Electrochemical Correlation Spectroscopy in Nanofluidic Cavities

Anal. Chem. 2009, 81, 8203–8212

Electrochemical Correlation Spectroscopy in Nanofluidic Cavities Marcel A. G. Zevenbergen, Pradyumna S. Singh, Edgar D. Goluch, Bernhard L. Wolfrum,† and Serge G. Lemay* Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands We introduce both theoretically and experimentally a new electrochemical technique based on measuring the fluctuations of the faradaic current during redox cycling. By analogy with fluorescence correlation spectroscopy (FCS), we refer to this technique as electrochemical correlation spectroscopy (ECS). We first derive an analytical expression of the power spectral density for the fluctuations in a thin-layer-cell geometry. We then show agreement with measurements using ferrocenedimethanol, Fc(MeOH)2, in water and in acetonitrile in microfabricated thinlayer cells with a ∼70 nm electrode spacing. The fluctuation spectra provide detailed information about the adsorption dynamics of Fc(MeOH)2, which cause an apparent slowing of Brownian motion. We furthermore observe high-frequency fluctuations from which we estimate the rates of adsorption and desorption. Conventional electrochemical techniques such as amperometry and cyclic voltammetry are extremely powerful methods for studying redox-active species. These approaches rely on measuring faradaic currents that are proportional to the flux of redox molecules impinging upon a suitably biased electrode. These techniques are macroscopic in the sense that they probe the average behavior of a large number of molecules: the discreteness of the individual molecules contributing to the total current is averaged out by the measurement process. Fluctuations around the mean merely add to the measurement noise and are therefore preferably kept as small as possible. In principle, however, additional microscopic information is contained in these fluctuations. Perhaps the best-known technique to exploit this fact is fluorescence correlation spectroscopy (FCS), which measures the fluctuations in fluorescence that result from freely diffusing molecules entering and leaving a microscopic illuminated volume of solution. FCS yields detailed information about microscopic mass transport, and can be used to determine local concentrations and diffusion coefficients as well as rate constants for reactions involving fluorescent molecules.1-3 Here we extend the concept of FCS to electrochemical measurements. By direct analogy with FCS, we refer to this approach as electrochemical correlation spectroscopy (ECS). This * To whom correspondence should be addressed. E-mail: [email protected]. † Current address: IBN-2, Forschungszentrum Ju ¨ lich GmbH, JARA-FIT, Germany. (1) Magde, D.; Webb, W. W.; Elson, E. Phys. Rev. Lett. 1972, 29, 705–708. (2) Eigen, M.; Rigler, R. Proc. Natl. Acad. Sci. U. S. A. 1994, 91, 5740–5747. (3) Schwille, P. Cell Biochem. Biophys. 2001, 34, 383–408. 10.1021/ac9014885 CCC: $40.75  2009 American Chemical Society Published on Web 09/10/2009

technique is based on measuring the statistical fluctuations in a faradaic current as a function of time. Since the fluctuations reflect underlying fluctuations in the density of redox molecules, they contain additional information that is otherwise inaccessible. Analysis of the spectrum or autocorrelation function of the fluctuations allows probing the dynamics of molecules undergoing independent Brownian motion. While they are in principle always present, faradaic current fluctuations are difficult to observe in traditional electrochemical experiments for two main reasons. First, the magnitude of the statistical fluctuations in the number of molecules present in a given volume scales as 〈N〉, where 〈N〉 is the average number of molecules present. Since the measured signal normally scales as 〈N〉, the relative magnitude of the fluctuations scales as 1/〈N〉 and becomes increasingly irrelevant as the size of the system increases. Second, only one or a few electrons are involved in typical redox reactions. In order to obtain measurably large signals, conventional measurements must therefore necessarily sum over a large number of elementary reaction events, which once again tends to average over fluctuations. Here we introduce an experimental system that circumvents both of these obstacles. Our approach is based on the nanoscale thin-layer cell (TLC) sketched in Figure 1a. This device consists of a thin, solution-filled channel whose floor and roof are composed of separately addressable electrodes. Redox-active molecules are repeatedly oxidized at one electrode and reduced at the other, leading to a factor ∼104 amplification of the contribution of each molecule to the faradaic current. Furthermore, the volume of the active region of the device is only 1 femtoliter, which ensures that fluctuations in N(t) can easily be resolved even at millimolar concentrations using standard bipotentiostats. The technique of ECS introduced here is partly related to electrochemical emission spectroscopy, which probes the properties of a corroding electrode by measuring current or voltage fluctuations. This has proven to be a powerful method for revealing microscopic details of the corrosion process such as uniform or localized corrosion, film rupture, pitting, and gas evolution.4-7 The key difference between the two techniques is that electrochemical emission spectroscopy probes fluctuations in the properties of the electrode, whereas ECS probes equilibrium fluctuations of freely diffusing electroactive species in solution. (4) Bertocci, U.; Gabrielli, C.; Huet, F.; Keddam, M.; Rousseau, P. J. Electrochem. Soc. 1997, 144, 37–43. (5) Gabrielli, C.; Keddam, M. Corrosion 1992, 48, 794–811. (6) Mansfeld, F.; Little, B. Corros. Sci. 1991, 32, 247–272. (7) Cottis, R. A. Corrosion 2001, 57, 265–285.

Analytical Chemistry, Vol. 81, No. 19, October 1, 2009

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Figure 2. Simplified geometry of the cavity used for modeling. Because of symmetry, only half of the channel is considered. It consists of an active region between the electrodes with length La and entrance channels with length Le. Coupling to the bulk reservoir is described by an escape rate κ. pa(x,t) and pe(x,t) are the probability densities in the active region and the entrance channels, respectively.

Figure 1. (a) Illustration of the device concept. Redox-active molecules undergo Brownian motion in a nanofluidic cavity with two parallel embedded electrodes. The two electrodes are biased with respect to a reference electrode immersed in the bulk solution. The molecules are repeatedly and reversibly reduced and oxidized, thereby shuttling multiple electrons between the electrodes. This leads to a giant amplification of the signal per molecule. The cavity is coupled to a bulk reservoir of fluid via two entrance channels and access holes. Therefore, molecules can diffuse freely in and out of the active region between the electrodes, resulting in intrinsic fluctuations in the faradaic current. (b) Scanning electron microscopy image of a device (top view). The active region between the electrodes has a length La ) 10 µm, whereas the entrance channels are Le ) 8 µm. The device is coupled to the bulk reservoir via two 1 × 1 µm access holes. Contacting wires for the top and the bottom electrode can be seen at the top and sides of the image, respectively.

