Quantitative nanoscale visualization of heterogeneous electron transfer rates in 2D carbon nanotube networks

Abstract

Carbon nanotubes have attracted considerable interest for electrochemical, electrocatalytic, and sensing applications, yet there remains uncertainty concerning the intrinsic electrochemical (EC) activity. In this study, we use scanning electrochemical cell microscopy (SECCM) to determine local heterogeneous electron transfer (HET) kinetics in a random 2D network of single-walled carbon nanotubes (SWNTs) on an Si/SiO₂ substrate. The high spatial resolution of SECCM, which employs a mobile nanoscale EC cell as a probe for imaging, enables us to sample the responses of individual portions of a wide range of SWNTs within this complex arrangement. Using two redox processes, the oxidation of ferrocenylmethyl trimethylammonium and the reduction of ruthenium (III) hexaamine, we have obtained conclusive evidence for the high intrinsic EC activity of the sidewalls of the large majority of SWNTs in networks. Moreover, we show that the ends of SWNTs and the points where two SWNTs cross do not show appreciably different HET kinetics relative to the sidewall. Using finite element method modeling, we deduce standard rate constants for the two redox couples and demonstrate that HET based solely on characteristic defects in the SWNT side wall is highly unlikely. This is further confirmed by the analysis of individual line profiles taken as the SECCM probe scans over an SWNT. More generally, the studies herein demonstrate SECCM to be a powerful and versatile method for activity mapping of complex electrode materials under conditions of high mass transport, where kinetic assignments can be made with confidence.

Within the family of nanostructured materials, carbon nanotubes (CNTs) have attracted particular attention because they are readily synthesised at low cost, have exceptional electronic properties, exhibit chemical and mechanical stability, and are amenable to a wide range of simple chemical functionalization routes (1–3). These characteristics have led to CNTs being considered ideal substrates for electronics (4), sensing systems (5, 6), electrocatalytic supports (7), and batteries (8). Furthermore, the different configurations in which CNTs can be arranged broaden their versatility and allow custom design of devices for specific applications. Individual single-walled carbon nanotubes (SWNTs) (9), 2D networks (10–12), and 3D nanostructures (13) have all been employed successfully.

Understanding heterogeneous electron transfer (HET) at CNTs is of considerable importance, due to the wide range of electroanalytical and electrocatalytic systems based on CNTs (14–17), and also because electrochemistry provides an attractive route to functionalize and tailor the properties of CNTs (18–20). Probing HET in 1D electrode materials is interesting fundamentally, given their inherent electronic structure and properties (21, 22). However, as we highlight herein, despite many studies aimed at characterizing HET at CNTs, substantial questions remain unanswered, such as the location and rate of HET.

A popular approach for studying electrochemistry at CNTs involves drop-casting the material onto an electroactive support (23, 24), but this makes it difficult to unambiguously identify the contribution from CNTs alone. These voltammetric studies have led to an interpretation that CNTs are active only at edge-plane-like defects in multiwalled nanotubes (MWNTs) and at the open oxygenated ends of MWNTs and SWNTs (23–25), with the sidewall inactive, even for simple redox couples. A separate approach has been to study CNTs on an inert substrate to ensure that the electrochemical (EC) signal measured can be uniquely attributed to the CNT material used. Both 2D networks of SWNTs (12, 26–28) and individual SWNT devices (29, 30) have been employed, and facile HET has been reported for several redox complexes at a majority of SWNTs assessed. Such studies suggest highly active sidewalls, but the measurements are typically averaged over many SWNTs and SWNT contacts (in the 2D networks) or over a length of μm to mm for individual SWNTs.

In order to more fully assess and understand the electrochemical behavior of SWNTs, it is necessary to measure HET of different components, e.g., sidewalls and ends, at very high spatial resolution, allowing the spread of activity to be identified. Herein, we use scanning electrochemical cell microscopy (SECCM) (31, 32), which utilizes a theta pipet probe filled with an electrolyte solution and a quasi-reference counter electrode (QRCE) in each channel, for the EC interrogation of a substrate. The meniscus created at the end of the probe defines a local and mobile EC cell, confining the measurement of the substrate to the dimensions of the pulled pipet (33), approximately 250 nm diameter herein (34). Using feedback protocols, the tip maintains a constant distance from the sample to produce EC maps (vide infra), specifically of redox reactions at the SWNT network biased at a defined potential with respect to the QRCEs. This allows EC data to be collected across a range of characteristic SWNT sites while accessing only a very small part of the electrode material at a time. In parallel, finite element method modeling (33) of SECCM allows the quantitative assignment of standard HET rate constants for the redox reactions of interest.

