Ultrahigh Frequency Voltammetry: Effect of Electrode Material and

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Ultrahigh Frequency Voltammetry: Effect of Electrode Material and Frequency of Alternating Potential Modulation on Mass Transport at Hot-Disk Microelectrodes Andrzej S. Baranski* and Aliaksei Boika† Department of Chemistry, University of Saskatchewan, 110 Science Place, Saskatoon, Saskatchewan, Canada, S7N 5C9 S Supporting Information *

ABSTRACT: Ultrahigh frequency voltammetry involves low scan rate voltammetric measurements with microelectrodes polarized by high-frequency large-amplitude alternating potential. The method provides a simple means for studying electrothermal and dielectrophoretic effects, which are important in micro and nanofluidic systems. The method also allows for indirect measurements of electrode impedance at gigahertz frequencies. This increases the upper frequency limit in impedance measurements about 1000 times. In this work we demonstrated, for the first time, that the effect of dielectric relaxation of water can be observed in a simple voltammetric experiment. The paper focuses on the description of electrothermal convection at ac heated disk microelectrodes as a function of frequency and provides a comparison of numerical simulations with experimental results.

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The polarization of microelectrodes by high-frequency largeamplitude alternating potential gives rise to a number of fascinating phenomena, which have never been studied before. The polarization of the solution resistance leads to intense Joule heating of solution near the electrode (in a volume of several femtoliters). More interestingly, the high temperature gradient around the microelectrode is accompanied by a very intense alternating electric field (>106 V/m for electrodes 1 μm in radius) which causes strong dielectrophoretic effects.7 In addition, an interaction of the temperature gradient with the potential gradient causes an intense flow of solution (called an electrothermal flow) around the microelectrode.7 The electrothermal flow, generated directly on microelectrodes employed in electrochemical detection, causes a very substantial enhancement of diffusion controlled currents. The electrothermal convection was discovered about 13 years ago in research done on electrohydrodynamics in microfluidic systems.9−12 These systems usually involve coplanar electrodes fabricated on a glass substrate. The theory of electrothermal convection was specifically designed for temperature distributions expected to occur in microfluidic systems and, in some conclusions, oversimplified the complexity of the relationship between the electrical field and the temperature gradient. This issue is discussed in detail in our paper. Ultrahigh frequency voltammetry provides a simple means for studying electrothermal and dielectrophoretic effects at very small electrodes. Our method is also rooted in the faradaic

n recent years, a new electroanalytical method, which we call ultrahigh frequency voltammetry (UHF voltammetry), started to emerge. Similar research called microwave electrochemistry was initiated 13 years ago by Compton and coworkers1−5 who used focused microwave radiation to heat up microelectrodes, while simultaneously performing voltammetric experiments. In this case, a microelectrode acts as an rf antenna, which becomes polarized by the electromagnetic radiation and creates a high intensity electrical field in solution around the electrode surface. In our laboratory, we use a high frequency ac generator which produces a few volt output, at frequencies between 0.1 MHz and 2 GHz. The output of this generator is connected to a microelectrode via a small capacitor or a high frequency transformer.6−8 In both cases, only a dc component of the electrode current is monitored because, at very high frequencies, ac current flows mainly through stray capacitance and is not characteristic of the electrode process. Besides measurements of ac currents at such high frequencies are very difficult. In general, Compton’s method differs from our method by the way of delivering electrical energy (wireless versus wired). It also focuses on high power excitations that produce large magnitude effects (such as heating, convection, or even cavitation and plasma formation5), which the authors have called microwave induced activation of the electrode. Their method is suitable for immediate applications in electrochemical analysis. We concentrate on low power excitations, which are more suitable for a theoretical description. We believe that our wired method allows for better control of experimental conditions and provides more insight into the phenomena occurring at ac heated microelectrodes. © 2011 American Chemical Society

Received: August 31, 2011 Accepted: December 28, 2011 Published: December 28, 2011 1353

