Coherence Resonance in a Single-Walled Carbon Nanotube Ion Channel

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Science  10 Sep 2010:
Vol. 329, Issue 5997, pp. 1320-1324
DOI: 10.1126/science.1193383

Oscillations in Carbon Nanotube Conductivity

Theoretical studies have suggested that protons can be conducted rapidly in water trapped inside carbon nanotubes. C. Y. Lee et al. (p. 1320) connected two aqueous reservoirs with opened, single-walled carbon nanotubes, half a millimeter long but only 1.5 nanometers wide, and observed a high, stable proton current under electroosmotic conditions arising from a single nanotube. The addition of alkali cations caused random pore blocking and oscillations in ion current, resembling events seen in biological ion channels.


Biological ion channels are able to generate coherent and oscillatory signals from intrinsically noisy and stochastic components for ultrasensitive discrimination with the use of stochastic resonance, a concept not yet demonstrated in human-made analogs. We show that a single-walled carbon nanotube demonstrates oscillations in electroosmotic current through its interior at specific ranges of electric field that are the signatures of coherence resonance. Stochastic pore blocking is observed when individual cations partition into the nanotube obstructing an otherwise stable proton current. The observed oscillations occur because of coupling between pore blocking and a proton-diffusion limitation at the pore mouth. The result illustrates how simple ionic transport can generate coherent waveforms within an inherently noisy environment and points to new types of nanoreactors, sensors, and nanofluidic channels based on this platform.

Stochastic resonance (1, 2) as a detection mechanism occurs when a subthreshold signal is amplified instead of obscured by environmental noise, allowing for discrimination despite stochastic interference. There are many examples of this in biological systems, including crayfish detecting incident pressure waves from predators (3), plankton detection by paddlefish (4), the cercal sensory system of crickets (5), and human visual perception (6) and balance control (7). The distinguishing attribute of these systems is that the signal-to-noise ratio of the sensory output increases to an optimum value with increasing noise, the signature of stochastic resonance. In 1995, Bezrukov and Vodyanoy (8) reported that a collection of alamethicin ion channels constituted the simplest experimental system capable of stochastic signal enhancement. It still remains an open experimental question whether a single, static ion channel alone can demonstrate this important function (9), or if stochastic resonance can be reliably incorporated into human-made devices. A synthetic ion channel with no complexity associated with multiple proteins and their conformational states may help to resolve this question. A nanopore formed from the interior of a single-walled carbon nanotube (SWNT) has captivated the interest of many theorists, who have predicted enhanced water permeation (10), large proton fluxes (11), icelike water phases (12), and high ion-rejection rates (13). However, it has proven difficult to experimentally observe the translocation of individual small ions, such as Na+ and K+, from a single isolated nanotube, despite success with macroscopic ensembles forming various membranes for gas and liquid separations (14, 15). We studied transport of single ions through the interior of a single 500-μm-long carbon nanotube and observed a proton flux of ~108 protons/s. This nanopore system can show oscillations in the electroosmotic current at constant bias, resulting in rhythmic, mode-locked frequencies of ion transport that arise from coherence resonance, a variant of stochastic resonance characterized by self-synchronization at an optimal noise level. This resonant transport substantially increases the throughput of a nanopore by a factor of 100. Liu et al. (16) recently designed a SWNT nanopore for DNA translocation using a much shorter length (2 μm) and higher electric field (~105 V/m), but they did not observe stochastic pore blocking, particularly from ions. Our results show that simple, single-ion transport from an isolated nanopore is all that is needed to produce a stochastic resonant mechanism, and they also open the door to new types of chemical nanoreactors, nanofluidic conduits, and single-molecule sensors.

We created the SWNT ion channels (diameter 1.3 to 2.3 nm, average 1.5 nm) with the use of an epoxy structure (Fig. 1) with two compartments bonded onto a Si/SiO2 wafer containing an array of chemical vapor deposition (CVD)–synthesized, ultralong aligned SWNTs (17). After filling with various aqueous ionic solutions (NaCl, LiCl, or KCl), we monitored the ion current with Ag/AgCl electrodes in either well and standard voltage-clamp techniques (18). We measured current traces for various applied electric fields (600 to 6000 V/m) across reservoirs connected by up to 45 nanotubes that are 500 μm long (Fig. 2A). We observed stochastic fluctuations in the ion current that are quantized, similar to observations of biological ion channels (19) and solid-state nanopores (20) as signatures of stochastic pore blocking.

