water transport through these larger MWNTs has recently been reported (11). Quantifying transport through an individual tube in a MWNT membrane is difficult, however, because MWNTs are prone to blockages, in particular by “bamboo” structures and catalyst particles that can migrate to and obstruct the nanotube interior (9, 12, 13). The consequence of such blockages is a marked reduction of the active membrane pore density. In contrast, there are few, if any, reports of “bamboo” structure formation or catalyst migration for SWNTs or double-walled carbon nanotubes (DWNTs). However, it is difficult to produce vertically aligned carbon nanotubes of this size (14, 15). The major challenges also lie in finding a conformal deposition process to fill the gaps in this nanotube array, as well as in designing a selective etching process to open up the nanotube channels without producing voids in the membrane.

We have developed a microelectromechanical systems (MEMS)–compatible fabrication process (Fig. 1A) for sub–2-nm nanotube pore membranes. The process uses catalytic chemical vapor deposition to grow a dense, vertically aligned array of DWNTs on the surface of a silicon chip (Fig. 1B), followed by conformal encapsulation of the nanotubes by a hard, low-pressure chemical vapor–deposited silicon nitride (Si3N4) matrix (Fig. 1C). Scanning electron microscopy (SEM) images show that our process produces gap-free membranes over the length scale of the whole chip. The excess silicon nitride is removed from both sides of the membrane by ion milling, and the ends of the nanotubes are opened up with reactive ion etching. The membranes remain impermeable to both liquids and gases until the very last etching step; this is further evidence that our fabrication process produces crack-free and void-free membranes. Transmission electron microscopy (TEM) plan-view images (Fig. 2, C to E) of a slice of the membrane also demonstrate that the silicon nitride coats the DWNTs conformally and does not leave any gaps between the outer surface of the nanotube and the silicon nitride. We also used the same nitride-encapsulation method to produce MWNT membranes (16).

To characterize the membrane pore size, we performed size exclusion measurements (Table 1) (16) and compared these results with electron microscopy (EM) data. DWNT membranes passed species with sizes up to 1.3 nm yet blocked 2-nm gold particles, which suggests that these membranes have pore sizes between 1.3 and 2 nm. Comparison of the water flow rates before and during filtration, coupled with the upper-limit estimate of the flux of 2-nm gold particles, suggests that less than 0.1% of the flux through the membrane can be attributed to pores larger than 2 nm (16). These size exclusion measurements are further supported by the evidence obtained through EM. The distribution of DWNTs, as measured by TEM, revealed an inner diameter average of 1.6 nm (Fig. 2B). TEM images (Fig. 2C) of the membrane also revealed that the only holes that span the thickness of the membrane are of a size that is consistent with the inner diameter distribution of as-grown DWNTs. A MWNT membrane, used as a reference, transported colloidal gold particles with diameters of 2 and 5 nm but excluded the 10-nm colloidal gold particles, in agreement with MWNT diameters of 6.5 ± 0.5 nm estimated by TEM. This result also suggests that pore clogging by particles smaller than the average pore size is unlikely for the solution concentrations used in these experiments. We conclude that the transport in our samples occurs exclusively through the inner pores of the carbon nanotubes spanning the membrane.

Table 1.

Size exclusion tests on DWNT and MWNT membranes and molecular fluxes (per unit membrane area) of analytes. Values denoted by “<” were derived from the limits of detection for our concentration measurements when we did not observe any Au particles in the permeate solution. Differences of three to four orders of magnitude between this limiting value and the flux of the next smallest species indicate that the given analyte did not pass through the membrane.

AnalyteAnalyte size (nm)DWNT membrane flux (molecules cm-2 s-1)MWNT membrane flux (molecules cm-2 s-1)
Ru2+(bipyr)3 1.3 5 × 1013 5 × 1013
Colloidal Au 1 2 ± 0.4 <2 × 109 1 × 1011
Colloidal Au 2 5 ± 0.75 <3 × 108 3 × 1010
Colloidal Au 3 10 ± 1 Not tested <4 × 107

The absolute gas flux through our membranes exceeded the flux predicted by the Knudsen diffusion model. As the dimensions of the pore shrink, the mean free path (λ) becomes larger than the channel dimensions (d) and the transport enters the molecular flow regime. In such situations, where particle-surface collisions dominate over particle-particle collisions, the Knudsen diffusion model (17) is frequently applied. Indeed, our pore geometries are characterized by Knudsen numbers (λ/d) of 10 to 70, which places them well into the free molecular transport regime. However, the flux measured through our membranes exceeded the flux predicted by the Knudsen model (18) by at least one to two orders of magnitude (Table 2). By comparison, a polycarbonate membrane (Nucleopore, Osmonics Inc.) revealed just a slight enhancement in flux. The single largest uncertainty in quantifying the flux through our membrane pores lies in determination of the active pore density (i.e., those nanotubes that are open and spanning the membrane). A pore density estimate of 2.5 × 1011 cm–2 was derived from the plan-view TEM images of the DWNT membrane (Fig. 2, C to E), and the enhancement factors that we report represent lower boundary estimates (19). The observed flow enhancement is most likely caused by the intrinsic smoothness of the nanotube surface, as predicted by MD simulations of gas flow through SWNTs (6, 2022). In atomically smooth pores, the nature of gas-wall collisions can change from purely diffuse (as in the Knudsen model) to a combination of specular and diffuse collisions (23), thus leading to observed faster transport.

Table 2.

Comparisons of experimental air flow rates observed for several DWNT membranes with Knudsen model predictions, and of experimental water flow rates with continuum flow model predictions. The differences among the three DWNT membranes are most likely the result of different numbers of pores opened in the fabrication process. Values for a polycarbonate membrane are provided as a reference. Pore diameters were determined from size exclusion measurements, TEM measurements, and (for polycarbonate) manufacturer's specifications. Pore density values are upper limits, as determined from TEM measurements and (for polycarbonate) manufacturer's specifications.