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rectification method,8,13 which, in the essence, involves indirect measurements of the electrode impedance. In the classical version, only measurements of dc current are performed, but we see a possibility of extending this method by including measurements of nonelectrical variables such as electrode temperature change and convectional flux. This allows for quantification of the solution resistance (which becomes frequency dependent at very large frequencies). In addition the apparent acceleration of irreversible redox processes and a distortion of quasi-reversible processes provide a means for quantification of the double layer impedance and the faradaic impedance, respectively.8 Altogether this allows for carrying out indirect impedance measurements, in the presence of huge ohmic drops, at gigahertz frequencies under voltammetric conditions. Normally impedance measurements cannot be carried out at frequencies higher than a few megahertz because of limitations caused by the electrode time constant and other technical problems.14−16 In our measurements, excitation frequencies can be so large that they approach the frequency of rotation of molecules in liquids. In this paper, we will show that by using this approach (although restricted to measurements of the solution resistance only) we can indeed obtain impedance related information about dielectric relaxation of water at gigahertz frequencies. But getting to this point would not be possible without the development of a numerical model of an ac heated disk microelectrode.17 The current paper discusses important conclusions arising from this model and shows that quantitative description of phenomena occurring at ac heated microelectrodes opens new possibilities for studying electrode processes. Our primary focus is on the description (both through numerical simulations and experiments) of two very important and interesting effects: the effect of the thermal conductivity of the electrode material and the effect of frequency on mass transport at hot-disk microelectrodes.

obtained using an inverted metallurgical microscope and Xli Digital imaging camera from XL Imaging Ltd. Numerical Simulation Program. Numerical simulation and plotting software (for plotting RGB color maps, contour lines, gradient lines and curved arrows, i.e., arrows which follow the gradient lines for a predefined distance) was devised by us from scratch; the program was written using Microsoft Visual Studio 2008. Details of the simulation program were previously published.17 For the reader’s convenience, we listed the major equations used in these simulations in the Supporting Information.

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RESULTS AND DISCUSSION AC Heating of Disk Microelectrodes Made of Different Materials. In order to perform numerical simulations of phenomena occurring at ac heated microelectrodes, the electrical field around the electrode has to be described; this was done using Newman’s analytical solution.18 Figure S-1 in the Supporting Information illustrates the distribution of ac potential around the microelectrode. We describe the alternating potential only in terms of its magnitude, so ϕac is a simple variable (rather than a phasor, which is a complex variable). It should be noted that according to eq [S-3] in the Supporting Information the rate of heating is proportional to |∇ϕac|2; therefore, the most effective heat generation occurs in close proximity to the circumference of the electrode (see Figure S-2 in the Supporting Information). Results of numerical simulations of the heating process can be summarized by considering a normalized temperature distribution map:

T̃(ρ̃, z ̃, t )̃ =

T (ρ̃, z ̃, t )̃ − T0 Tmax(t )̃ − T0

(1)

where T0 is the bulk temperature, Tmax is the maximum temperature of the system at a given moment, and ρ̃ = (ρ/r0), z̃ = (z/r0), and t ̃ = (t/r02) are distance and time variables normalized with respect to the electrode radius, r0, z is the distance coordinate perpendicular to the electrode surface. It can be shown that the T̃ (ρ̃,z̃,t)̃ function is independent of the electrode radius and independent of heating parameters (the magnitude of ac potential applied to the electrode and electrical conductivity of the solution); also, when the steadystate is reached, it becomes independent of time. However, the normalized temperature distribution strongly depends on heat diffusion coefficients (DH) of involved phases. DH = κ/(Csd) where κ is the heat transfer coefficient, cs is the heat capacity of the medium, d is the density of the medium; all values used in calculations are taken from literature.19 In this study, we focused exclusively on the effect of the heat diffusion coefficient of the electrode material. In all our considerations, thermal properties of two other materials: the electrode insulator (which is assumed to be made of glass with DH = 5.64 × 10−3 cm2/s) and the electrolyte solution (with DH = 1.43 × 10−3 cm2/s) remain the same. In Figure 1, steady-state normalized temperature distribution maps of bismuth, with DH = 0.066 cm2/s, and gold, with DH = 1.27 cm2/s, microelectrodes are shown. In these figures, the normalized temperature is represented by color: maximum (1) by red, minimum (0) by blue, and middle value (0.5) by green; values between 1 and 0.5 are represented by proportional blends of red and green and values between 0.5 and 0 are represented by proportional blends of green and blue. In the Supporting Information, similar figures (Figure S-3) for