The focus herein was 2D networks of SWNTs grown by chemical vapor deposition (CVD) on insulating substrates (SiO₂). In addition to the pristine and low-defect nature of such SWNTs (26, 35), this arrangement exposes a large quantity of characteristic sidewalls, cross-points and ends for investigation, and a random population of semiconducting and metallic SWNTs (36). The use of an inert substrate avoids any possible EC contribution from the substrate. Two-dimensional networks of SWNTs are also of interest for at least two further reasons: (i) There is now ample evidence that such an electrode arrangement is optimal for maximizing signal to noise in voltammetric and amperometric measurements (26, 37), and so understanding the intrinsic activity is valuable; and (ii) this arrangement presents a rather challenging array of closely spaced active elements for electrochemical imaging and highlights the capabilities of SECCM in resolving such complexity. More generally, we also show that the configuration of SECCM and an SWNT on an inert substrate leads to ultrahigh mass-transport rates that allow incredibly high HET rate constants to be quantitatively determined while also permitting distinction between different models of EC activity.

Results and Discussion

A scheme of the SECCM setup is depicted in Fig. 1A. Taking into account the characteristic separation between SWNTs, it was essential that the SECCM probe was of the order of ca. 250 nm in diameter (SEM image in Fig. 1A), since this dimension defines the size of the EC cell (33) when the electrolyte at the end of the pipet comes into contact with the substrate. At this length scale, individual SWNTs could be resolved, and it should be possible to distinguish between sidewalls, nanotube ends, and crossing SWNTs. The 2D SWNT network is comprised of randomly distributed interconnected SWNTs, as seen in Fig. 1B. Each SWNT is grown from a single catalytic nanoparticle that is embedded at one end of the SWNT (38, 39). The density of SWNTs was above the metallic percolation threshold (40), so after establishing macroscopic electrical contacts to the network, the sample was ready to be used without any need of postprocessing cleaning steps. This was particularly important because the number of defects present in the SWNTs remains at the intrinsic value. Naturally, SWNTs contain no edge-plane-like sites of the type held responsible for the EC response in MWNTs (41); also, for SWNTs grown by CVD, the ends are likely to be closed, which might be expected to lead to very slow HET kinetics (24, 25). The only other type of defect site that one could reasonably consider is point defects in the sidewall, identified through electrodeposition (42). These have a spacing of 100 nm–4 μm (averaging approximately 400 nm) along the sidewall.

Fig. 1.

Depiction of the SECCM setup and the 2D SWNT network samples on which studies were performed. (A) Schematic (not to scale) of the SECCM setup, showing the SECCM tip with QRCEs over a 2D SWNT network sample connected as the working electrode with a gold contact. FE-SEM image of a typical SECCM tip and the geometry for the FEM is also shown [3D simulation, making use of the symmetry plane perpendicular to the tip septum (33)]. (B) Representative AFM image of SWNT network samples, with an expanded image of a junction and a bundle splitting. Catalyst particles can be seen in the AFM image but are not connected to the SWNTs and are electrochemically inactive.

Samples were routinely characterized with atomic force microscopy (AFM), confirming a typical coverage of SWNT of approximately 4 μm/μm² (SWNT length / substrate area) and heights of approximately 1 nm, with high monodispersity (vide infra). Some SWNT bundles were also observed, as shown in the zoomed image of Fig. 1B, where the splitting of a bundle is seen. Further analysis of the sample with Raman spectroscopy (Fig. S1) confirmed the presence of high quality SWNTs.

Two simple one-electron processes with widely different redox potentials were employed: ferrocenylmethyl trimethylammonium (FcTMA⁺) oxidation and ruthenium (III) hexaamine () reduction (in phosphate buffer pH 7.2 as supporting electrolyte). Fig. 2 A and B show EC current maps, together with linescans, for these processes, with the SWNT substrates biased at the formal potential of the redox couple (Fig. S2). In each case, the EC activity of the substrate is clearly similar to the characteristic topography of SWNT network geometries. The data show that the EC activity is mostly uniform along the length of SWNTs.

Fig. 2.

