Steady-State Voltammetric Response of the Nanopore Electrode

tions detailing the steady-state response of the truncated cone- shaped nanopore ..... nanopore electrode approaches zero, while the current at the co...
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Anal. Chem. 2006, 78, 477-483

Steady-State Voltammetric Response of the Nanopore Electrode Bo Zhang, Yanhui Zhang, and Henry S. White*

Departments of Chemistry, University of Utah, Salt Lake City, Utah 84112

The steady-state voltammetric response of the truncated conical-shaped glass nanopore electrode is presented. Analytical theory, finite-element simulations, and experimental measurement of the diffusive flux of a redox molecule through the pore orifice demonstrate that the steady-state current decreases rapidly as the pore depth increases and then asymptotically approaches a constant value when the pore depth is ∼50× larger than the pore orifice. The asymptotic limit of the steady-sate current is only a function of the pore orifice radius and the cone angle of the pore and has a finite value for all cone angles greater than zero. Experimental confirmation of the predicted dependence on pore depth is obtained using nanopore electrodes with 100-1000 nm orifice radii, by measuring the steady-state voltammetric current corresponding to the oxidation of ferrocene in acetonitrile solutions containing an excess of supporting electrolyte. We report voltammetric studies, theory, and computer simulations detailing the steady-state response of the truncated coneshaped nanopore electrode (referred to as the “nanopore electrode”). The nanopore electrode, schematically depicted in Figure 1, is a Pt disk electrode embedded in the bottom of a cone-shaped glass pore. A previous report from our laboratory described the fabrication of nanopore electrodes with orifices as small as a few tens of nanometers.1 These electrodes are being developed in our laboratory as a platform for investigating molecular transport through nanometer-scale orifices.2 A characteristic of the nanopore electrode, which offers potential advantages in both fundamental and analytical measurements, is that the largest mass transport resistance of the electrode is localized at the pore orifice. This feature is a consequence of the combination of (i) the convergent radial flux of redox-active molecules (and ions) from the bulk solution to the disk-shaped orifice (analogous to the flux to a disk-shaped microelectrode)3 * Corresponding author: (e-mail) [email protected]. (1) Zhang, B.; Zhang, Y.; White, H. S. Anal. Chem. 2004, 76, 6229. (2) See: (a) Liu, N.; Dunphy, D. R.; Atanassov, P.; Bunge, S. D.; Chen, Z.; Lopez, G. P.; Boyle, T. J.; Brinker, C. J. Nano Lett. 2004, 4, 551. (b) Harrell, C. C.; Kohli, P.; Siwy, Z.; Martin, C. R. J. Am. Chem. Soc. 2004, 126, 15646. (c) Sun, L.; Crooks, R. M. J. Am. Chem. Soc. 2000, 122, 12340. (d) Kuo, T.-C.; Cannon, D. M., Jr.; Chen. Y.; Tulock, J. J.; Shannon, M. A.; Sweedler, J. V.; Bohn, P. W. Anal. Chem. 2003, 75, 1861. (e) Li, N.; Yu, S.; Harrell, C. C.; Martin, C. R. Anal. Chem. 2004, 76, 2025. (f) Kasianowicz, J. J.; Henrickson, S. E.; Weetall, H. H.; Robertson, B. Anal. Chem. 2001, 73, 2268. (g) Bayley, H.; Cremer, P. S. Nature 2001, 413, 226. (h) Saleh, O. A.; Sohn, L. L. Nano Lett. 2003, 3, 37, and references therein, for representative studies of transport in different pore structures. 10.1021/ac051330a CCC: $33.50 Published on Web 12/03/2005

© 2006 American Chemical Society

Figure 1. Drawn-to-scale schematic of the nanopore electrode described in the text with a ) 96 nm, d ) 5.96 µm, and ap ) 827 nm.