We first theoretically derive the form of the ECS fluctuation spectrum for our thin-layer-cell geometry. We then present ECS measurements using nanofluidic thin-layer cells with a 71 nm spacing between the electrodes. We show that the form of the spectra is consistent with the theoretical predictions. Surprisingly, however, at low concentrations of redox species the diffusive motion of individual redox molecules slows down and additional high-frequency fluctuations appear. We interpret these results in terms of adsorption of the redox species, which cannot be resolved in steady-state measurements. Theory. Here we derive the theoretically predicted fluctuation spectrum for ECS in our devices. We first derive an analytical expression for the spectrum in terms of a single parameter characterizing mass transport between the device and an outside bulk reservoir. We then numerically evaluate the value of this parameter for our device geometry. The fluctuations could alternately be described in terms of autocorrelation functions, as is conventionally done in FCS. We find, however, that the spectral density representation is more analytically tractable in the ECS case due to the sharp boundaries of the active region. a. Analytical Form for the Fluctuation Spectrum. We consider a thin-layer cell with electrode spacing z. The typical time for a redox-active molecule to complete one cycle of the redox cycling process, that is, to diffuse from one electrode to the other and back, is τc ) z2/D where D is the diffusion coefficient.8 Normal measurement electronics lack the resolution to detect these individual electron shuttling events, and only detect the average current due to multiple shuttling events. The corre(8) Fan, F. R. F.; Bard, A. J. Science 1995, 267, 871–874.

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sponding current for one molecule, ip, is given by ip ) ne/τc ) neD/z2, where n is the number of electrons transferred in the redox reaction. When N molecules are present in the active region between the electrodes, the total current is simply Nip. If N varies slowly in time, the total instantaneous measured current, i(t), is thus given by i(t) ) N(t)ip. This simplification is possible so long as N(t) varies on a time scale much longer than the shuttling time τc. In the experiments presented below this condition is easily satisfied since τc is a few µs, whereas we are mostly interested in the range 0-10 Hz and never sample above 1 kHz.9 We aim to calculate the power spectral density, Si( f ), of the faradaic current. We follow the approach of Bezrukov et al.,10 which was originally developed to describe the influence of translocating molecules on the ionic current through ion channels. We employ a one-dimensional geometry model consisting of an active region with length La connected at either end through an entrance channel of length Le to a bulk reservoir, as sketched in Figure 2. Coupling between the entrance channels and the reservoirs is characterized by an escape rate, κ, as discussed below. The current power spectral density can be written as Si(f ) ) 〈N〉ip2SN(f )

(1)

Here 〈N〉 is the average number of molecules in the active region and SN( f ) is the power spectral density of the number of redox active molecules present in the active region, normalized for a single molecule. Equation 1 indicates that the noise power increases linearly with increasing 〈N〉. The frequency dependence of the spectrum is however independent of 〈N〉, as expected for an ensemble of molecules undergoing independent random walks. The Wiener-Khintchine theorem10 relates the power spectral density to its autocorrelation function, C(t): SN(f ) ) 4

∫

∞

0

C(t) cos(2πf t)dt

(2)

The autocorrelation function is itself defined as the integral over the molecule probability density in the active region, pa(x,t): C(t) )

∫

0

-La/2

pa(x, t)dx

(3)

(9) The discreteness of the electronic charge leads to additional shot noise. This contribution to the fluctuations is however negligible compared to diffusion noise at the low frequencies investigated here. (10) Bezrukov, S. M.; Berezhkovskii, A. M.; Pustovoit, M. A.; Szabo, A. J. Chem. Phys. 2000, 113, 8206–8211.

The probability density is the probability of finding the molecule at position x, subject to the initial condition that the molecule is uniformly distributed in the active region at t ) 0. Evaluating Si( f ) thus reduces to evaluating pa(x,t) and substituting into eqs 1-3. The probability densities pa(x,t) in the active region -La/2 < x < 0 and pe(x,t) in the entrance channel 0 < x < Le obey the one-dimensional diffusion equation, ∂2pa,e ∂pa,e )D 2 ∂t ∂x

(4)

with initial conditions pa(x,0) ) 2/La and pe(x,0) ) 0. Because of symmetry, the outward flux at x ) -La/2 is 0. Coupling to the reservoir is represented by a radiation boundary condition at the entrance of the channel in which the rate for leaving the channel is characterized by an escape rate κ:11 D

∂pa ∂x

|

x)-La/2

) 0; D

∂pe ∂x

|

x)Le

) -κpe(Le, t)

(5)

In addition, the probability density and the flux are continuous at the boundary between the active region and the entrance channel: pa(0, t) ) pe(0, t);

∂pa ∂x

|

) x)0

∂pe ∂x

|

(6)

x)0

Figure 3. Power spectral densities predicted by eq 10 for κ ranging from 0 to ∞. Depending on κ, the solution exhibits different plateaus at low frequencies. For κ f 0 the height of the plateau S0 is given by eq 13, whereas for κ f ∞, S0 is given by eq 17. At high frequencies all the curves exhibit a f -3/2 power-law decay which is characteristic for diffusive fluctuations. The red curve is the relevant limit for the experiments.

The power spectral density can then be calculated12 using SN( f ) ˆ (s)}, as derived from eqs 1 and 2, to yield ) 4Re{C

Si(f ) )

〈N〉ip2 La( πf )3/2

Taking the initial conditions into account, the temporal Laplace transform (s ) 2πif ) of eq 4 is given by ∂2pâ ∂2pê 2 spâ ) D 2 ; spê ) D 2 La ∂x ∂x

pâ(x, s) )

[

2 1sLa

]

1 cosh β x + La [κ cosh(βLe) + βD sinh(βLe)] 2 1 1 κ cosh β Le + La + βD sinh β Le + La 2 2 (8)

(( ((

))

))

((

))

where β ) (s/D)1/2. Integration over the area between the electrodes yields the Laplace transform of the autocorrelation function, ˆ (s) ) C

)

∫

0

-La/2

pâ(x, s)dx

[ (

1 1s

. 1 Laβ 2 Laβ

( )

2 sinh

)

×

κ cosh(βLe) + βD sinh(βLe) 1 1 κ cosh β Le + La + βD sinh β Le + La 2 2

((

))

((

1 [D3/2πfF(θ, aθ, bθ) + κ2√DG(θ, aθ, bθ) + 2

√πfDκH(θ, aθ, bθ)]/[πfD(cosh θ - cos θ) + 1 2 κ (cosh θ + cos θ) + √πfDκ(sinh θ - sin θ)] (10) 2

(7)

The solutions to above equations are simple exponents; the solution satisfying the boundary conditions specified by eqs 5 and 6 is

))

]

×

where F(θ, aθ, bθ) ) sinh θ + sin θ - sin aθ cosh bθ sin bθ cosh aθ - sinh bθ cos aθ - sinh aθ cos bθ G(θ, aθ, bθ) ) sinh θ - sin θ - sin aθ cosh bθ + sin bθ cosh aθ - sinh bθ cos aθ + sinh aθ cos bθ H(θ, aθ, bθ) ) cosh θ - cos θ + cosh aθ cos bθ cos aθ cosh bθ - sin aθ sinh bθ - sinh aθ sin bθ θ)

πfD (2L + L ); a ) 2L + L ; b ) 2L + L 2Le

La

e

a

e

a

e

a

(11) Figure 3 shows the power spectral density Si( f ) predicted by eqs 10 and 11 for values of the escape rate κ ranging between 0 and ∞ for the channel geometry used in the experiments (La ) 10 µm, Le ) 8 µm, z ) 70 nm). Below we describe in further detail several limits of interest. At high frequencies, as specified below for each case, eq 10 simplifies to a characteristic “diffusion tail” for all values of κ:

Si(f f ∞) ) 〈N〉ip2

√D La(πf )3/2

(12)

(9)

(11) We use the term “escape rate” for consistency with Bezrukov. In electrochemical parlance, κ is a mass-transfer coefficient.