Experimental data obtained using SECCM of the SWNT network. SECCM images (2.5 μm × 2.5 μm) of SWNT networks at the formal potential for FcTMA^+/2+ (2 mM) (A) and (5 mM) (B). Representative trace (A, B) and retrace (B) linescans are shown below.

The EC map for FcTMA⁺ oxidation (Fig. 2A) shows only a small variation in current across the nanotubes. This is consistent with the similar density of states of metallic and semiconducting nanotubes (29) at this positive potential, so that electrochemistry with this mediator is relatively insensitive to the electronic nature of the SWNT. Although a highly active SWNT network is also evident for reduction (Fig. 2B), there also appears to be a small proportion of SWNTs with lower ET activity (for examples, see the left hand side of the line profile in Fig. 2B, the EC current histogram in Fig. 3A; Fig. S3). This could be because the formal potential of lies in the charge depletion region of some of the semiconducting SWNTs (43), diminishing HET activity (44); and because some SWNTs may be highly defective or poorly connected (45) in the network.

Fig. 3.

Summary of experimental peak EC current data and simulation results. (A) The SWNT height distribution from the AFM images. Histograms of peak current populations from SECCM images for reduction and FcTMA⁺ oxidation from regions of the image where individual SWNT peaks were identified. For reduction the peak current population is also shown for points where more than one nanotube is under the tip (crossed SWNT). (B) and (C) Working curves of EC current vs. standard rate constant for HET at a fully active individual SWNT of height 1 nm with and FcTMA⁺ as the redox mediator, respectively. The simulations were at the formal potential and a transfer coefficient of α = 0.5 was assumed.

SECCM also acquires three complementary maps (Figs. S4, S5, S6) simultaneously with the EC maps: z-piezo displacement, ion conductance, and the AC component of the ion conductance (Materials and Methods, SI Text). These extra data can be viewed to confirm the stability of the meniscus size and the constancy of the tip-sample separation during an SECCM scan. To further ensure that the images did not contain artifacts, and to demonstrate the high reproducibility of the method, we recorded some trace and retrace EC maps (example, Fig. S3). We show, along with the trace image in Fig. 2B, examples of trace and retrace linescans, with each line comprising points obtained every 4 nm. The trace and retrace linescans overlay, confirming the consistency of the measurements, accurate tracking of the surface, the absence of blocking or fouling of the SWNT, and of meniscus dragging effects.

To assign standard HET rate constants, k₀, a finite element method model of the SECCM system was developed (33) (Materials and Methods, SI Text). We approximated the cylindrical SWNT (diameter, d_nt) by a band electrode in the plane of the inert substrate, with an equivalent width w_nt = (π/2)·d_nt and a length defined by the substrate-meniscus contact area, as shown in Fig. 1A. A band approximation is reasonable (46), especially when the HET kinetics are close to surface-limited (vide infra). When the SWNT lies across the center of the SECCM meniscus, the relative orientation between the SWNT and the SECCM probe barrels (defined by the orientation of the septum) was found to be negligible (SI Text). Thus, unless stated otherwise, the SWNT was considered to be parallel with the septum between the SECCM probe barrels (Fig. 1A). Based on our previous experimental studies (31–33), typical tip-substrate separations in SECCM are between 25% and 50% of the tip radius, and we considered a tip-substrate separation of 50 nm in the present study, which was most consistent with the ion current response (35). Further simulations (Fig. S7) proved that the effect on the current at the formal potential of changing the tip-substrate separation by up to ± 20 nm was less than 3% for a constant ion current.

SECCM images were analyzed to extract current values where the pipet most likely passed over individual SWNTs and cross-points of SWNTs in networks. These data are summarized in Fig. 3 as histograms of peak currents in the images at these locations. A corresponding AFM of SWNT height (equivalent to diameter, d_nt) is also shown for comparison. The working curves of EC current vs. log(k₀) (with a transfer coefficient of α = 0.5) for d_nt = 1 nm (w_nt = 1.57 nm) used to analyze these data are shown in Fig. 3 B and C (pipet radius of 125 nm). These curves indicate that, if SWNT heights are well-defined, k₀ approaching 30 cm s^-1 may be accessible (where the current is ca. 90% of the maximum transport-limited value and so distinguishable from the limit). The modal value of the current distribution for the individual SWNTs with (Fig. 3A) is i_we = 4 pA (± 1 pA accounts for 40% of the values) yielding k₀ = 4 ± 2 cm s^-1. For FcTMA^+/2+, a modal current of i_we = 1.8 pA (± 0.2 pA accounts for 42% of the values) yields k₀ = 9 ± 2 cm s^-1. Confidence in these assignments is high because these kinetic values are so far away from the reversible limit with the high mass-transport rates of SECCM. Interestingly, the value determined for is of the order of that found on gold nanoelectrodes, k₀ = 13.5 ± 2 cm s^-1 (47). For FcTMA^+/2+, k₀ = 4 ± 2 cm s^-1 has been found for individual SWNT devices, and we previously estimated k₀ > 1.0 ± 0.6 cm s^-1 or k₀ > 2 ± 1 cm s^-1 for this couple (37, 48).