Fig. 1

Fabrication of SWNT ion channels. An epoxy structure with two compartments is bonded onto a substrate with ultralong and aligned CVD-grown nanotubes (scanning electron microscope image). Plasma etching removes the SWNT at the reservoir bottoms, opening both ends of nanotubes that span the divider between compartments. Both compartments are filled with ionic solution, and the electroosmotic current through the nanotubes is monitored. The epoxy structure serves as a mask during plasma etching and as an efficient inhibitor of exterior transport along the nanotube (figs. S4 to S6). UV, ultraviolet.

Fig. 2

Detection of individual cation transport through the interior of a single nanotube. (A) Stochastic blockage and recovery of the proton flux through the SWNT interior is observed at varied electric fields and with different ion types. Several control experiments reveal that cations in NaCl, LiCl, and KCl are the pore blockers. The top trace (1 M NaCl, 300 mV) shows three distinct pore states as indicated in three peaks in an all-point histogram (right), suggesting that the blocking events are from two nanotubes. Careful adjustment of the bias results in transitions observed between only two states (second trace and all others) and allows for the study of the single-tube phenomenon. (B) Histograms of the conductance decrease upon pore blocking, ΔG (left), and ion mobility (right) for the tested ions. Proton conductivity, estimated from ΔG = 200 pS and assuming a tube diameter of 1.5 nm, is 5 × 102 S/cm. The ion mobility is 100 times higher than the value in bulk water. (C) The conductance change with blocking, ΔG, decreases with ion mobility, as expected. (Inset) Small mobility region is zoomed in for NaCl.

The step height, or the current decrease (ΔI), is the amount of reduced flux of charge-carrying species upon pore blocking. The velocity and, thus, the mobility of the blocker (in case the blocker translocates through the nanotube) can be estimated from the width of the downward peaks, or the dwell time, τdwell. Each peak in an all-point histogram (Fig. 2A, right) corresponds to a single pore state. The presence of three major states in the top trace in Fig. 2A suggests that the events are from two nanotubes. We note that there is a threshold voltage that must be exceeded for pore-blocking events to be observed. For each sample, there is an optimal voltage to observe blocking events, and we tune to this region with each experiment. By carefully adjusting the bias to the low field threshold of this region, we could regularly observe current fluctuations between only two states with conserved ΔI (remaining Fig. 2A traces), indicative of the single-tube phenomenon (21). The inactive nanotubes in the system are either permanently blocked by impurities or remain below the requisite field to clear low-mobility blocking species within the observation window of the experiment. In this way, transport across a single nanotube can be studied, as indicated by the observation of a two-state Coulter effect. This two-state stochastic blocking phenomenon is evidence of single-molecule transport in a singular, isolated channel across a diverse range of fields: from patch clamp experiments on biological ion channels (19) to synthetic nanopores (20).

A series of control and comparison experiments reveal that the dominant ion conductor consists of protons, with cations from the added electrolyte as the dominant blocking species. First, the unblocked current levels decrease with an increase in salt concentration (fig. S7), which is consistent with previous reports of highly unfavorable K+ or Cl partitioning into the interior of an uncharged nanotube (13, 15). Hence, only H+ and/or OH may be the majority charge carriers. Several theoretical studies indicate that proton conduction along the hydrogen-bonded water network inside the SWNT should be faster than in bulk water (11). The proton conduction can occur via a “hop-and-turn” Grotthuss mechanism along the water chain (22) and does not require physical translocation of protons. The current levels are enhanced by the addition of HCl (1 M) and only upon addition to the anode (+) (fig. S8). Higher conductance change upon pore blocking (ΔG) at acidic pH further supports our claim (fig. S9). Conversely, Fig. 2 shows that the blocking events change markedly with the type and concentration of added electrolyte, with the cation appearing to be the dominant blocker. We fed cations of successively larger hydration radius (with the anion unchanged as Cl) to the nanotube ion channel and found that 1 M tetramethylammonium (TMA) chloride [crystallographic diameter of TMA+ ~ 0.7 nm (23)] yielded no blocking events (fig. S10), even though Cl should be small enough to translocate. The etched ends of nanotubes from the oxygen plasma bear carboxylic acid groups with pKa of ~4.5 (where Ka is the acid dissociation constant) (24). The repulsion of anions is consistent with these observations. Impurities and other artifacts are also readily ruled out (2528).

