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J. Phys. Chem. C 2009, 113, 9878–9883
On the Applicability of Conventional Voltammetric Theory to Nanoscale Electrochemical Interfaces Yu Sun, Yuwen Liu, Zhixiu Liang, Lu Xiong, Aili Wang, and Shengli Chen* Hubei Electrochemical Power Sources Key Laboratory, Department of Chemistry, Wuhan UniVersity, Wuhan 430072, China ReceiVed: March 15, 2009; ReVised Manuscript ReceiVed: April 21, 2009
The voltammetric responses of Pt disk electrodes 5-50 nm in radii in the presence of excess inert electrolyte were investigated to verify the applicability of the conventional diffusion-based voltammetric theory to nanoscale electrochemical interfaces. A so-called “inverted heat-sealing” procedure was introduced in the electrode fabrication process to eliminate the possible tiny interstice between the glass sheath and electrode wire that could severely distort the voltammetric curves of nanometer-szied electrodes. Linear relations between the limiting currents (iL) and the concentrations of electroactive ions (ca) were found at electrodes as small as 5 nm, seemingly inferring that the classic voltammetric theory is applicable at such small electrodes. However, a delicate analysis on the dependences of iL on the electroactive size of electrode and the charge carried by the electroactive ions revealed that the iL ∼ ca linearity is altered from the predication of the conventional voltammetric theory as the size of electrode approaches nanometer scales (e. g., 20 nm, but the deviation is clearly seen at electrodes of a few nanometers, agreeing with the experimental observation shown in Figure 2b. The present PNP-based calculations thus indicate that the linear iL ∼ ca dependence in the presence of EDL effect at nanoscale electrochemical interfaces is due to the small change of the interfacial potential distribution with ca, which results in a linear variation of electromigration rate of electroactive ions with ca. This is actually easy to understand when considering that the variation of ca makes a negligible change in the total ion concentration in the solution in the presence of excess inert electrolyte, therefore little change in the potential distribution (EDL structure). Failure of eq 1 has also been reported by Krapf et al.14 on quasispherical nanoelectrodes fabricated by filling gold into nanometer-sized pores drilled in thin SiN membranes with a focused electron beam. The deviations reported by these authors are rather significant. It was shown that a strong nonlinear dependence of the transport-limited current on the concentration of electroactive ion (e.g., FcTMA+) occurs at electrodes below 10 nm. They believed that such a pronounced deviation is due to the breakdown of the continuum and mean-field conditions at nanoscale electrochemical interfaces, whereas the results in the present study show that there remains a linear iL ∼ ca dependence at electrodes smaller than 10 nm, which can be welldescribed in terms of the overlap effect of EDL and CDL and is predictable by the continuum and mean-field based PNP theory. Usually, continuum and mean-field approximations may break down when the liquid is confined into narrow spaces like nanosized gaps, channels, or pores. For the interface of individual open nanoelectrodes where the semi-infinite transport condition applies, the continuum assumption should still apply since no space confinement of solution occurs.
Figure 3. The calculated interfacial potential and concentration distributions under limiting transport condition at an electrode 5 nm in radius for different concentrations of Fe(CN6)3- in 1.0 M KCl. The insets are the calculated iL/r0 ∼ ca dependence for different r0 values.
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Sun et al. whereas the ratio gradually departs from unity as the electrode size goes below 10 nm. The smaller the electrode size, the greater the deviation is. Since excess inert electrolyte is presented, the observed EDL effect must be a result of reduced electrode size. The inset in Figure 4 gives the values of iL/d for the one-electron reduction of Fe(CN6)3- and Ru(NH6)3+, and the one-electron oxidation of the neutral species ferrocene at electrodes of different sizes. For electrodes smaller than 10 nm, the values of iL/d significantly deviate from 1 for charged reactants and opposite deviations occur for anion and cation reactants, respectively, clearly indicating the failure of eq 1. 4. Further Discussion
Figure 4. The ratio between the values of id-normalized limiting current for the reduction of Ru(NH6)3+ and Fe(CN6)3- in 1.0 M KCl solution. Inset: The ratio between the measured limiting current and that calculated according to eq 1 for differently charged electroactive species at electrodes of various sizes.
The present results seemingly agree somewhat with the recent study by Sun and Mirkin.13 They showed that fitting voltammetric responses of nanometer-sized disk electrodes with the diffusion-based voltammetric theory produces kinetic parameters for a range of heterogeneous electron transfer reactions very similar to those obtained at macroscopic electrodes, implying that no major deviation from the conventional voltammetric theory occurs at nanometer sized electrodes. The iL ∼ ca data in Figure 2 have been used to evaluate the value of diffusion coefficient according to eq 1. The D value given by the iL ∼ ca linearity for the electrode of 5 nm is about 80% of that given by the electrode 9.7 µm in radius. This is indeed a minor difference that could be easily considered a scattering of experimental data, therefore leading to a conclusion that the diffusion-based voltammetric theory is applicable at electrodes of nanometer sizes. However, the iL/r0 ∼ ca dependence and the theoretical analysis given above tend to tell that the altered value of D is most probably due to enhanced EDL effects at nanoscale electrochemical interfaces. 3.4. The Dependence of the Limiting Current on the Charge of Electroactive Species. To further confirm the failure of eq 1 at the interface of nanosized electrodes, we investigated the dependence of the voltammetric current on the charge carried by the electroactive species. According to eq 1, the limiting current is only related to the electron number involved in the reaction, having nothing to do with the charge of redox species. However, the situation would change when the EDL effect arises since the value of D′ would change with the charge number carried by the reactant in addition to the electrode size as indicated by eqs 4 and 5. Furthermore, opposite EDL effects are expected for anion and cation reactants. The anion reduction (or cation oxidation) will be inhibited while the cation reduction (or anion oxidation) will be enhanced. That is, the limiting current (iL) for cation reduction at a nanosized electrode would be larger than the limiting diffusion current (id) calculated with eq 1, and decreased limiting current would be obtained for anion reduction. Thus, a comparison between the id-normalized limiting current (iL/d ) iL/id) for electroactive ions with opposite charges will be an ideal way to manifest the EDL effect since the deviation is significantly magnified. In Figure 4, the ratio between the value of iL/d for the oneelectron reduction of a highly charged cation, Ru(NH6)3+, and that of a highly charged anion, Fe(CN6)3-, is plotted against the electrode radius. It can be seen that the ratio between the two normalized limiting currents is about 1 for electrodes larger than 20 nm, in accordance with the predication of eq 1,
It should be pointed out that for electrodes with very small size (e.g.,