ˆ (s) was evaluated using MapleSoft Maple 12. (12) The real part of C


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This limit reflects a random walk of particles in the vicinity of the sharp boundary between the active region and the entrance channel, and is independent of details of the device geometry.13 For the case κ ) 0, that is, perfectly reflecting end points or “particles in a box”, the solution presented here simplifies to the expression obtained previously by van Vliet and Fasset.14 This solution features a constant plateau in Si( f ) at low frequencies. The magnitude of this plateau, which we label S0, is given by

S0 )

〈N〉ip2 4LaLe2 D 6Le + 3La

(13)

The crossover from the plateau described by eq 13 to the f -3/2 diffusion tail of eq 12 occurs at a frequency

ftail )

(

D 6Le + 3La π 4La2Le2

)

2/3

(14)

For the case κ > 0, transfer of molecules between the device and the bulk reservoir leads to additional fluctuations at low frequencies compared to the κ ) 0 case, leading to a κ-dependent increase in the low-frequency plateau amplitude S0:

S0 ) 〈N〉ip2

(

2LeLa 2La La2 + + 3D D κ

)

(15)

At high frequencies, f > ftail, the universal “diffusion tail” is recovered. At intermediate frequencies, Si(f ) exhibits a complex form characterized by two plateaus corresponding to the time scales for diffusion inside the device and to the bulk reservoir. In the limit κ f ∞, however, the two plateaus merge into a single one. For this limiting case, which is reached when κ . D/(Le + La/6), Si( f ) is well-described by the simpler form that interpolates between S0 and the diffusion tail,

Si(f, κ f ∞) )

S0 1 + (f/f0)3/2

(16)

with eq 15 reducing to

S0(κ f ∞) )

〈N〉ip2 2 (L + 6LeLa) 3D a

(17)

In this limit the crossover frequency from a flat plateau S0 to a f -3/2 tail occurs at frequency f0 (with f0 < ftail) given by f0 )

(

3 D π L2(L + 6L ) a a e

)

2/3

(κ f ∞)

(18)

As will be shown in the next section, the κ f ∞ limit is the relevant limit for our experiments. (13) The parameter La only enters in the form 〈N〉/La, which is simply the linear concentration of molecules at this interface between the active region and the entrance channel. (14) Burgess, R. E. Fluctuation Phenomena in Solids; Academic Press: New York, 1965.

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We also note that in the limit Le f 0, eqs 10 and 17 reduce to the expressions obtained by Bezrukov and co-workers10 for this case,

Si(f ) )

〈N〉ip2 La(πf )3/2

× [D3/2πfκ2√D(sinh θ - sin θ) +

2√πfDκ(cosh θ - cos θ)]/[πfD(cosh θ - cos θ) + 1 2 κ (cosh θ + cos θ) + √πfDκ(sinh θ - sin θ)] (19) 2 and

S0 ) 〈N〉ip2

(

2La La2 + 3D κ

)

(20)

Summarizing, eq 10 provides a general analytical expression for the faradaic current fluctuations arising from the finite number of molecules in a one-dimensional channel coupled to a reservoir. In the experimentally relevant limit κ f ∞, the power spectral density takes the simple form of eq 16 and consists of a plateau at low frequencies and a f -3/2 dependence at high frequencies. b. Finite-Element Calculations of the Escape Rate K. The value of the escape rate κ can be determined from the selfconsistency condition that the flux of molecules exiting the device in the analytical one-dimensional model, jout ) κpe(Le), must correspond to the solution of the mass-transport problem for the three-dimensional geometry of the 1 × 1 µm × 550 nm access hole connecting the actual device to the external reservoir. To determine the value of κ, we numerically solved the time-independent diffusion equation, ∇2c ) 0, using commercial finite-element software (Comsol Multiphysics 3.4). The geometry employed in the calculations, shown in Figure 4a, represents the 3.5 µm long and 2 µm wide section of the channel in the vicinity of the access holes. The channel height was 71 nm, corresponding to the experiments described later in this article. The bulk reservoir of fluid was simulated by a 5 µm radius hemisphere with a fixed concentration of zero imposed on its boundary. The entrance to the channel was held at a fixed boundary concentration c0 ) 1 mM to represent molecules exiting the channel. The calculated concentration profile is shown in Figure 4a and a magnification of the area near the access hole is shown in Figure 4b. The dashed area in Figure 4b represents the entrance channel, which is described by the one-dimensional analytical model of the previous section. The calculated flux corresponding to Figure 4 was 2.5 × 10-16 mol/s for a value of the diffusion coefficient of D ) 6.7 × 10-10 m2/s, which corresponds to Fc(MeOH)2 in water. This flux can be converted into κ of the one-dimensional model by dividing by c0 × Ac, where Ac is the cross-sectional area of the channel, 71 nm × 2 µm, resulting in κ ) 1.8 × 10-3 m/s. This is 30 times greater than the quantity D/(Le + La/6) ) 6 × 10-5 m/s, indicating that for our device geometry the κ f ∞ limit (eqs 16-18) applies. The same conclusion holds for Fc(MeOH)2 in acetonitrile (ACN); here κ ) 4.8 × 10-3 m/s, reflecting the 2.7× higher value of D for this case. The above calculation assumes a uniform cross-sectional flux of molecules and ignores the slight fringing of the concentration profile near the side wall. Since the value of κ is well into the

Figure 4. Steady-state concentration profiles determined by finiteelement calculations. The white dashed regions represents the actual channel which was modeled analytically in the previous section. (a) Simulated geometry of the access hole (black lines) and cross-section of the three-dimensional concentration profile (color scale). The entrance of the channel was held at a concentration c0 ) 1 mM (blue) and the outer boundary of the hemisphere was held at c0 ) 0 mM (red) to represent the bulk reservoir. (b) Magnification of the concentration profile near the entrance hole. Only ∼3 µm of the 33 µm long channel can be seen. The yellow channel exiting on the right is a dead volume also present in the real fabricated devices.