Figure 2. Schematic of the flux of a redox species into the pore.

and (ii) the divergent radial flux of molecules from the orifice to the electrode. The flux of molecules and ions from the bulk solution to the electrode, Figure 2, thus obtains a maximum value at the orifice that may be orders of magnitude larger than the flux on the electrode surface. In addition, because steady-state radial fluxes on both sides of a disk-shaped orifice are theoretically predicted, the net flux (and thus current) at the nanopore electrode is also anticipated to display a true steady state.4 The steady-state response of the nanopore electrode is reported herein, with emphasis on the dependence of the voltammetric currents on the pore depth, d, and the half-cone angle of the pore, θ (see Figure 1). Analytical theory, finite-element computer simulations and experimental voltammetry are used to determine the voltammetric conditions that correspond to the steady-state response. We show that the steady-state voltammetric current asymptotically approaches a constant value when the depth of the cone-shaped pore is ∼50× larger than the radius of the pore orifice, a. The asymptotic limit is also a consequence of the radial flux pattern within a conical pore. This finding is important in that (3) Oldham, K. B. J. Electroanal. Chem. Interfacial Electrochem. 1988, 250, 1-21. (4) Chapman, A, J. Fundamentals of Heat Transfer; Macmillan Publishing Co.: New York, 1987.

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it defines the minimum pore depth required to fabricate nanopore electrodes that exhibit nearly identical steady-state voltammetric responses. EXPERIMENTAL SECTION Chemicals. Ferrocene, (Fc, Strem Chemical), reagent grade CaCl2, NaCN, NaOH, and tetra-n-butylammonium hexafluorophosphate (TBAPF6, Aldrich) were used without purification. All aqueous solutions were prepared using 18 MΩ‚cm water from a Barnstead E-pure water purification system. Acetonitrile (HPLC grade, J. T. Baker) was stored over 3-Å molecular sieves. Nanopore Electrode. Nanopore electrodes were fabricated according to our previous report1 with a minor modification. Briefly, a 2-cm-length piece of 25-µm-diameter Pt wire (Alfa-Aesar, 99.99%) was electrically contacted to a 7.6-cm-long W rod (0.254mm diameter, FHC, Inc.) using Ag conductive epoxy (DuPont). The Pt/W wire ensemble was heated in an oven at 120 °C for ∼15 min to dry the Ag epoxy. The end of Pt wire was electrochemically etched to a sharp point in 6 M NaCN/0.1 M NaOH solution. The etched Pt wire was inserted with care into a 5-cm length of glass capillary (Dagan Corp., Prism glass capillaries, SB16, 1.65-mm outer diameter, 0.75-mm inner diameter, softening point 700 °C) leaving ∼4 mm between the tip and the end of glass tube. The glass tube was then melted around the Pt tip using a H2-O2 flame. An optical microscope was used to check the sealing and ensure that no air bubbles were trapped near the metal tip. After sealing the Pt in the glass, the W rod was secured to the glass capillary with insulating epoxy (Dexter). The end of the capillary containing the sealed Pt wire was polished to make a nanometer-sized Pt disk electrode.1 A nanopore electrode is prepared by electrochemically etching the sealed Pt wire. In our previous report, we used a 15% CaCl2 solution to rapidly etch the Pt wire to depths of ∼10 µm. In the current study, a procedure that slowly etches the Pt was also developed in order to more precisely control the etching rate for generating shallow pores. A 0.05 M H2SO4 solution was used instead of the 15% CaCl2 solution employing an etching procedure adapted from Libioulle et al.5 Positive 15-V pulses of 16-µs duration at a repeating frequency of 4 kHz were applied to the Pt nanodisk relative to a large carbon electrode. Etching times of 2-10 s were found to remove between 100 and 600 nm of Pt. The 15% CaCl2 solution described in the previous report was also used to generate deeper pores, as noted in the Results and Discussion section. All etching procedures were performed with the cell placed in an ultrasonic bath (Sonicor Instruments Corp., Copiague, NY, model SC-40) in order to enhance transport rates in to and out of the pore during etching. After generating the pore, the electrode was thoroughly rinsed with water and ethanol. Prior to electrochemical measurements, the nanopore electrode was sonicated in a CH3CN solution containing 0.1 M TBAPF6 and 5 mM Fc for at least 2 min. Scanning electron micrographs (SEMs) of a disk-shaped Pt electrode and the orifice of a nanopore electrode, of dimensions comparable to the electrodes employed herein (∼100 nm), were published in our initial report.1 Electrochemical Measurements. A one-compartment, twoelectrode cell was employed with the cell and preamplifier in a (5) Libioulle, L. Houbion, Y. Gilles, J.-M. Rev. Sci. Instrum. 1995, 66, 97.