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EXPERIMENTAL SECTION Reagents. All solutions were prepared using Millipore water and ACS grade chemicals. Methyl viologen (Sigma-Aldrich) as well as all other chemicals were used without any further purification. Electrochemical Cell. In most experiments a standard three-electrode electrochemical cell was used. The auxiliary and pseudoreference electrodes were made of a platinum wire ∼0.3 mm in diameter (Alfa Aesar). In some cases, a standard Ag| AgCl|KCl(sat.) reference electrode was employed. The disk working electrodes (12.5 μm in radius) were prepared by sealing Pt or Au microwires (Goodfellow Metals Ltd.) into glass tubing (World Precision Instruments). A lead was made by connecting a copper wire with a microwire with the aid of a small piece of Pb−Sn solder. Then the electrode tip was cut and the electrode was polished with 3 and 0.3 μm finishing films on a Micropipet Beveller (WPI, model 48000) to achieve a mirrorlike surface. Bismuth microelectrodes were made by filling up a glass tube (0.5 mm i.d., 5 mm o.d., about 10 cm in length) with molten bismuth (Alfa-Aesar, Puratronic). When the metal solidified over the entire length of the tube, the middle part of the tube (about 1 cm in length) was heated up in a gas burner flame to soften the glass, then the tube was pulled. Bismuth electrodes were cut and polished the same way as noble metal electrodes. Experimental Setup. The same electronic setup as previously described7 was used in experiments. Videos were 1354

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the Supporting Information. For steady-state heating, the change of the electrode temperature can be predicted by the following equation, which was obtained by matching a large number of numerical simulations:

ΔTel,ss = Tel,ss − T0 = KET σVRs 2

(2)

where Tel,ss is the steady-state electrode temperature, σ is the electrical conductivity of the solution, VRs is the root-meansquare (rms) magnitude of the alternating potential which develops across the solution resistance, and KET is a parameter which depends on the heat diffusion coefficient of the electrode (values are given in Table 1). Equation 2 together with the ratios of ΔTmax,ss/ΔTel,ss (given in Table 1) can also be used to predict the steady-state temperature of the hot spot in the solution. For a TNC electrode, we obtained KET = 88.5 K W−1 cm−1; on the basis of an analytical solution6 derived for a spherical electrode, this coefficient should be equal to 1/(2κ) = 83.6 K W−1 cm−1. In the cited paper, we pointed out that the analytical solution obtained for a spherical electrode should also be valid for a hemispherical and a disk electrode if these electrodes conduct heat only toward the solution. Our simulation results for a TNC electrode is 6% larger than the theoretical value, but the error decreases to just 2% when a finer simulation grid is used (with the spatial increment equal to r0/16). Unfortunately, a decrease of the spatial increment by 2 increases the simulation time 16 times; therefore, we could not use this grid for all the simulations. Simulation results for metal electrodes can be, in principle, compared with experimental data. There is however one major difficulty: the potential drop which develops across the solution resistance cannot be measured directly, and at high frequencies it cannot even be estimated with high accuracy (because of problems with the inductance of the electrode lead and with the stray capacitance of the electrode).20 Recently, we performed measurements of ac heating of relatively large platinum electrodes (62.5 μm in radius) at 25 MHz (at such low frequency problems with inductance and capacitance are negligible). On the basis of these experiments, we determined KET for platinum as 11.5 ± 0.5 K W−1 cm−1; this value is about 13% higher than the one obtained by numerical simulations. However, in this case, the error increases to 16% when a finer simulation grid is used. It is possible that in real systems there is another (minor) source of heating, perhaps due to dielectric relaxations in the double layer; this issue is discussed in the last section of the paper. Electrothermal Convection: Direction of the Flow and the Effect of Frequency. Pivotal work by Ramos et al.9 provided the basic theory of electrothermal convection, but

Figure 1. Normalized temperature distribution map for an ac heated bismuth (A) and gold (B) microelectrodes. The electrode crosssection (parallel to the electrode axis) is shown by two vertical lines. Temperature is represented by color: maximum (1) by red, minimum (0) by blue, and middle value (0.5) by green. The heating time was 1 ms/μm2 (the spatial dimension applies to the electrode radius), which practically means reaching the steady-state condition.