limit of perfect absorption (κ f ∞) and thus insensitive to the exact value of κ, this slight fringing is expected to be inconsequential. In our procedure, the time-dependent probability density in the channel is matched to the steady-state concentration profile in the access hole; eq 10 is therefore only valid on time scales longer than the time to reach steady-state in the access hole. As discussed by Bezrukov et al.,10 for shorter time scales the escape rate becomes time dependent, κ(t). Steady-state in the access hole is reached in a time ∼h2/D where h ) 550 nm the height of the access hole. In the frequency domain, eq 10 is thus valid in water for f < D/2πh2 ) 350 Hz and in ACN for f < 940 Hz.10 The bulk of our data presented below are in the range f < 10 Hz, for which this approximation holds very well. For the frequency range investigated in Figure 8, where the upper limit is 1 kHz and the solvent is ACN, this may introduce a slight error in the predicted curve for the highest frequencies. EXPERIMENTAL SECTION A detailed description of the device fabrication was given elsewhere.15-17 In short, a multilayer Pt/Cr/Pt structure was deposited and patterned on an insulating substrate and passivated using SiO2. Two 1 µm2 access holes were etched through the SiO2 capping layer to the Cr layer. In a final step the Cr layer (15) Wolfrum, B.; Zevenbergen, M. A. G.; Lemay, S. G. Anal. Chem. 2008, 80, 972–977.

was etched, creating a thin cavity for fluid between the Pt electrodes. The device is a direct implementation of the geometry of Figure 2 with La ) 10 µm and Le ) 8 µm. A scanning electron microscopy image of a finished device is shown in Figure 1b. After the chromium etch, the devices were rinsed extensively with deionized water and filled with 0.5 M H2SO4. To ensure clean electrode surfaces, the potential of both electrodes was swept between -0.2 and 1.2 V vs Ag/AgCl and stable voltammograms were obtained showing the characteristic Pt electrochemistry in sulphuric acid.17 Ferrocenedimethanol, Fc(MeOH)2, was purchased from Acros (cat. no. 382250010), hexaamineruthenium trichloride, Ru(NH3)63+, from Johnson Matthey (cat. no. 190245), the supporting electrolyte tetrabutylammonium hexafluorophosphate, TBAPF6, from Fluka (cat. no. 86879), potassium chloride, KCl, from Sigma-Aldrich (cat. no. P3911), chromium etchant (Selectipur) from BASF and sulphuric acid, H2SO4, from J. T. Baker (cat. no. 6057). All solutions were prepared in 18.2 MΩ cm deionized water (Millipore) or HPLC-grade acetonitrile (ACN, Sigma Aldrich, cat. no. 34998). A bulk reservoir was formed using a tapered PDMS cell. This cell had a ∼200 µm opening at the bottom to make contact between the device and the electrolyte, and a 0.6 cm opening at the top in which a reference electrode was positioned. This electrode was an aqueous Ag/AgCl reference (3 M NaCl, RE-6, BASi) or a nonaqueous Ag/Ag+ reference (Ag wire in contact with 0.1 M TBAPF6 and 10 mM AgNO3, MF-2062, BASi) for measurements in water and in ACN, respectively. A commercial bipotentiostat (CHI832b, CH Instruments) was used for measurements. RESULTS Cyclic Voltammetry. Figure 5a shows cyclic voltammograms for two typical devices. The first measurement was in aqueous solution with 1.2 mM Fc(MeOH)2, 2 M KCl as supporting electrolyte, and 5 mM H2SO4 to prevent the electrodes from fouling. The second measurement was performed in acetonitrile with 1.0 mM Fc(MeOH)2 and 0.1 M TBAPF6 as supporting electrolyte. The potential of the top electrode, Et, was swept between 0.05 and 0.45 V for the voltammograms in water (black curves) and between -0.1 and 0.4 V for ACN (red curves). For both cases, the potential of the bottom electrode was held at a reducing potential (Eb ) -0.1 V for ACN and Eb ) 0.05 V for water). In water, at high oxidizing overpotential applied to the top electrode, the current reached a steady-state diffusion-limited plateau ilim. The current through the bottom electrode, ib, which was recorded simultaneously, was essentially the exact inverse of the current through the top electrode, it, as expected for redox cycling. In our geometry, the diffusion profile is well approximated by a one-dimensional diffusion profile in the direction perpendicular to the plane of the electrodes,17 yielding

ilim ) 〈N〉ip )

nFADcB z

(21)

Here F is Faraday’s constant, A is the area of overlap between the electrodes (15 µm2), and cB the bulk concentration. The Analytical Chemistry, Vol. 81, No. 19, October 1, 2009

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Figure 5. (a) Cyclic voltammograms (scan rate 10 mV/s) for two devices filled with 1.2 mM Fc(MeOH)2 and 2 M KCl in water (black curves) and 1.0 mM Fc(MeOH)2 and 0.1 M TBAPF6 in ACN (red curves). The potential of the top electrode, Et, was cycled while the potential of the bottom electrode, Eb, was held at a reducing potential (Eb ) 0.05 V in water, Eb ) -0.1 V in ACN). The current through the top, it electrode was measured simultaneously with the current through the bottom electrode, ib. The black and red dotted lines are fits to the Nernst and Butler-Volmer equation, respectively. The potentials are with respect to the Ag/AgCl and Ag/Ag+ reference electrodes in water and ACN, respectively. (b) Current versus bulk concentration of Fc(MeOH)2. For water the fitted diffusion-limited current, ilim, is plotted (black squares); for ACN the current at Et ) 0.4 V is plotted (red circles). The solid lines are fits to lines with slope 1.

height of the plateau was determined by fitting the voltammogram with the Nernst equation with ilim and the formal potential E0′ as fitting parameters (black dotted line, fit indistinguishable from the data in Figure 5a). The fitted value E0′ ) 0.242 V in water was in exact agreement with the formal potential obtained from a voltammogram recorded with a commercial 5 µm radius Pt disk (BASi) and the same Ag/AgCl reference electrode. The voltammograms in ACN exhibited higher current levels, as expected since the diffusion coefficient of Fc(MeOH)2 is 2.7 times higher in ACN (D ) 6.7 × 10-10 m2/s for water and D ) 1.8 × 10-9 m2/s for ACN). Unexpectedly, however, the current did not reach a steady-state diffusion-limited plateau at high overpotential. Instead, a linear increase of the current with applied potential was observed in this regime. We can rule out ohmic drops as the origin of this behavior since the conductivity of the 0.1 M TBAPF6 supporting electrolyte (9.2 mS/cm18) is of the same order as used previously and we have shown that ohmic drops are negligible under these conditions.17 Another possible origin for the linear current dependence at high overpotential in ACN is that the current might still be electrontransfer limited due to the high diffusion coefficient. We therefore fitted the voltammograms with the Butler-Volmer (16) Goluch, E. D.; Wolfrum, B.; Singh, P. S.; Zevenbergen, M. A. G.; Lemay, S. G. Anal. Bioanal. Chem. 2009, 394, 447–456. (17) Zevenbergen, M. A. G.; Wolfrum, B.; Goluch, E. D.; Singh, P. S.; Lemay, S. G. J. Am. Chem. Soc. 2009, 131, 11471–11477. (18) Bowyer, W. J.; Engelman, E. E.; Evans, D. H. J. Electroanal. Chem. 1989, 262, 67–82.