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Faraday cage. A Ag/AgxO wire (1-mm diameter) or Ag/AgCl electrode was used as the reference/auxiliary electrode. A Chem-Clamp (Cornerstone Series) voltammeter-amperometer (Dagan Corp.) and a Princeton Applied Research 175 Universal Programmer were used to perform voltammetric measurements. The high-sensitivity preamplifier (50 pA/V-10 nA/V) of the potentiostat was used in all experiments. The ChemClamp was interfaced to a PC computer through a National Instruments (NI) PCI-6040 multifunction I/O and NI-DAQ card and a NI BNC-2090 analog breakout accessory. Slow-scan voltammetric data were recorded using in-house virtual instrumentation written in LabVIEW 6.0. Fast-scan waveforms (1-1000 V/s) were recorded with a LeCroy 9410 digital oscilloscope interfaced to a computer through the NI USB-to-GPIB controller. A NI GPIB instrument driver was modified in-house to control the instrument and transfer the collected waveforms. Fast-scan voltammograms were averaged (50 waveforms) to increase the signal-to-noise ratio. No additional filtering of the data was performed. Scanning Electron Microscopy. SEM images of the etched Pt wires were obtained using a Hitachi S3000-N microscope. Finite-Element Simulations. The voltammetric responses of nanopore electrodes were simulated using Femlab v2.3 software (Comsol, Inc.) operated on a Dell Dimension XPS (Pentium 4 CPU, 3.2 GHZ, 2 GB RAM). Details of the simulation procedure, the model geometry, and accuracy were previously given in ref 1. RESULTS AND DISCUSSION Analytical Expression for the Steady-State Voltammetric Current. An approximate expression for the diffusion-limited steady-state current at a nanopore electrode is given by eq 1,

ilim ) 4nFaDC*

[

(1 + (d/a) tan θ) (4d/aπ) + (1 + (d/a) tan θ)

]

(1)

where C* and D are the bulk concentration and diffusivity, respectively, of the redox-active molecule, F is Faraday’s constant, n is the number of electrons transferred per molecule, and the geometrical parameters d, a, and θ are defined in Figure 1. Equation 1 is readily derived from the general definition of the limiting current, ilim ) nFC*/RMT, where RMT is the steady-state mass-transfer (diffusion only) resistance6 comprising the internal pore resistance, Rin, and the external solution resistance, Rex. The latter is equivalent to the mass transport resistance at a disk electrode, Rex ) 1/4Da.7 The internal pore resistance has both axial and radial components (i.e., r and z in a cylindrical coordinate system); we have not been able to obtain a general expression for this resistance. However, for small values of θ, the radial component of Rin may be ignored. The result of this approximation is Rin ) d/Daπ(a + d tanθ) (see Supporting Information for the derivation of Rin), which, when combined with the above expression for Rex and RMT, yields eq 1. Finite-element computer simulations described in the following section demonstrate that eq 1 is valid to within 5% when θ < 20°. In the experiments reported below, 7 < θ < 12°; thus, the use of eq 1 in the analysis of our data is quite reasonable. (6) Cussler, E. L. Diffusion, Mass Transfer in Fluid System, 2nd ed.; Cambridge University Press: New York 1997. (7) Saito, Y. Rev. Polarogr. 1968, 15, 177.