MembranePore diameter (nm)Pore density (cm-2)Thickness (μm)Enhancement over Knudsen modelView inline(minimum)Enhancement over no-slip, hydrodynamic flowView inline(minimum)Calculated minimum slip lengthView inline(nm)
DWNT 1 1.3 to 2.0 ≤0.25 × 1012 2.0 40 to 120 1500 to 8400 380 to 1400
DWNT 2 1.3 to 2.0 ≤0.25 × 1012 3.0 20 to 80 680 to 3800 170 to 600
DWNT 3 1.3 to 2.0 ≤0.25 × 1012 2.8 16 to 60 560 to 3100 140 to 500
Polycarbonate 15 6 × 108 6.0 2.1 3.7 5.1
• View inline* From (18).

• View inline From (26).

• View inline From (29).

• We found that single-component selectivity for most of the gases exhibited the expected inverse-square-root scaling of molecular mass (Fig. 3, inset) with the exception of hydrocarbons, whose selectivities were higher. This result is not surprising for a molecular diffusion process because it reflects the mass dependence of molecular velocity (note that the nature of wall collisions would not affect the mass scaling). Of all the measured gases, only the hydrocarbons deviated from this pattern, exhibiting a higher selectivity (Fig. 3) in both DWNT and MWNT membranes. Interestingly, a reference polycarbonate membrane with a pore size of 15 nm did not show this deviation. We attribute the deviation to the preferential interaction of hydrocarbons with the carbon nanotube sidewalls. The hydrocarbon transport enhancement most likely results from surface diffusion or possibly a solubility/diffusion mechanism (24). Pulse mass analysis of various organic compounds has shown strong adsorption of hydrocarbon molecules (e.g., hexane) on SWNTs relative to more polar molecules (e.g., ethanol) (25). It is noteworthy that the hydrocarbon selectivity we observe in these single-component experiments may be more pronounced for practical gas separation problems where mixtures are involved (20).

Our membranes also transported water across the carbon nanotube channels at rates that cannot be accounted for by continuum flow models. The measured water flow rates reveal a flow enhancement (Table 2) that is more than three orders of magnitude faster than the no-slip, hydrodynamic flow as calculated from the Hagen-Poiseuille equation (26). Breakdown of this continuum model is not surprising for channels 1 to 2 nm in size. If we take the formalism used for gases and define a mean free path in liquids as the molecular diameter (e.g., ∼0.3 nm for H2O), the Knudsen number for a 1- to 2-nm pore is 0.15 to 0.3. These values lie on the border between “slip flow” and “transitional flow.” In this size regime, where the pore is only ∼7 water molecules in diameter, continuum theory concepts such as a velocity profile may be difficult to define. For this reason, MD simulations are often used for the prediction of water flows through carbon nanotube pores with diameters on the order of 1 nm (4, 5). However, the computational expense of MD simulations, as well as observations of a finite fluid “slipping velocity” at hydrophobic interfaces (27), have motivated attempts to use meso- and macroscopic flow models to simulate flow through SWNTs (28). These simulations calculated a corresponding “slip length” that describes the noncontinuum behavior of a liquid near the pore walls. If we apply a similar formalism for the flow through our sub–2-nm nanotube membranes, we calculate (29) slip lengths as large as 1400 nm (Table 2). These values are almost three orders of magnitude larger than the pore size and are on the order of the overall size of the system (pore length). In contrast, the polycarbonate membrane with a pore size of 15 nm reveals a much smaller slip length of just 5 nm. This comparison suggests that slip-flow formalism may not be applicable to water flow through sub–2-nm carbon nanotubes, possibly because of length scale confinement (30) or partial wetting of the carbon nanotube surface (31).

Our observed water flux compares well with that predicted by the MD simulations (5). The simulations predict a flux of 12 water molecules through 1 nm2 of the nanotube cross-sectional area in 1 ns; our measured flux, extrapolated to the simulation pressure drop, corresponds to 10 to 40 water molecules nm–2 ns–1 (32). The MD simulations attributed the observed high water flow rates to the formation of water “wires” in the confined space inside the nanotube. The strong dependence of the structure of the water in the nanotube on diameter (33) indicates that small differences in nanotube diameter can have large effects on transport. Therefore, it is unclear whether the mechanism proposed by MD is responsible for the high water flow rates observed with the larger nanotubes used in our experiments, or whether the flow enhancement can be attributed simply to the presence of a nearly frictionless surface.

Membrane permeability provides a figure of merit for membrane performance for practical applications. Despite having an order of magnitude smaller pore size, the enhanced flow rate per pore and the higher pore density render the sub–2-nm membranes superior to conventional polycarbonate membranes in both air and water permeability (34) (Fig. 4).

Supporting Online Material

Materials and Methods

Fig. S1

## References and Notes

1. For gas flow in the Knudsen regime, the overall membrane flow rate can be determined from $Math$ where M is molecular weight, R is the universal gas constant, T is absolute temperature, d is pore diameter, Vm is the molar volume, Δp is the pressure drop, L is the thickness of the membrane, σ is the areal pore density, and A = 89·π(25 μm)2 = 175,000 μm2 is the total area of the membrane.
2. The Hagen-Poiseuille equation is $Math$ where QHP is the volumetric flow rate, Δp is the pressure drop, d is the pore diameter, μ is the water viscosity, and L is the membrane thickness.
3. With the inclusion of a slip-flow correction, the Hagen-Poiseuille equation becomes $Math$ where Ls is defined as the slip length, $Math$ where Uwall is the axial velocity at the wall, and dU/dr is the radial velocity gradient at the wall (or shear rate).
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