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equation with ilim, the transfer coefficient R, and the heterogeneous rate constant k0 as fitting parameters (red dotted line in Figure 5a).17 The formal potential E0′ ) 0.082 V was independently determined by fitting the perfectly Nernstian voltammogram recorded with a standard Pt ultramicroelectrode and the same nonaqueous Ag/Ag+ reference electrode. A good fit was obtained with a value k0 ) 6.6 ± 0.8 cm/s, which is consistent with previously reported values.17,19,20 The fitted value for the transfer coefficient, R ) 0.93 ± 0.02, was however unusually high.21 Importantly, the shape of the voltammogram does not affect the analysis of the fluctuations presented below. In Figure 5b, the diffusion-limited current ilim is shown as a function of cB. In water, for each investigated concentration, four or more pairs of forward and backward traces of the voltammgram were fitted with the Nernst equation. The mean ilim from these traces is shown in Figure 5b (black squares) while the error bars represent the standard deviation for the four fits. ilim scales linearly with cB over three decades. From the linear fit (solid black line) and eq 21 we determined the height of the cavity to be z ) 71 ± 7 nm. Since in ACN no true steadystate diffusion-limited current was obtained, an accurate determination of the device height could not be made. The range of currents observed in the plateau region of the voltammogram translates into estimates of the cavity height ranging between z ) 54 ± 7 nm and z ) 76 ± 7 nm. Since all devices were fabricated simultaneously from the same wafer, we use z ) 71 nm obtained in water as device height. For the analysis of the fluctuations discussed below, the current value at 0.4 V vs Ag/AgCl was used for ilim since this was the potential applied to the top electrode during the amperometric measurements. For the smallest concentration investigated, there were approximately 600 molecules present in solution in the active region of the devices. Current Fluctuations. The current is proportional to the number of redox-active molecules in the active region of the device that are free to diffuse and shuttle electrons, N. Since the active region is connected to a reservoir, however, N(t) fluctuates around the average value 〈N〉. To observe these fluctuations, we performed amperometric measurements under redox cycling conditions; typical time traces are shown in Figure 6. The black lines represent simultaneously measured currents in water for the top electrode, it(t) (Et ) 0.45 V, Eb ) 0.05 V), and the bottom electrode, ib(t), while the red lines were recorded in ACN (Et ) 0.4 V, Eb ) -0.1 V). For both cases, the fluctuations in it(t) are perfectly anticorrelated with fluctuations in ib(t), a signature of the redox cycling process. This demonstrates that the observed fluctuations are not due to background instrumental noise, but rather originate from the faradaic process. In Figure 6b the corresponding power spectral densities (PSDs) for the time traces in (a) are shown together with the power spectral density of a time trace recorded in water for which (19) Sun, P.; Mirkin, M. V. Anal. Chem. 2006, 78, 6526–6534. (20) Velmurugan, J.; Sun, P.; Mirkin, M. V. J. Phys. Chem. C 2009, 113, 459– 464. (21) During extended measurements, we also observed a slight increase of the current due to evaporation of the solvent. This cannot account for the nonideal shape of the voltammograms, however, since this shape was essentially identical during subsequent forward and backward sweeps. Furthermore, the time scale for evaporation (0.5% current increase per 100 s, the time to record one voltammogram) was much slower compared to the observed current increase of 4% between 0.3 and 0.4 V (10 s).

Figure 6. (a) Amperometric (current-time) traces recorded under redox cycling conditions (black curves 1.2 mM Fc(MeOH)2 in water, Et ) 0.45 V, Eb ) 0.05 V; red curves 1.0 mM Fc(MeOH)2 in ACN, Et ) 0.4 V, Eb ) -0.1 V) revealing fluctuations in the faradaic current due to fluctuations in the average number of molecules in the channel. Fluctuations in the current through the top electrode, it(t), and the bottom electrode, ib(t), are perfectly anticorrelated, as expected. The curves are offset for clarity. (b) Power spectral densities (PSDs) for the time traces in (a) showing the characteristics for diffusion noise: a plateau S0 at low frequencies and a power-law decay with slope -1.5 at high frequencies. A PSD in the absence of redox cycling is also shown (gray squares) that characterizes the background noise of the electronics. The solid lines are fits to eq 16.

Figure 7. Power spectral densities for various concentrations of Fc(MeOH)2 recorded in water (a) and acetonitrile (b). The solid green lines are drawn corresponding to eq 10 without a fitting parameter. The solid orange lines correspond to a smaller effective diffusion coefficient in the channel, Deff, which was obtained separately for each curve by fitting the crossover frequency f0, as discussed in the text. Deff is shown as a function of concentration Fc(MeOH)2 in water (c) and ACN (d). For both cases, Deff is smaller than the bulk diffusion coefficient (D ) 6.7 × 10-10 m2/s for water and D ) 1.8 × 10-9 m2/s for ACN).

Et ) Eb ) 0.05 V (no redox cycling, gray squares).22 The latter characterizes the background noise of the measurement electronics in the absence of redox cycling and is, in the frequency range investigated, two or more orders of magnitude smaller than the corresponding PSDs in redox cycling mode. The form of the PSDs exhibits the characteristics of diffusion noise: a plateau Si( f ) ) S0 at low frequencies and a power-law decay Si( f ) ∝ 1/f 3/2 at high frequencies. Fitting Si( f ) with the expression valid in the κ f ∞ limit, eq 16, yields excellent agreement (solid blue lines). Figure 7 shows PSDs recorded both in water and ACN for different concentrations of Fc(MeOH)2. All the PSDs show a plateau S0 at low frequency and the f -3/2 power-law decay at higher frequencies, as predicted by theory. Surprisingly, however, the crossover frequency, f0, decreases gradually from 0.6 Hz at the highest concentrations investigated to 0.1 Hz at the lowest concentrations, both in water and in ACN. Furthermore, at low concentrations (1.2 µM in water and below 1 mM in ACN), a second plateau starts to develop above 1 Hz. To further investigate the second plateau in Figure 7, we performed additional experiments at higher frequencies. Figure 8 shows PSDs for 1 mM and 10 µM Fc(MeOH)2 in ACN up to 1 kHz. The spikes at 50 Hz and higher harmonics are due to power-line interference and are characteristic for the CHI

potentiostat employed.23 The 10 µM PSD (green squares) reveals that the second plateau is part of a Lorentzian-like spectrum with a power-law decay having an exponent near -2 at high frequencies (the actual exponent of the power-law decay is -2.4). At 1 mM Fc(MeOH)2, this plateau has shifted to slightly higher frequencies. The plateau manifests itself only for the lower concentrations of Fc(MeOH)2 in the data of Figure 7 because for the higher concentrations (>12 µM in water and 1.0 mM in ACN), the second plateau remains masked by the f -3/2 power-law decay. Additional measurements were also performed on Ru(NH3)63+ in water. These measurements exhibited the same trends as

(22) A background PSD recorded in ACN (Et ) Eb ) -0.1V) was indistinguishable from the background PSD in Figure 6.