In the limit d f 0, the nanopore electrode geometry reduces to that of a disk electrode of radius a, and eq 1 yields

id)0 lim ) 4nFaDC*

(2)

the well-known expression for the steady-state limiting current at a disk-shaped electrode.7 Conversely, in the limit d f ∞, corresponding to a very deep pore, eq 1 reduces to

idf∞ lim ) 4nFaDC*

θ [4/πtan+ tan θ]

(3)

df∞ A key result of this analysis is that ilim is independent of d, a consequence of the radial divergent flux within a conical-shaped pore, as discussed in the introduction. Equation 3 provides accurate values of the nanopore electrode current for finite values of d (eq 1) when a/d , tanθ. More precisely, for typical values of θ obtained in preparing nanopore electrodes (∼10°), the current (eq 1) asymptotically approaches the depth-independent value (eq 3) to within 10% when d/a g 49.2 (∼50). Thus, the current at the nanopore electrode is essentially independent of pore depth when the pore depth is at least 50× greater than the radius of the pore orifice. For instance, for a pore with a 20-nm-radius orifice, any pore depth greater than ∼1 µm will yield the same steady-state limiting current. Finite-Element Simulations of the Steady-State Response. Figure 3A shows finite-element simulations (using Femlab software) of the steady-state voltammetric response of a nanopore electrode. In this example, the orifice radius (a ) 100 nm) and half-cone angle (θ ) 20°) were held constant, and the voltammetric response was computed as a function of the normalized pore depth, d/a. The voltammetric response of the nanopore electrode displays the characteristic sigmoid shape expected of steady-state Nernstian behavior and is, in fact, identical to the reversible diffusion-controlled response of a disk-shaped microelectrode8 (which corresponds to the curve labeled d/a ) 0 in Figure 3A). (See Supporting Information for a comparison of disk and nanopore electrode voltammetric wave shapes.) Consistent with the above analytical expressions, the simulations in Figure 3A show that the limiting current, ilim , decreases rapidly with decreasing d/a, approaching a constant value for d/a > 50. Figure 3B shows this behavior more quantitatively in d)0 a plot of simulated values of ilim/ilim versus d/a as a function of d)0 θ. The quantity ilim/ilim , eq 4, represents the limiting current of

ilim /id)0 lim )

1 + (d/a) tan θ 4d/aπ + (1 + (d/a) tan θ)

(4)

a nanopore of orifice radius a, normalized to the limiting current at a microdisk electrode of the same radius. An interesting feature of eq 4, which is readily obtained from eqs 1 and 2, is that ilim/ d)0 ilim is a function of the normalized depth d/a and not the absolute values of either d or a. Thus, the set of curves in Figure 3B applies universally to nanopore electrodes of artbitrary orifice radius. In addition, when examining the experimental dependence (8) Zoski, C. G. In Modern Techniques in Electroanalysis; Vanysek, P., Ed.; John Wiley and Sons: New York, 1996; Vol. 139, pp 241-312.

Figure 3. (A) Simulated steady-state voltammetric response of a 100-nm radius, θ ) 20° nanopore electrode as a function of the normalized pore depth, d/a. (B) Theoretical (lines, eq 1) and simulated d)0 for a 100-nm-radius nanopore electrode (symbols) values of ilim/ilim as a function of θ and d/a. The simulated i-V curves employed values of D ) 2.47 × 10-5 cm2/s, C* ) 5 mM, and n ) 1

of ilim on d, it is very convenient to consider the normalized values d)0 d)0 versus d/a, where ilim and a are of these quantities, i.e., ilim /ilim measured prior to etching the Pt disk to generate the nanopore. d)0 Values of ilim/ilim in Figure 3B show a relatively strong dependence on θ, with smaller currents obtained as the pore geometry approaches that of a straight cylinder (θ ) 0°). In the limit d f ∞, the asymptotic limit of eq 4 is given by

idlimf∞/id)0 lim )