The SWNT proton channels demonstrate unusually high conductivity compared with sputtered Si ion channels (29), which are typically ~104 times less conductive for KCl, even at larger diameters. The difference highlights the contribution of the hydrogen-bonded water network (12) in the former that contributes substantially to proton conduction. The typical current blocking of 100 pA corresponds to 6.25 × 108 protons/s. This value is comparable to the maximum rate reported from water-filled gramicidin proton channels (2.2 × 109 protons/s) under strongly acidic conditions (30). The result suggests highly efficient proton conduction through SWNT, enhanced by the expected accumulation of protons near the negatively charged pore mouth (30). The high aspect ratio of our nanochannel (>250,000) appears critical for the detection of single ions in aqueous solution. Despite some similarities with biological ion channels, the transport mechanisms in the nanotube are distinct in that they do not conduct or gate ions by conformational changes in pore structure (31).

The necessity of partially shedding hydration shells imposes an energetic threshold on ions entering a nanochannel (13). In protein ion channels this barrier is minimized by carbonyl oxygen mimicking the hydration environment (32), whereas in carbon nanotubes, the activation energy must be supplied externally. The observed minimum threshold voltage of 100 mV in our study falls into the range of theoretically predicted free energy of permeation: 80 to 150 meV (13). Ensemble measurement by Fornasiero et al. (15) reported the percentage of ion rejection by a nanotube membrane at varied ion concentration and valence; the rejection of anions was more efficient because the pore mouth was negatively charged. The ability of our system to count single ions allows precise measurement of the separation factor (proton rate/alkali cation rate). The complete rejection of ions can be achieved below the threshold, and the minimum separation factor at suprathreshold voltage is as high as 6 × 107. This high rejection rate reinforces the notion that membranes based on carbon nanotubes may have applications to desalination of sea water and water purification.

The blocking events provide information about hydrated-ion transport in the carbon nanotube, a subject of intense theoretical investigation (13, 33). The histograms (Fig. 2B) of the blocking currents (ΔG) show Gaussian distributions for Li+ and K+ (with too few events for Na+) and mean values that scale as Na+ > K+ > Li+. The ratio of the areas between the hydrated-ion cross section and that of the nanopore provides an estimate of the conductance change upon ion partitioning, ΔG, relative to the value at complete blockage, ΔGmaxEmbedded Image(1)Here, dion is the ion diameter; dtube is the diameter of the nanotube, and σC-O is the Lennard-Jones parameter for C-O interaction (0.0319 nm) (10). For example, K+ produces ΔG values that vary between 52 and 323 pS (with the latter value taken as ΔGmax); therefore, dion/dtube varies between 0.39 and 0.98, assuming dtube = 1.5 nm. For Li+, the ΔG range is 7.5 to 145 pS, and dion/dtube is 0.22 to 0.98; for Na+, ΔG is 120 to 420 pS, and dion/dtube is 0.52 to 0.98. The ΔG value is smallest for the Li+ ion, which has the smallest crystallographic diameter and the largest hydrated diameter in bulk. It is expected that ΔGNa+ > ΔGK+, due to stronger hydration around Na+, which generally requires larger pore size as observed in protein ion channels (34), although Carrillo-Tripp et al. (35) predict higher hydration numbers for K+. A rigid, synthetic nanopore capable of single-ion detection, as introduced in this work, may resolve outstanding questions about ion-hydration numbers in a confined space (36).

This ordering of the blocking currents is also consistent with the data shown in the mobility histograms in Fig. 2B, which should instead scale inversely with dion/dtube. Here the Na+ mean (5 × 10−8 m2/Vs) is much smaller than that of Li+ and K+ with values of 8 × 10−6 m2/Vs. The Gaussian distribution of τdwell leads to asymmetry in the mobility distribution (mobility ~ 1/τdwell). The smallest ΔG values accompany a large variation in mobility for Li+, which has a small ionic diameter (0.12 nm). However, the mobility of Na+ is nearly three orders of magnitude smaller by comparison, suggesting sterically hindered transport inside the nanotube. The values are two orders of magnitude higher than the cation mobility of bulk water and the theoretical mobility through a 3-nm-diameter SWNT (37) (both ~10−8 m2/Vs). Theoretical treatments describe this high ion mobility as arising from the atomically smooth surface of the SWNT interior and velocity slip at the wall (10). Plotting ΔG versus mobility at 1 M (Fig. 2C) reveals an inverse relation, as expected for Li+ (red) and K+ (black), and describes the mean value of Na+ (blue).