(23) These spikes are also observed with no electrode connected, indicating that the interference occurs internally in the potentiostat.

Figure 8. Power spectral densities for 1 mM (black squares) and 10 µM (green squares) Fc(MeOH)2 in ACN up to 1 kHz. The second plateau in the spectrum is part of a Lorentzian type spectrum (the solid orange lines are fits to eq 24 that consists of a diffusive part and a Lorentzian showing approximate agreement).


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the data of Figure 7, but the shift of f0 was so pronounced that the plateau at low frequencies could no longer be resolved, precluding further analysis as discussed below. These additional measurements are presented in the Supporting Information. DISCUSSION The data of Figure 6 indicate that, at the highest concentrations investigated, the measured power spectral densities Si( f ) follow the universal f -3/2 form for diffusion near a sharp boundary. This form has been predicted by several authors in different limits,10,14,24-27 and has been reported previously in light scattering experiments on colloidal suspensions,28 in electrolyte-filled capillaries,29-31 in the electronic current through thin films,32-34 in GaAs field-effect transistors,,35 and by the present authors in an earlier study of redox cycling in fluidic devices.36 Figure 7a and b show that, surprisingly, the shape of the spectra depends on concentration, and in particular that the value of the frequency f0 at which Si(f ) crosses over from a flat plateau to a f -3/2 decay, depends on the concentration of redox species. This behavior is completely unexpected if the redox molecules are assumed to diffuse independently. In this case changes in concentration are predicted to lead to changes in the overall magnitude of the fluctuations (a vertical shift in the curves in Figure 7a and b), but the frequency dependence is expected to remain identical. A dependence of the shape of Si(f ) on concentration therefore indicates an unexpected form of interactions between redox molecules. In order to better understand this behavior, we first compare the measured power spectral densities Si( f ) with the theoretical model presented in the Theory section. The solid green lines in Figure 7a and b represent the predictions of eq 10 for the highest concentration measured. These curves employ no fitting parameter, instead using the known device geometry and the numerically determined values of the escape rate (κ ) 1.8 × 10-3 m/s for water and κ ) 4.8 × 10-3 m/s for ACN). The model correctly predicts the magnitude of the low-frequency plateau S0, but it overestimates the crossover to the f -3/2 regime compared to the experimental data. Specifically, the model predicts f0 ) 1.4 Hz for water and 3.7 Hz for ACN, which contradicts the fitted values of f0 ) 0.58 Hz for water and 0.66 Hz for ACN. Redoxactive molecules thus appear to move slower in the thin-layer cells than expected from the diffusion equation. (24) (25) (26) (27) (28) (29) (30)

(31) (32) (33) (34) (35) (36)

Lax, M.; Mengert, P. J. Phys. Chem. Solids 1960, 14, 248–267. MacFarlane, G. G. Proc. Phys. Soc. B 1950, 63, 807–814. Richardson, J. M. Bell System Tech. J. 1950, 29, 117–141. Berezhkovskii, A. M.; Pustovoit, M. A.; Bezrukov, S. M. J. Chem. Phys. 2002, 116, 6216–6220. Voss, R. F.; Clarke, J. J. Phys. A: Math. Gen. 1976, 9, 561–571. Green, M. E. J. Membr. Biol. 1976, 28, 181–186. van den Berg, R. J.; de Vos, A.; van der Boog, P.; de Goede, J. Resistivity fluctuations in ionic solutions. Proceedings of the 8th International Conference on Noise in Physical Systems and 1/f Noise, xv, 1986; pp 213-216. Weissman, M.; Feher, G. J. Chem. Phys. 1975, 63, 586–587. Scofield, J. H.; Webb, W. W. Phys. Rev. Lett. 1985, 54, 353–356. Rogers, C. T.; Myers, K. E.; Eckstein, J. N.; Bozovic, I. Phys. Rev. Lett. 1992, 69, 160–163. Otten, F.; Kish, L. B.-G.; Granqvist, C.; Vandamme, L. K. J.; Vajtai, R.; Kruis, F. E.; Fissan, H. Appl. Phys. Lett. 2000, 77, 3421–3422. Duh, K. H.; Zhu, X. C.; van der Ziel, A. IEEE Electron. Device Lett. 1984, 5, 202–204. Zevenbergen, M. A. G.; Krapf, D.; Zuiddam, M. R.; Lemay, S. G. Nano Lett. 2007, 7, 384–388.

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Empirically, the data can still be described well by eq 10 if we assume that the diffusion coefficient in the channel, Deff, is reduced to a value below its bulk value; the only other parameters that influence f0 are the devices dimensions La and Le, but these are independently known and are of course independent of concentration. From the fitted value of f0 and eq 18 we estimate Deff ) 2.8 × 10-10 m2/s for water and Deff ) 3.2 × 10-10 m2/s for ACN, which is 2.4× and 5.6× smaller than the bulk diffusion coefficient, respectively. The solid orange lines in Figure 7 represent the prediction of eq 10 with the sole free parameter Deff obtained separately for each concentration by fitting f0 and using eq 18. Figure 7c and d show the values of Deff obtained through this procedure as a function of cB in water and ACN, respectively. The theoretical model describes very well the complete PSDs recorded in water down to 12 µM and the 1 mM PSD in ACN. It also describes the low-frequency parts of the PSDs up to 1 Hz at lower concentrations in both solvents. In these cases however, a second plateau starts to develop above 1 Hz, as noted above. We propose that these phenomena are caused by dynamic adsorption and desorption of the molecules to the electrodes. Such adsorption is expected to have two effects on the PSD. First, it slows down the motion of molecules along the channel since each molecule spends some time on the surface rather than freely diffusing. Second, it introduces extra fluctuations at frequencies corresponding to the adsorption/desorption rate. The shape of the spectrum becomes dependent on concentration since the degree of adsorption depends on the concentration of adsorbing species present in the bulk. Although we have used a different derivative in the experiments presented here, adsorption of ferrocene onto platinum and other electrodes has been reported previously, both in acetonitrile37,38 and in aqueous solutions,39 and the extent of adsorption was found to depend on solvent, electrode material, and composition of the supporting electrolyte. It is often ignored in electrochemical studies, however, as it has minimal impact on steady-state voltammograms. Its direct influence on the fluctuation spectra thus provides a valuable additional means of studying adsorption. We initially concentrate on the first effect, the slowing down of diffusion along the channel. Adsorption causes the apparent D to decrease because intermittent adsorption effectively slows down the diffusive random walk undergone by each molecule. The underlying assumption is that molecules adsorbed on the surfaces are immobile or diffuse much more slowly than in bulk solution. The slowing down of diffusion due to adsorption can be simply described in the limit where the adsorption and desorption times are much shorter than the typical time for diffusing through the device. In this limit the actual adsorption and desorption events are averaged, and diffusion is slowed down by a factor corre-