tan θ 4/π + tan θ

(5)

df∞ d)0 For θ ) 0 °, eq 5 predicts that ilim /ilim ) 0, which is the correct result for an infinitely deep cylindrical pore.8,9 However, for θ ) 90°, corresponding to a disk-shaped pinhole in an infinitely thin df∞ d)0 membrane, eq 1 predicts that ilim /ilim ) 1, which is exactly twice the correct value of 0.5. This error is a consequence of ignoring the radial transport component in deriving eq 1. Thus, as noted above, eqs 1-5 are limited to θ < 20°. The advantage of a conical-shaped pore, relative to a straight cylindrical-shaped pore,9 is demonstrated by the results of Figure 3B. For deep pores, the steady-state current at a cylindrical-shaped nanopore electrode approaches zero, while the current at the conical pore approaches a limiting nonzero value that is a sig-

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nificant fraction of the current expected at a disk electrode of same radius. For the values of θ experimentally realized in our study, the nanopore electrode limiting currents in the limit d f ∞ are at least 15% of the corresponding value at the disk electrode prior to etching the Pt. Experimental Studies of the Steady-State Voltammetric Response. The remainder of the article focuses on verifying the above analytical expressions and numerical results. In addition to presenting experimental verification of eq 1, we also demonstrate that a set of nanopore electrodes can be prepared that generates essentially identical limiting currents, without precise knowledge of the pore depth. This is an obvious result from above, df∞ as ilim quickly approaches an asymptotic limit as d increases. To demonstrate the validity of eq 1, it is necessary to prepare electrodes with fully characterized nanopore geometries. This requires experimental measurement of at least three of the four geometrical quantities: d, a, ap, and θ (see Figure 1). The procedure for measuring these quantities was described in detail in our previous report1 and is briefly summarized here: (i) Pore Orifice Radius. The value of a is also equal to the radius of the Pt disk prior to etching the Pt to generate the pore. The disk radius is determined by measuring the steady-state d)0 diffusion-limited current, ilim , in a solution containing millimolar concentrations of a redox-active molecule. The accuracy in d)0 determining a based on ilim depends on whether the geometry of the pore walls after etching faithfully reproduces the shape of the original Pt wire surface. We previously demonstrated, using SEM, that the orifice radius is within 20% of the original Pt disk radius after etching the Pt in the CaCl2 solution and that both pore orifice and Pt disk have disk geometries.1 (ii) Half-Cone Angle. Values of θ are determined to within 1° by optical and electron microscopy of the etched Pt wire prior to sealing it in glass. The angle θ is constant except at the very apex of the sharpened wire. Limitations of a varying θ near the tip of the wire are discussed below in context of the experimental results. (iii) Radius of Pt Disk at the Bottom of the Pore. At sufficiently high scan rates in a voltammetric experiment, the current becomes limited by planar diffusion of redox molecules initially present in the pore and adjacent to the Pt surface. The simulated volammetric response of the nanopore electrode, Figure 4A, is identical to that of a shielded macroscopic planar electrode in this limit.10 The value of ap is determined from the slope of a plot of the voltammetric peak current, ip, versus the square root of scan rate, ν1/2, according to the classical expression for ip, at a shielded planar electrode under diffusion control:10

ip ) (2.69 × 105)n3/2D1/2C*ν1/2πap2

(6)

(iv) Pore Depth. A value of d is obtained by geometry if a, θ, and ap are previously measured by the above methods. Alterna(9) (a) Bond, A. M.; Luscombe, D.; Oldham, K. B.; Zoski, C. G. J Electroanal. Chem. 1988, 249, 1. (b) West, A. C.; Newman, J. J. Electrochem. Soc. 1991, 138, 1620. (c) Henry, C. S.; Fritsch, I. Anal. Chem. 1999, 71, 550. (d) Mirkin, M. V.; Fish, G.; Kokotov, S.; Palanker, D.; Lewis A. Anal. Chem. 1997, 69, 1627. (e) Amatore, C.; Save´ant, J. M.; Tessier, D. J. J. Electroanal. Chem. 1983, 147, 39. (f) Tokuda, K.; Morita, K.; Shimizu, Y. Anal. Chem. 1989, 61, 1763. (10) Bard, A. J.; Faulkner, L. R. Electrochemical Methods. 2nd ed.; John Wiley & Sons: New York, 2001.