The samples used for the present study were found to have a narrow range of SWNT heights, as shown in Fig. 3A (blue histogram). To investigate how this variation might impact on the EC response, simulations were performed for the range of nanotube heights (converted to w_nt in the model) for d_nt between 0.8 nm and 2.75 nm, using k₀ = 4 ± 2 cm s^-1 for and 9 ± 2 cm s^-1 for FcTMA^+/2+. The results, presented in Fig. S8, indicate that the EC response changes with nanotube diameter, from 3.5 pA to 7.5 pA for and from 1.7 pA to 2.4 pA for FcTMA^+/2+. This variation is thus a plausible explanation for the main part of the distributions of peak current, in which more than half of the values lie, although we recognize that this is a simple analysis and that k₀ may change with nanotube height (22).

On the other hand, particularly for , there is a small but detectable population of low currents that cannot be accounted for simply from the height distribution. This can reasonably be attributed to the presence of some semiconducting SWNTs (vide supra) and SWNTs with high resistance drop within parts of the ensemble. Indeed, SWNT densities from SECCM (2.6 ± 0.3 μm/μm²) are lower than densities from AFM (3.8 ± 0.4 μm/μm²). While this difference could partly be due to the lower spatial resolution of SECCM, it is also reasonable to postulate a proportion of SWNTs not being electrically connected (26) and low HET rates at some semiconducting nanotubes (vide supra).

One of the characteristic features of 2D SWNT networks is the presence of nanotube junctions (vide supra). The mapping feature of SECCM allowed us to readily view the EC activity of such arrangements. We observed a slight increase in the modal current compared to the individual SWNT case (Fig. 3A), but taking account of the extra active area (compare individual and crossed SWNT working curves in Fig. 3B), the corresponding k₀ value was similar, k₀ = 3 ± 2 cm s^-1.

It has been suggested (24, 25, 41, 49, 50) that sidewalls are electrochemically inactive in CNTs and that only edge-plane-like sites and open oxygenated nanotube ends are active. Our work shows this earlier hypothesis is incorrect. For the SWNT network, we have no obvious edge-plane sites, and the ends are most likely to be closed (38). We see high and similar activity across different sites in SWNT networks.

We now consider whether other possible defects in the sidewall could be solely responsible for the SECCM observations. Given the defect density attributed from selective electrodeposition (42), vide supra, the SECCM pipet would be expected to encounter only one defect when passing over an SWNT. We thus consider this case, simulating a defect (at most) as a square of length 1 nm positioned in the plane of an inert substrate, for . The resulting working curve (Fig. 4A) of EC current vs. log(k₀) at the formal potential provides two important observations: (i) the discernible range of k₀ values is now up to 1,000 cm s^-1, due the very high mass transport to a nm-scale electrode; and (ii) the maximum mass-transport limited current for a 1 nm sized defect is 0.46 pA (for k₀ > 1,000 cm s^-1), an order of magnitude lower than the modal value observed experimentally, and this is for an unfeasibly high rate constant.

Fig. 4.

Comparison of experimental data with simulated data for fully active SWNT sidewall, one defect and three defects. (A) Working curves of EC current vs. standard rate constant for HET of a fully active individual SWNT, one point defect and three point defects (with the SWNT central under the pipet meniscus). (B) Experimental (points) for the current response of an SECCM probe translated over a portion of an individual SWNT, compared to simulations for full sidewall activity (blue, k₀ = 4 cm s^-1), one active defect (red, k₀ = 100 cm s^-1) and three active defects (green, k₀ = 100 cm s^-1).