different electrode materials are shown, but they are zoomed (×4) on the electrode and contain temperature field lines (these lines start at equal intervals at the frame of the figure and then move, pixel by pixel, in the direction of the local temperature gradient). In the Supporting Information we also provided a brief discussion of the heating rate. The results shown in Figure 1 clearly indicate that the maximum temperature is reached in the solution and not at the electrode surface. Furthermore, the higher the heat diffusion coefficient of the electrode wire the larger the difference is between the maximum temperature in the hot spot and the temperature at the surface of the electrode. This happens because in our system heat is generated in the solution by ohmic polarization, and the electrode acts as a heat sink. Additional information in this regard is given in the third row of Table 1; the only exception is a hypothetical electrode with DH = 0 for which the hot spot is on the surface of the electrode. In the case of this electrode (which we call TNC, i.e., thermally nonconductive), it was also assumed that the electrical insulator surrounding the electrode has zero thermal conductivity. The normalized distance of the hot spot from the electrode surface (zmax/r0) is also shown in Table 1. These data indicate that the higher the thermal conductivity of the electrode, the further into the solution the point of the maximum temperature is located. The fact that the maximum temperature in the hot spot is reached in the solution and not at the electrode surface (with the exception of a hypothetical TNC electrode) means that the temperature gradient near the electrode is reversed. It should also be noted that the temperature gradient lines around the electrode follow a quite complicated pattern (as shown in Figure S-3 in the Supporting Information). The exception again is the TNC electrode, for which the temperature gradient lines are parallel to the potential gradient line shown in Figure S-1 in

Table 1. Characteristic Parameters for Hot Disk Microelectrodes Made of Different Materials TNCa

Bi

Pt

Au

0 1.0 0 88.5 (85.2)b −3.59 × 10−5 3.41 × 10−6

0.066 1.30 0.52 22.8 −2.18 × 10−6 2.58 × 10−6

0.250 2.06 0.68 10.0 (9.6)b 1.05 × 10−5 2.41 × 10−6

1.27 4.30 0.77 3.8 1.59 × 10−5 2.42 × 10−6

electrode material 2 −1

heat diffusion coefficient [cm s ] ΔTmax,ss/ΔTel,ss zmax/r0 parameter for ΔTel,ss (KET) [K W−1 cm−1] low frequency parameter for volume flow rate (KFR)c [Ω cm3 s−1 V−4] high frequency parameter for volume flow rate (KFR)d [Ω cm3 s−1 V−4] a

Hypothetical material which does not conduct heat; in this case also the electrical insulator has no thermal conductivity. bValues in parentheses were calculated for the spatial increment Δh = r0/16, in all other cases Δh = r0/8 was used. cCalculated for 0 Hz. dCalculated for 2 GHz and σ = 0.0129 Ω−1 cm−1. 1355

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some conclusions derived from this theory oversimplified the effect of frequency. The authors postulated that in an electrolytic solution affected by a potential gradient and a temperature gradient, the electrothermal force acting on the solution is produced. The direction of this force is independent of the direction of the electrical field (so it does not change sign when an alternating potential is used) and it is described by the following equation:

1 ⎡⎛⎜ dσ dε ⎞⎟ ε ⎢ − (∇T ∇ϕac) 2 ⎣⎝ σdT εdT ⎠ 1 + (ωτ)2 ⎤ 1 dε ∇ϕac + |∇ϕac|2 ∇T ⎥ 2 dT ⎦

Figure 2. Electrothermal convection around Bi (A) and Au (B) microelectrodes heated at 100 MHz. Color indicates the magnitude of the linear flow (red, high; green, medium; blue, low), lines with arrows show the direction of the flow. Dashed lines in part A illustrate how the characteristic volume flow rate, uc, is defined (details are given in the text).

fE̅ = −

(3)

where ∇ϕac and ∇T are gradients of the electrical potential and temperature, respectively, τ = ε/σ is the space charge relaxation time, ω is the angular frequency, ε is the dielectric permittivity, and σ is the electrical conductivity of the solution. The first term in eq 3 describes a frequency dependent Coulombic force, which dominates at low frequencies, and the second term describes a frequency independent force on a dielectric. As temperature increases, the solution conductivity (σ) increases and the dielectric permittivity (ε) decreases, so dσ/dT and dε/dT have the opposite sign but |dσ/dT| > |dε/dT|. At these bases, Ramos et al.9 concluded that there must be a crossover frequency at which the electrothermal force changes sign by going through zero. The crossover frequency should depend on the concentration of the electrolyte (which determines σ) and it was calculated from eq 3 as