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Figure 4. Simulated voltammetric response of a 50-nm-radius pore. The parameters a ) 50 nm, d ) 25 µm, and θ ) 20° were held constant in all simulations, while the sweep rate was varied. (A) Voltammetric response at (A) high sweep rates (ν ) 0.05-100 V/s) and (B) low sweep rates (ν ) 0.01-10 mV/s).

tively, at intermediate scan rates, the nanopore electrode behaves as a thin-layer electrochemical cell,10 exhibiting a symmetrical cathodic and anodic voltammetric peaks corresponding to redox molecules initially present in the pore, Figure 4B. This thin-layer response is superimposed on the steady-state diffusion current (eq 1), and the peak current is approximated by1

ip ) ilim +

n2F2ν CV 4RT b p

(7)

where R is the gas constant, T is absolute temperature, and Vp is the pore volume. If any pair of geometrical parameters is known (e.g., θ and ap), then d can be computed from the value of Vp obtained from a linear plot of ip versus ν. The reader is referred to ref 1 where the details of the pore geometry measurements are further discussed, and estimates of precision and accuracy based on example applications are presented. We note here that independent measurement of all four geometric parameters yields a self-consistent and unique electrode geometry (i.e., the value of d computed from a, θ, ap and geometrical relations is in agreement with the value measured by the “thin-layer” method outlined above in (iv)). As an example of using the above pore geometry analysis, Figure 5A shows the voltammetric response (ν ) 20 mV/s) of a

Figure 6. (A) Voltammetric peak current for 96-nm-radius orifice nanopore electrode plotted as a function of ν. Inset: ν from 0.1 to 20 V/s. (B) Same data plotted as a function of ν1/2. (Data from Figure 5, but not all voltammetric curves are presented in Figure 5)

Figure 5. (A) Steady-state voltammetric response at 0.020 V/s of a 96-nm-radius orifice nanopore electrode in CH3CN/0.1 M TBAPF6 containing 5 mM Fc before and after etch. (B) Transient voltammetric response of the same electrode at ν ) 0.1-500 V/s.

disk electrode in CH3CN/0.1 M TBAPF6 containing 5 mM Fc. This electrode was prepared from a sharpened Pt wire with θ measured to be 7°. The diffusion-limited steady-state current, d)0 ilim , is 460 pA, and using eq 1, the radius a is computed to be 96 nm. The disk electrode was then etched in a 15% CaCl2 solution for ∼10 s, cleaned very carefully, and briefly sonicated in a 5 mM Fc solution to introduce the solution into the pore, as described in the Experimental Section. Figure 5B shows the voltammetric response of the nanopore electrode as a function of scan rate (0.1 e ν e 500 V/s). The background-corrected peak current, ip, was measured at each scan rate and plotted versus ν (Figure 6A, both main curve and the inset) and versus ν1/2 (Figure 6B). Inspection of these plots shows that the voltammetric response is dominated by a radial diffusion at slow scan rates (i.e., the limiting current is independent of ν, see Figure 6A inset), displays thin-layer cell behavior at intermediate scan rates (peak current is proportional to ν, see dashed line in Figure 6A inset), and dispays planar diffusion-controlled behavior at even higher ν (ip is proportional to ν1/2, dashed line in Figure 6B). The slope of the straight region of the inset of Figure 6A is 0.022 nA‚s‚V-1, which, when combined with eq 6, corresponds to Vp ) 4.7 × 10-12 cm-3. Using θ ) 7°, the pore depth is then calculated to be 5.9