Next we considered an increase in the number of side-wall defects, up to three in the area under interrogation, to simulate a maximum possible EC response for an SWNT with defects spaced ca. 100 nm. The EC current vs. log(k₀) working curve for this case yields a maximum current i_we = 1.5 pA, but only for k₀ > 1,000 cm s^-1. This rate constant is unfeasibly high, yet this current and lower values are only evident in < 10% of the nanotubes visualized. Thus, the overwhelming majority of SWNTs have an EC activity that can only reasonably be explained based on considerable intrinsic SWNT sidewall activity.

The high spatial resolution of SECCM during scanning (see Materials and Methods) allowed tracking of EC currents across individual SWNTs. The resulting peak profiles also indicate that the EC current response is best explained by HET activity intrinsic to SWNT sidewalls. Fig. 4B compares a simulation for scanning over a fully active SWNT (w_nt = 1.5 nm and k₀ = 4 cm s^-1) with a typical, and frequently observed, experimental example linescan for reduction. It is evident that there is a good match to the shape and magnitude of the profile, assuming a uniformly active tube. Note that such profiles provide an excellent test of the uniformity of SWNT activity as the SWNT length accessed changes in a systematic manner as the SECCM meniscus scans over it.

Simulations of moving the SECCM probe across an SWNT containing one and three sidewall defects with an exaggerated value of k₀ = 100 cm s^-1 were also performed and compared (Fig. 4B). Besides the large difference in the EC response compared to the sidewall active case, there is also a significant difference in the shape of the line scan profile, with an abrupt change in current for the defect, depending on whether or not a defect is accessed. Furthermore, there is negligible variation of the EC current response for the single active site. Note, additionally, that the defect simulations are for the most favorable case where the defect is located central to the scanned meniscus of the SECCM pipet. On this basis, we can further deduce that the most appropriate model for electrochemistry at pristine, untreated SWNTs is one where the large majority of the sidewalls display high intrinsic EC activity.

Conclusions

We have shown that the intrinsic EC activity of a complex electrode, comprising a random network of interconnected SWNTs on an Si/SiO₂ substrate, can be resolved by SECCM. By measuring the EC currents as a function of probe position at the formal potential of two redox couples, we have found that the majority of SWNTs are more or less uniformly active on the length scale of high resolution SECCM. The images herein provide striking quantitative information on EC activity across different types of sites in SWNTs (sidewall and ends) and in 2D networks (cross-points). Quantitative information has not only come from the analysis of peak currents in SECCM maps (when the SWNT lies under the center of the SECCM pipet) but also from the analysis of line profiles in which the line shape across an SWNT running across the Si/SiO₂ substrate perpendicular to the linescan direction can be fitted to a single rate constant for a uniformly active SWNT.

We have also demonstrated that SECCM accesses very high mass-transport rates, which allows the ready assignment of rate constants with high accuracy, and proves that a high proportion of the SWNTs detected had k₀ in the narrow range 4 ± 2 cm s^-1 for and 9 ± 2 cm s^-1 for FcTMA^+/2+. Furthermore, for both couples, all SWNTs detected by SECCM show k₀ > 0.1 cm s^-1. Thus, in practical terms, SWNT sidewalls will show fast HET for most common electrochemical techniques that operate with much lower mass transport coefficients than these standard rate constants. As such, SWNTs should be viewed as highly active electrochemical materials, at least for outer-sphere redox couples of the type studied herein. This new understanding of electrochemistry at SWNTs clarifies misconceptions that have arisen from different views in the literature on the electroactivity of SWNT materials; also, it has implications for related materials such as graphene and graphite.

Materials and Methods

Synthesis of SWNT Network Samples.

SWNTs were grown by catalytic chemical vapor deposition (cCVD) on silicon/silicon oxide substrates (IDB Technologies Ltd., n-type Si, 525 μm thick with 300 nm of thermally grown SiO₂) as reported elsewhere (12). Fe nanoparticles served as the metal catalyst (35), deposited from aqueous solutions of horse spleen ferritin (Sigma Aldrich), diluted from the original concentration in a ratio of 1∶200. To establish macroscopic electrical contact to the SWNT networks, 100 nm Au (with an adhesion layer of 4 nm Cr) was thermally evaporated (Moorfield Minibox evaporator) on the SWNT samples, partly covered by a stencil shadow mask. AFM characterization of the SWNT networks was performed using a Veeco Enviroscope AFM (Bruker) with Nanoscope IV controller, in tapping mode. Raman spectra were acquired with a 514 nm Ar laser microRaman (Renishaw inVia, United Kingdom) and a 50x optical lens.