(37) Rivera, I. M.; Cabrera, C. R. J. Electrochem. Soc. 1993, 140, L36–L38. (38) Daschbach, J.; Blackwood, D.; Pons, J. W.; Pons, S. J. Electroanal. Chem. 1987, 237, 269–273. (39) Bond, A. M.; McLennan, E. A.; Stojanovic, R. S.; Thomas, F. G. Anal. Chem. 1987, 59, 2853–2860.

sponding to the fraction of the time that each molecule spends adsorbed on the surface:40

Deff ) D

cB cB + 2Γ/z

(22)

Here Γ is the surface concentration of adsorbates; the factor of 2 comes from assuming that the molecules can adsorb on both electrodes.41 Γ is itself a function of cB since the surface is in diffusive equilibrium with the solution. Such a model has also been introduced recently to describe the diffusion of proteins in glass nanofluidic channels probed by FCS.42 Equation 22 directly shows how adsorption becomes increasingly relevant as the channel height z is reduced and the surface-to-volume ratio increases.43 A consequence of adsorption is that the total number of molecules present in the active region, 〈Ntot〉, is greater than the number expected from the bulk concentration, 〈N〉 ) NAzAcB. Here NA is Avogadro’s number. A number of additional molecules, 〈Na〉 ) 2NAΓA, is adsorbed on the surface, yielding 〈Ntot〉 ) 〈N〉 + 〈Na〉 ) (D/Deff)〈N〉. Each molecule only contributes to mass transport while it is desorbed; hence each molecule contributes a current ip,eff ) ip(〈N〉/〈Ntot〉) ) ip(Deff/ D). The steady-state diffusion-limited current is thus unaffected by adsorption since ilim ) 〈Ntot〉 × ip,eff ) 〈N〉 × ip. In particular, eq 21, from which the height of the cavity was deduced, remains valid in the presence of adsorption.44 While it does not affect the steady-state diffusion-limited current, adsorption is detectable by chronoamperometry in some cases. Unwin et al. demonstrated the existence of slow transients in chronoamperometric measurements using ferrocene and a scanning electrochemical microscope in positive feedback mode.45,46 These authors could explain their results by assuming a 20% difference of the (effective) diffusion coefficient between oxidized and reduced ferrocene, and noted that adsorption of the oxidized species could also be responsible for the observed transient. Those data are consistent with our observations, but since the electrode spacing in our devices is ∼20× smaller, the effect of adsorption on Deff is far more pronounced here. (40) Deff/D ) τd/(τa + τd) which reflects the average time a molecule spends adsorbed, τa, and desorbed, τd. This is equal to the ratio 〈N〉/(〈N〉 + 〈Na〉) where 〈N〉 and 〈Na〉 are the numbers of desorbed and adsorbed molecules, respectively. Substituting 〈N〉 ) NAAzcB and 〈Na〉 ) 2NAAΓ, where NA is Avogadro’s number, and dividing the numerator and denominator by NAAz yields equation 22. A more sophisticated way of modeling the role of adsorption is for example via the McNabb and Foster equation,56 a modified diffusion equation that takes adsorption explicitly into account via trapping and release rates. However the resulting equations can be solved analytically only for a limited set of boundary conditions. For simplicity we therefore employ the model of eq 22. (41) We make this assumption for simplicity; a more detailed model of potentialdependent adsorption could easily be introduced at this stage. (42) Durand, N. F.; Dellagiacoma, C.; Goetschmann, R.; Bertsch, A.; Ma¨rki, I.; Lasser, T.; Renaud, P. Anal. Chem. 2009, 81, 5407–5412. (43) This explains why the influence of adsorption on the fluctuations was not as pronounced in our earlier devices with z ≈ 300 nm.36 (44) An unbound molecule traverses the cavity in a time given by the bulk diffusion coefficient. If this time becomes comparable to the electron-transfer rate, the total current can still be electron-transfer limited. Our analysis of electron-transfer kinetics17 is therefore also unaffected. (45) Martin, R. D.; Unwin, P. R. Anal. Chem. 1998, 70, 276–284. (46) Martin, R. D.; Unwin, P. R. J. Electroanal. Chem. 1997, 439, 123–136.

Figure 9. Surface concentration of adsorbed Fc(MeOH)2, Γ, as a function of bulk concentration, cB, for water (black squares) and ACN (red circles). The solid lines are fits to the Freundlich isotherm, eq 23.

The surface concentration of adsorbates, Γ, can be deduced using eq 22 and the values of Deff determined from fitting the PSDs. In Figure 9, Γ is shown as a function of cB both in water (black squares) as in ACN (red circles). Γ increases monotonically with cB in both cases. The Freundlich adsorption isotherm47 describes adsorption to a so-called heterogeneous surface for which the affinity varies among sites and is given by

()

Γ ) Kf

cB c0

n

(23)

where Kf is the Freundlich adsorption coefficient, c0 a reference concentration of 1 mM and n the Freundlich exponent. The solid lines in Figure 9 are fits to eq 23 showing good agreement with the experimental data. These fits yield Kf ) 4.9 × 10-8 mol/ m2, n ) 0.72 for water and Kf ) 1.7 × 10-7 mol/m2, n ) 0.62 for ACN. Bond and co-workers39 estimated the surface coverage for a monolayer ferrocene to be 4.6 × 10-6 mol/m2 based on crystallography data on the area of the C5H5 ring.48 The measured Γ in Figure 9 is therefore considerably less than a monolayer coverage, even at the highest concentration investigated in water that is near the solubility limit of Fc(MeOH)2. These findings are comparable with the surface coverage for ferrocene determined by Bond and co-workers,39 who found that Γ varied between 1.6 × 10-7 mol/m2 and 9.7 × 10-7 mol/ m2 in various supporting electrolytes for glassy carbon and Au electrodes. The large charging current for Pt electrodes prevented Bond et al. from obtaining surface coverage for this type of electrode; since ECS measurements are performed in the steady state, however, interference from charging currents is circumvented here. We now turn to the rapid fluctuations that are expected to arise from individual adsorption/desorption events. These fluctuations then cause the current generated by each molecule in the active region to oscillate between a value of ip when the molecule is desorbed and 0 when the molecule is adsorbed. This causes (47) Lyklema, J. Fundamentals of Interface and Colloid Science Volume 1: Fundamentals; Academic Press: London, 1991. (48) We note that the surface area covered by Fc(MeOH)2 is slightly bigger because of the two methanol groups. (49) Nestorovich, E. M.; Danelon, C.; Winterhalter, M.; Bezrukov, S. M. Proc. Natl. Acad. Sci. U. S. A. 2002, 99, 9789–9794. (50) Zara, A. J.; Machado, S. S.; Bulhoes, L. O. S.; Benedetti, A. V.; Rabockai, T. J. Electroanal. Chem. 1987, 221, 165–174. (51) Bashkin, J. K.; Kinlen, P. J. Inorg. Chem. 1990, 29, 4507–4509. (52) Fehlhammer, W. P.; Moinet, C. J. Electroanal. Chem. 1983, 158, 187–191.