In some instances, we observed durations in which the cation-blocking events developed highly synchronized, rhythmic patterns (three of which are shown for each ion type in Fig. 3A). From the all-point histograms, it is clear that these oscillations occur from single-nanotube transport. A power spectrum from fast Fourier transform (FFT) of the resulting current output (Fig. 3B) reveals the oscillator frequencies as 0.046 Hz for NaCl, 2.7 Hz for LiCl, and 8.1 Hz for KCl. These instances appeared to be metastable and particularly sensitive to environmental perturbation, but otherwise could be observed for durations lasting several minutes (figs. S11 to S15). Despite extensive work examining synthetic analogs of the ion channels, sustainable current oscillation (28, 38) produced from transport of single ions has not been observed. The productivity of the nanopore in terms of single-molecule processing dramatically increases during these periods, counting hundreds of individual cations per minute.

Fig. 3

Metastable oscillations of the electroosmotic current are observed for durations lasting several minutes. (A) Two defined peaks in the all-point histograms indicate that the phenomena arise from single-tube pore-blocking events. The frequency of the oscillations varies with the type of ion. (B) Sharp peaks in a power spectrum of the electroosmotic current are pronounced at the oscillation frequencies.

These oscillations are caused by a coupling between the stochastic pore blocking and a proton-diffusion limitation that develops at the pore mouth (Fig. 4A). The calculated proton mobility through the nanotube was substantially greater than that of the bulk solution [supporting online material (SOM) text]. Thus, an unobstructed proton current in the nanotube will necessarily deplete the proton concentration at the pore mouth. Coupling occurs because the depletion increases the relative blocking-ion concentration, thus increasing its probability of partitioning into the nanotube, an extremely rare event otherwise. Once the nanotube is blocked, the proton concentration at the pore mouth increases rapidly while the blocking cation traverses the nanotube. When the blocker emerges from the other side, the proton current is restored, and a subsequent blocking event is suppressed by the initially high concentration of protons relative to blocking cations. Reformation of the diffusion limitation as the proton flow is restored then allows this cycle to repeat indefinitely. We constructed a stochastic simulation (39) (Fig. 4B) with six equations and rate constants [diffusion of H+ into (ks) and out of (ksd) the pore mouth region, partitioning of H+ (k1) and blocking cation (k2) into the nanotube, and the exit of these two species from the nanotube (k1d and k2d for H+ and blocker, respectively)] (SOM text) with a Gaussian dwell-time distribution known from our experiments. The blocking rate constant (k2) is small compared with the proton value (k1). This system reconstructs the oscillation frequencies observed experimentally for certain values of the proton-exchange rate constant (ks) and also the Gaussian distribution in open-channel lifetime (τopen) observed during the oscillations events in Fig. 3A.

Fig. 4

Stochastic simulation illustrating the influence of coherence resonance on the oscillatory behavior. (A) Six reaction steps define this system: protons and blocking cations entering and transporting across the channel, as well as proton exchange between bulk and pore-mouth region. (B) Summary of the six reactions. (C) Trace of the number of protons at near-pore region (Hnp, black) and ion occupancy in the nanotube (Itube, blue) at varied rate constants of proton-in to near-pore (ks). Histograms of open-channel lifetimes, τopen (red), show Gaussian distributions at only optimal values of ks (in resonance) and random, exponential distributions at larger and smaller values of ks (off resonance). The autocorrelation function, C(τ), shown on the right is pronounced only when the system is in resonance. (D) Sharp peaks in the FFT appear only within the resonance window (red). (E) Characteristic correlation time (τC, red) and coefficient of variation (black) plotted versus ks, a noise-defining parameter in our system, show that the optimal noise level exists for maximum characteristic correlation time or minimum coefficient of variation, corresponding to a maximum signal-to-noise ratio in the context of stochastic resonance.