µm. The radius of the Pt disk, ap, is then computed to be 820 nm. An independent value of ap ) 827 nm, which does not require knowledge of d, is obtained from the slope of the linear region of the plot of ip versus ν1/2 (Figure 6B) and eq 6. The good agreement between these values is an indicator of the reliability of the values of d reported in the following sections. For commonly employed scan rates (0.001-1000 V/s), the above method is limited to pores that are sufficiently deep to exhibit well-defined thin-layer or planar diffusion behavior as the scanned rate is varied. For very shallow pores (e.g., d < 500 nm), we estimated values of the pore depth by adjusting the value of d in numerical simulations to match the experimental scan rate dependence of the voltammetric peak current and wave shape. Reference 1 gives an example of this latter method of analysis. Steady-State Voltammetric Response as a Function of d)0 Pore Depth. Figure 7 shows a plot of ilim/ilim versus d/a for two nanopore electrodes with orifice radii of 156 and 89 nm, and θ ) 12.5 and 9°, respectively. These data were obtained by measuring ilim as a function of d, between repeated etchings of the Pt wire to obtain different pore depths. Table 1 presents the raw data for the normalized results shown in Figure 7, as well a description of the different etching conditions. To obtain very shallow pores or small increments (500-1000 nm) in d between each etch, the Pt disk was etched in H2SO4 solutions using a pulsed method, rather than in the CaCl2 solution, as described in the Experimental Section. Also shown for comparison are the theoretical plots of d)0 ilim/ilim versus d/a (eq 3, solid lines) based on the measured values of a and θ. As anticipated from theory and simulation described above, d)0 ilim/ilim decreases rapidly with increasing d/a for shallow pores, but then approaches a constant value that is in reasonably good agreement with the theoretical expectation. However, the experid)0 mental values of ilim/ilim data are significantly larger that theoretical values for small values of d/a. A likely reason for this Analytical Chemistry, Vol. 78, No. 2, January 15, 2006

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d)0 Figure 7. ilim/ilim as a function of the normalized pore depth, d/a, for 156- (θ ) 12.5°) and 89-nm-radius (θ ) 9°) nanopore electrodes. Theoretical steady-state voltammetric response as a function of d/a for θ ) 9° and 12.5° are shown as the solid lines.

Table 1. Steady-State Limiting Current as a Function of Pore Depth no. etch

d (nm)

0 1a 2a 3a 4a 5a 6b 7b

0 660 1,180 1,870 2,330 3,190 8,840 10,800

a ) 156 nm d)0 0 760 () ilim ) 4.2 495 7.2 389 11.4 336 14.1 304 19.4 241 53.6 165 65.6 195

1.00 0.65 0.51 0.44 0.40 0.32 0.22 0.26

0 1a 2a 3b 4b

0 -c 730 7,030 8,500

a ) 89 nm d)0 0.00 420 () ilim ) -c 398 7.95 338 76.4 65 93.4 81

1.00 0.95 0.80 0.15 0.19

d/a

ilim (pA)

Figure 8. (A, B) SEM images of a Pt tip etched in a 6 M NaCN solution containing 0.1 M NaOH. (C) Schematic diagram of the Pt tip showing the variation in half-cone angle near the end of the Pt tip.

d)0 ilim/ilim

a Pulsed electrochemical etch in 0.05 M H SO . b AC etch in 15% 2 4 CaCl2. c Unable to determine pore depth using fast-scan voltammetry (see text).