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additional fluctuations compared to the predictions of eq 10-11, where it was assumed that each molecule contributes a constant current ip when it is in the active region. Adsorption and desorption of molecules is most simply described as binding to independent sites via a two-state Markov process.49 Adsorption in such a model is characterized by two parameters, τa and τd, that correspond to the average residence time adsorbed on the surface and in the desorbed state, respectively. The ratio of these parameters specifies the fraction of molecules that are adsorbed at any given time and is related to the previously defined quantities by τa/τd ) 〈Na〉/〈N〉 ) D/Deff - 1. The time scale of the fluctuations is set by the absolute magnitude of τa and τd. In the Supporting Information, we derive an approximate theoretical form for the PSD including adsorption (at the level of this two-state model), and show that it is approximately given by Si(f ) )

S0 1 + (f/f0)

3/2

+

S2 1 + (f/f2)2

(24)

Here S2 and f2 are the amplitude and the crossover frequency of a second plateau, respectively. f2 reflects the rate of binding and unbinding, and is given by f2 ) (τa + τd)/2πτaτd. The magnitude of this plateau is given by S2 ) 〈N〉i2p/τd(πf2)2. The solid orange lines in Figure 8 are fits to the form of eq 24 showing approximate agreement with the data. In particular, the model captures the double-plateau structure of the data. From the fitted values of f2 and Deff for the 10 µM spectrum, we extract τd ) 6 ms and τa ) 115 ms. This is still ∼8 times shorter than the typical time a molecule spends in the cavity given by 1/2πf0 ) 0.9 s. This validates the assumption that in the low-frequency regime, the actual adsorption and desorption events are averaged and the spectrum can be described by Deff. The above description of adsorption based on a two-state model is approximate. In particular, the Freundlich isotherm indicates that there exists some degree of heterogeneity among the binding sites, leading to site-dependent values for τd and τa. Therefore, the spectrum cannot be fully characterized by a two-state Markov process as assumed in our simplified model. This is further evidenced by the experimental observations, which show that the form of the PSD (for example the slope of the decay at high frequencies) is itself dependent upon the value of cB. Furthermore, the measured amplitude ratio S2/S0 ) 0.020 is an order of magnitude smaller than the expected ratio of S2/S0 ) 0.17 based on the above equations and the measured value of f2. Similarly, the measurements at 1 mM yield τd ) 1.7 ms, τa ) 18 ms, and S2/S0 ) 0.006, smaller than the predicted value of S2/S0 ) 0.055. Furthermore, we cannot at present distinguish which form (oxidized or reduced) of Fc(MeOH)2 adsorbs, because both forms are present in the cavity in approximately equal amount (50%) during redox cycling. Finally, we also neglected that the oxidized forms of ferrocene and ferrocene derivatives have been reported to be unstable in aerated solutions. The rate of decomposition depends on the derivative employed, the solvent and the supporting electrolyte.50-55 While data for Fc(MeOH)2 are (53) Barker, A. L.; Unwin, P. R. J. Phys. Chem. B 2001, 105, 12019–12031. (54) Hurvois, J. P.; Moinet, C. J. Organomet. Chem. 2005, 690, 1829–1839.

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unavailable, reported lifetimes for other Fc derivatives in ACN are in the range of minutes, which is only expected to affect the PSD at frequencies below 0.01 Hz. Decomposition is also usually accompanied by electrode fouling resulting in a decrease of the diffusion-limited current,38,53-55 which was not observed in our experiments. Decomposition on time scales comparable to or smaller than the typical residence time in the active region would also lead to a suppression of the diffusionlimited current below its expected value, which again was not observed. For these reasons we believe that decomposition of the oxidized form of ferrocenedimethanol does not significantly impact the results presented here. For cases where decay is rapid, however, reduced diffusion-limited currents and a different form for the PSD are expected. The data and analysis presented here illustrate that ECS provides a powerful new approach for exploring surface-solute interactions and solute stability; further work will be required to obtain a full quantitative description. CONCLUSION We have introduced electrochemical correlation spectroscopy (ECS), a technique that measures fluctuations in the number of reversibly redox-active molecules present in an active volume. The measured power spectral densities of these fluctuations exhibited the characteristics predicted by theory for freely diffusing molecules: a plateau S0 at low frequencies and a f -3/2 power-law decay at higher frequencies. Additional features in the spectra reflected adsorption of the redox-active molecules. These features included a concentration-dependent effective diffusion coefficient in the channel due to intermittent adsorption and additional fluctuations at high frequencies due to individual adsorption/desorption events. Both the adsorption isotherm and the individual rates of adsorption and desorption could be estimated. These findings illustrate the usefulness of ECS as a local probe of mass transport with the ability to reveal additional microscopic information that is difficult or impossible to extract from conventional steady-state or amperometric measurements. The analysis presented here has focused on the TLC geometry, but the approach can straightforwardly be extended to other systems. In particular, we envision that ECS could also be performed with scanning electrochemical microscopy (SECM) in feedback mode. This combination could prove to be a particularly powerful tool, allowing spatially resolved studies of solute-surface interactions. ACKNOWLEDGMENT We thank Diego Krapf, Bernadette M. Quinn, Hendrik A. Heering, and Cees Dekker for discussions and general support. This work was funded by NanoNed and NWO. B.W. was funded by the DFG. E.D.G. acknowledges the U.S. National Science Foundation for support under IRFP Grant No. 0754396. SUPPORTING INFORMATION AVAILABLE Power spectral densities for Ru(NH3)63+ in water. Derivation of eq 24. Additional discussion of device geometry. This material is available free of charge via the Internet at http://pubs.acs.org. Received for review July 6, 2009. Accepted August 7, 2009. AC9014885 (55) Zotti, G.; Schiavon, G.; Zecchin, S.; Favretto, D. J. Electroanal. Chem. 1998, 456, 217–221. (56) McNabb, A.; Foster, P. K. Trans. Metall. So. AIME 1963, 227, 618–627.

Electrochemical Correlation Spectroscopy in Nanofluidic Cavities

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