The signature of stochastic resonance is one where the signal-to-noise ratio of the system maximizes at the optimal level of noise or random fluctuations. In a typical demonstration, an external periodic forcing is superimposed with white Gaussian noise, with a resulting increase in the signal-to-noise ratio at an optimal noise level. In this work, the mechanism is best described by a coherence resonance (40), because the forcing function is intrinsic to the system dynamics. This phenomenon is illustrated in Fig. 4C by tuning the rate of proton diffusion into the near-pore region (ks), which controls the inherent noise in the system. A value of ks that is too large or too small to create the resonance condition results in a blocking frequency that is Poisson-like and erratic. As ks approaches an inherent resonance condition, the distribution of the open-channel lifetime becomes Gaussian and narrow, and the blocking events occur with a locked frequency. This mathematical system can reproduce the frequency values of the oscillations (Fig. 4D) by examining the FFT of the blocking-cation occupancy.

For a quantitative description of coherence resonance in this mechanism, we calculate a characteristic correlation time, τC, from the normalized autocorrelation function, C(τ) (40)Embedded Image(2)Here, C(τ) represents the expected value of the product of number of protons at the near-pore region (Hnp) and time-shifted Hnp and, therefore, becomes pronounced as the two are correlated in time and periodic. (In the above equation, Embedded Image is deviation from the average value of Hnp, and t is time.) τC at resonance corresponds to the interval between blocking events. Another measure of the coherence resonance is the coefficient of variation (41), defined in terms of the open-channel lifetime (τopen)Embedded Image(3)Both measures above are closely related to the signal-to-noise ratio (fig. S19). A summary of the coherence resonance appears in Fig. 4E, where the characteristic correlation time, as well as the coefficient of variation, is plotted versus ks. There is an optimal rate of proton diffusion into the near pore (ks) that maximizes the coherent signal and allows for oscillatory output.

The coherence-resonance condition dramatically increases the single-molecule processing rate of the nanotube (480 ions/min in resonance versus 5 ions/min off resonance). This result has implications for other nanopore systems, including DNA translocation and sequencing and Coulter detectors. The large proton conductivity opens possibilities for new types of proton-conducting membranes for fuel cells and catalysis. Our demonstration of single-ion detection in aqueous solution may allow for ultrasensitive detection technologies for environmental or biological monitoring of ion types differentiated by their dwell times. Our experimental system also provides an ionic resonator or waveform generator to the emerging family of devices that use ions for information processing, such as ionic transistors (42) and rectifier circuits (43). However, the ability to manipulate single small molecules (albeit, only ion manipulation is demonstrated in this work) may allow for the construction of chemical-reactor devices that process a single molecule at a time. The resonant transport condition necessarily ensures that only one molecule occupies the nanotube at any given time, which may be useful for manipulation and feedback control schemes, as well as coupling to external measurement devices for cataloging the single-molecule efflux.

Supporting Online Material

Materials and Methods

SOM Text

Figs. S1 to S19


References and Notes

  1. Materials and methods are available as supporting material on Science Online.
  2. Alternative explanations for pore blocking are easily rejected. The possibility of ion clusters blocking the pores is unlikely because the diameter of such clusters measured by dynamic light scattering is ~500 nm, which is too large to block these pores. Precipitation of ions shown by Powell et al. (28), which might block the pore entrance, does not occur because of the high solubility product (Ksp) of NaCl, KCl, and LiCl. Impurities in the solution were ruled out; water alone does not cause pore blocking; and no blocking from the TMA-Cl suggests that impurities in salts have little effect. The absence of blocking events from the water and TMA-Cl removes the possibility of nanobubbles discussed by Smeets et al. (26) as well. Ag+ and Cl dissolve from Ag/AgCl electrodes with Ksp of 1.8 × 10−10, with [Ag+] about five orders of magnitude smaller than [Na+]; therefore, their contribution to pore blocking is negligible. Once an ion blocks the pore, the ion must translocate to the other side, because the driving force of diffusing back to the solution is relatively weak compared with the high electric field across the blocker.
  3. Powell et al. observe oscillations from divalent ions precipitating at the pore mouth of a 2- to 6-nm Si nanopore (28). These oscillations are distinct from those observed in this work in that they are formed from an unstable (precipitating) system, which is difficult to sustain without reversible formation and dissolution of the precipitates.
We thank J. Brauman, J. Collins, N. Maheshri, K. Gleason, T. Swager, and C. Song for helpful discussions. We are grateful for funding from the Institute for Soldier Nanotechnology at the Massachusetts Institute of Technology supported by the U.S. Army Research Office under contract W911NF-07-D-0004 and a fellowship to M.S.S. from the Sloan Foundation. M.S.S. is grateful for an Office of Naval Research Young Investigator Award, as well as Career and PECASE Awards from the NSF.

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