high frequencies is shown in Figure S-6 in the Supporting Information. Of course, the difference in the behavior of these two types of the electrode material is related to the distribution of the temperature gradient, which directly affects the distribution of the electrothermal force; this issue is discussed in the Supporting Information in the section entitled “Direction of the electrothermal force”. To double check the validity of our solution, we also calculated the work performed by the electrothermal force on a unit volume of the solution, along the calculated flow lines. An example of results of such calculations for a gold electrode is shown in Figure S-7 in the Supporting Information; in the direction of the flow, the work is always positive. The predicted flow direction for all electrode materials discussed in this paper (except TNC, which obviously, does not exist) was confirmed experimentally. In Video S-1 in the Supporting Information, we recorded changes in the shape of the diffusion layer of the reduction product of 20 mM methyl viologen in 0.2 M KCl at a 12.5 μm Pt electrode. A snapshot from this video is shown in Figure S-10A in the Supporting Information. To avoid errors in the interpretation of the video, we calculated the distribution of the electrode reaction product in the presence of electrothermal convection (Figure S-11 in the Supporting Information) and then the transmittance of light through the solution around the microelectrode (Figure S-12 in the Supporting Information). All this clearly suggests that at ac heated Pt microelectrodes, at low frequency (∼150 MHz), the radial direction of the flow is towards the electrode center and the axial flow is out of the electrode. Bismuth is an easily available metal with low thermal conductivity. According to our simulations, the electrothermal convection on an ac heated Bi microelectrode at low frequencies is relatively weak, but it shows a different pattern than that observed at Pt or Au electrodes; this was also confirmed by experiments. First we performed experiments with relatively large Bi electrodes (about 60 μm in radius); in this case the radial flow is clearly out of the center, the opposite of one seen at Pt (see Video S-2 and Figure S-10B in the Supporting Information). However, the buoyancy convection at such a large electrode may be significant (the effect of buoyancy convection is discussed in the Supporting Information). In our experiments, the gravitational force was perpendicular (±5°) to the plane of the video, but still a shade of doubt remains. Later we recorded Video S-3 in the Supporting Information (snapshot in Figure S-10C in the Supporting Information) for a Bi microelectrode about 15 μm in radius. In this case, the

∂σ

ε 1 T fc = 2 ∂∂ε 2πτ σ

∂T

(4)

However, clearly this is true only when the angle between two vectors ∇ϕac and ∇T is always equal to zero. In general, Coulombic forces and forces on a dielectric do not necessarily follow the same directions; therefore, they cannot cancel each other completely. Coulombic forces follow the directions of the electrical field lines (to be exact, the direction of the force is the opposite to the projection of the temperature gradient on the electrical field lines). On the other hand, forces on a dielectric follow the direction of the temperature gradient. The distribution of Coulombic forces and forces on a dielectric for hot TNC and Au microelectrodes is shown in Figure S-4 in the Supporting Information. According to mathematics, the flow of the solution occurs along the contour lines of a stream function, i.e., a function which is related through the double laplacian to the curl of the force (equations are given in the Supporting Information). Numerical simulations indicate that the effect of frequency on the direction of the solution flow is different for electrodes made of materials with low thermal conductivity (TNC and Bi) and ones made of materials with high thermal conductivity (Pt and Au). The results of the flow rate simulation for Bi and Au disk microelectrodes heated at low frequency are shown in Figure 2. The flow direction at a Bi electrode is the opposite of the one at a Au electrode and the flow direction changes when frequency increases only at a Bi electrode (see Figure S-5 in the Supporting Information). The distribution of the linear velocity of solution around two other microelectrodes heated at low and 1356