Solutions.

Solutions were prepared using Milli-Q water (Millipore Corp.) and consisted of either 2 mM trimethyl(ferrocenylmethyl)ammonium (as the hexafluorophosphate salt, synthesized in house) (51) or 5 mM hexaamineruthenium (III) chloride salt (Sigma Aldrich, 98%). In each case phosphate buffer pH 7.2 (Sigma Aldrich) served as supporting electrolyte.

SECCM Instrumentation and Protocols.

Tip fabrication.

SECCM tips were pulled from quartz theta capillaries (o.d. 1.2 mm, i.d. 0.9 mm, Sutter Instrument) in a laser puller (P-2000, Sutter Instrument) to yield pipets of ca. 250 nm internal diameter at the end. Tip dimensions, including taper angle, were measured accurately using field emission-scanning electron microscopy (FE-SEM; ZeissSUPRA 55-VP) as shown, for example, in Fig. 1A. The outer walls of the tips were silanized with dimethyldichlorosilane (Fluka).

Instrumentation.

Two instruments were used, the first of which has been previously reported in detail (31–33). The second instrument was similar, but had the SECCM tip mounted on a one-axis piezoelectric positioner (P-753.31C, Physik Instrumente), with the sample positioned vertically below the tip in a humidity cell on a two-axis piezoelectric stage (P-622.1CD, Physik Instrumente). The SECCM tip was oscillated normal to the surface (200 Hz, 30 nm peak to peak) by an ac signal generated from a lock-in amplifier (SR830, Stanford Research Systems). The AC component of the barrel current (used for distance control) was measured using the same lock-in amplifier and recorded through the FPGA card (7852R, National Instruments).

Imaging procedure.

For the studies herein, the bias between the two QRCEs in the SECCM pipet was 500 mV, inducing an ion current between the two barrels. The conductance cell was floated with respect to the working electrode surface, held at ground, so that the driving force for HET was at the formal potential for the particular redox couple of interest. A meniscus forms at the end of the SECCM tip and acts as the electrochemical cell that is scanned across the sample (31–33).

The first instrument (31) used for the FcTMA^+/2+ studies produced an image with data points evenly spaced (typically 50 nm). The second instrument employed a continuous scanning method, providing higher spatial resolution in the tip-scan direction. The SECCM tip speed was 300 nm s^-1, and data were acquired at a rate of 78 Hz, yielding a spatial resolution of ca. 4 nm in the direction of the line scan. This imaging procedure was used for studies. Data analysis was performed in Matlab (R2010b, Mathworks). Points where the tip was directly above a carbon nanotube were identified as peaks in the substrate current.

Simulations.

Following the methodology we have reported recently for finite element method (FEM) modeling of mass transport within an SECCM probe (33), we approximated the SECCM probe geometry to a circular-based cone of radius 125 nm. In contrast to our previous studies (33), we applied a two-step solver for computational efficiency. Initially, the potential field and ion migration current between the two QRCEs in the SECCM pipet were determined in the absence of a working electrode reaction by solving the steady-state Nernst–Planck equations:

[1]

[2]

where D_j, c_j, z_j, and u_j are the diffusion coefficient, concentration, charge, and ionic mobility of species, j, respectively. F is the Faraday constant, and V is the electric field (provided by the bias applied between QRCE₁ and QRCE₂, E_f, as depicted in Fig. 1A). The boundary conditions were as outlined previously (33) and the potential field was calculated for a solution containing either 2 mM FcTMA⁺ as the salt or 5 mM as the chloride salt in 50 mM phosphate buffer at pH 7.2 using ion mobilities and diffusion coefficients obtained from the literature (52, 53) and summarized in Table S1.

To determine the EC response at the formal potential, we considered the majority of the substrate to be insulating, with the SWNT defined as a rectangle of width w_nt in the plane of the substrate with a length determined by the substrate-meniscus contact area. The width of the rectangle was related to the height of the SWNT, d_nt, as defined earlier. Butler-Volmer kinetics were applied to the SWNT electrode, assuming a transfer coefficient α = 0.5, with the remainder of the surface (Si/SiO₂) inert. Mass transport to the working electrode surface was calculated by solving Eq. 1 for the electrode reactant and product species, for a typical barrel current of ca. 1 nA, using protocols described in full (33).