discrepancy is the variation in θ near the very end of the pore. Panels A and B of Figure 8 show SEM images of a typical Pt tip etched in a solution containing 6 M NaCN and 0.1 M NaOH. One can clearly observe in the SEM images that θ at the very end of the tip (within a few µm) is considerably larger (∼20°) than the value one would measure from optical microscopy of the Pt wire (θ ∼ 8°). Figure 8C shows an overly simplified “two-angle” model of the variation in θ. Assuming that the variation in θ is reproduced in the shape of the nanopore glass wall, and since larger values d)0 of θ give rise to corresponding larger ilim/ilim (Figure 3B), the increase in θ near the pore opening would tend to lead to larger d)0 values of ilim/ilim at small values of d/a. For large values of d/a, the variation in θ very near the opening of the pore should have d)0 a significantly smaller influence on ilim/ilim , and the value of θ determined by optical microscopy will be sufficient in predicting d)0 ilim/ilim . These expectations appear to be born out by the data shown in Figure 7, as well as additional results presented in the next section. Preparing Electrodes of Arbitrary Pore Depth That Exhibit Nearly Identical ilim/id)0 lim . A perceived difficulty in preparing 482 Analytical Chemistry, Vol. 78, No. 2, January 15, 2006

d)0 Figure 9. ilim/ilim for the oxidation of Fc at nine nanopore electrodes as function of d/a. Data were recorded in CH3CN/0.1 M TBAPF6 solutions containing 5 mM Fc. The solid line corresponds to d)0 the theoretical values of ilim/ilim for θ ) 9° (as estimated by optical microscopy.) Orifice radii are indicated on the figure.

nanopore electrodes is the lack of control over the pore depth during the etching of the Pt. Although d/a can be measured as described above, this step is not essential in predicting the magnitude of the voltammetric current, as the theory and d)0 is very weakly dependent on simulations indicate that ilim/ilim d/a for values greater than ∼10 (see Figure 3B). Thus, assuming d/a > 10, the normalized limiting current is independent of the pore depth within the typical limits of the overall measurement accuracy (we estimate a typical error of at least ∼10% in voltammetric experiments using electrodes of nanoscale dimensions, albeit with very high precision). This weak dependence virtually eliminates the need to control (or measure) pore depths d)0 in obtaining predictable and reproducible values of ilim/ilim using d)0 for different electrodes. For instance, Figure 9 shows ilim/ilim the oxidation of Fc at nine nanopore electrodes, prepared by etching the Pt in CaCl2 solutions for sufficient time (10-20 s) to satisfy d/a > 10, but otherwise without effort to precisely control the final value of d/a. Values of orifice radii ranged from ∼100 to 1000 nm (denoted on the figure) and values of pore depth ranged from ∼2 to 20 µm. The average half-cone angle of this set of electrodes was determined by optical microscopy to be 9 ( 1° Following etching to produce the pores, the value of d/a for each

electrode was estimated by transient voltammetric measurements, as described above. d)0 Of the nine electrodes, six yield values of ilim/ilim are clustered around an average value of ∼0.19, and which are also within error of the theoretical prediction (Figure 9, solid line). d)0 The remaining three electrodes exhibit values of ilim/ilim that are significantly larger than predicted, but still within a factor of 2 of theory. We have not attempted to determine why these three electrodes yield slightly higher currents. While further improvement in the reproducibility is clearly desired, these preliminary results indicate that nanopore electrodes, with orifices in the 1001000-nm range, exhibit predictable and reproducible limiting currents without the need to precisely control or measure the pore depth. CONCLUSIONS We have demonstrated using theory, simulations, and experiment that the diffusion-limited steady-state current of a conicalshaped nanopore electrode asymptotically approaches a constant value when the pore depth is ∼50× larger than the pore orifice.

The asymptotic limit is only a function of the radius of the pore orifice and the cone angle of the pore and has a finite value for all cone angles greater than zero. ACKNOWLEDGMENT This research was supported by the National Institutes of Health, the Defense Advanced Research Projects Agency, and the American Chemical Society, Division of Analytical Chemistry Fellowship, sponsored by the Society of Analytical Chemists of Pittsburgh. SUPPORTING INFORMATION AVAILABLE Derivation of eq 1 and a comparison of the nanopore and nanodisk electrode wave shapes. This material is available free of charge via the Internet at http://pubs.acs.org. Received for review July 26, 2005. Accepted October 24, 2005. AC051330A

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