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The minimum occurred at a frequency predicted by eq 4 only for the TNC electrode, and for Bi it occurred at a much lower frequency. In the case of Pt and Au electrodes, there is no change of the flow direction with frequency. In our previous paper,7 we reported experimental measurements of the limiting current enhancement factor for a platinum microelectrode in 0.1 M KCl as a function of frequency. These experimental data also did not indicate the reversal of the flow direction between 50 and 1650 MHz, but we did not understand the reason at that time. In Figure 3, our old experimental data for Pt are plotted as filled circles with error bars and they agree quite well with the results obtained by simulations. The agreement is even better when the dielectric relaxation of water is taken into account (dashed line in Figure 3). The effect of the dielectric relaxation of water is discussed later in this paper. The characteristic volume flow rate (uc) is a useful parameter for describing the rate at which the solution is “pumped” around the electrode. It should be noted that the volume flow rate through the horizontal cross-section shown in Figure 2A is exactly equal, but opposite in sign, to the volume flow rate through the vertical (cylindrical) cross-section also shown in Figure 2A; this comes from the continuity principle and it is strictly obeyed by our simulations. The characteristic volume flow rate can be linked to the limiting current enhancement factor χ = ((ilim,h)/(ilim)) −1, where ilim is a steady-state limiting current without ac heating and ilim,h is a steady-state limiting current with ac heating). The increase of the limiting current is chiefly determined by the electothermal convention with other factors (like a change of the diffusion coefficient) playing a marginal role. Let us assume for a moment that all the depolarizer carried by the flowing solution into the zone marked by dashed lines in Figure 2A undergoes an electrochemical reaction at the electrode surface. In this case the extra influx of the depolarizer (counted in mol/s) is ucC (where C is the concentration of the depolarizer; we will assume here that it is equal to the bulk concentration, Cb, although this is not completely true). At the same time, the normal rate of the diffusional transport of the depolarizer (in mol/s) to a disk electrode is described by the well-known equation: 4Dr0Cb (where D is the diffusion coefficient, r0 is the electrode radius, and the other symbols are the same as before). Hence, the enhancement factor should be

motion of the solution is less spectacular but the conclusion is the same. The color cloud produced by the electrode reaction product broadens as the result of convection, which is consistent with the radial flow out of the electrode center. In order to provide some quantitative description of the relationship between the heat diffusion coefficient of the electrode material and the flow rate, we calculated, for each electrode, a characteristic volume flow rate (uc). It describes a volume of solution passing in 1 s through a cross-section equal to the surface area of the electrode and positioned above the electrode at the distance of one electrode radius (this crosssection is shown in Figure 2 A as a dashed line). It can be shown by matching the results of many simulations that

uc(f ) = KFR (f )σr0VRs 4

(5)

where KFR is a frequency ( f) dependent coefficient and other symbols have the same meaning as in previous equations. Equation 5 can be combined with eq 2 giving

uc(f ) =

2 KFR (f ) ΔTel,ss r 0 σ KET2

(6)

Values of the coefficient KFR are given in Table 1 for two frequencies: f = 0 and f = 2 GHz. Negative values indicate the flow of the solution toward the electrode (along the electrode axis), and positive out of the electrode (along the electrode axis). It should be noted that values for f = 0 have a theoretical importance, but obviously the ac heating of microelectrodes cannot be performed at f = 0; the lowest frequency which can be used in experiments is probably about 20 MHz. However, in relatively concentrated solutions (> 0.1 M), values of KFR for f = 0 and f = 20 MHz are practically the same. Figure 3 shows the frequency dependence of the normalized characteristic flow rate (|uc( f)/uc(0)| obtained from our simulations for four different electrode materials. The simulations were carried out for an electrolyte with σ = 0.0129 Ω−1 cm−1 (i.e., corresponding to 0.1 M KCl). Since absolute flow rates are plotted, the crossover frequency (if ever reached) is indicated by a minimum on a curve, but the minima were observed only for TNC and Bi electrodes.

χ=

ucC u ≈ c 4Dr0Cb 4Dr0

(7)

In reality, this equation gives a large overestimate of the enhancement factor because C/Cb < 1 and a large fraction of the depolarizer does not react and is carried away by the stream of the solution. Nevertheless, numerical simulations of steadystate voltammograms (such as shown in Figure S-8 in the Supporting Information) indicate that the enhancement factor is indeed proportional to uc. Therefore, the above equation can be rewritten as

Figure 3. Solid lines illustrate the effect of frequency on the normalized characteristic flow rate obtained by numerical simulations for four different electrode materials in 0.1 M KCl. Markers with error bars represent previously published experimental results for a Pt electrode in 0.1 M KCl (a normalized enhancement factor vs frequency).7 The dashed line represents a relationship between the normalized enhancement factor and frequency predicted by eq 16, which takes into account the dielectric relaxation of water.

χ = Kenh

uc 4Dr0

(8)

In all the cases which we examined, the proportionality coefficient Kenh = 0.214 ± 0.07, and it seems to be independent of the flow rate (except for very small flow rates, which give χ < 0.05), frequency, and the electrode material; this coefficient 1357

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must be determined by the geometry of the system. A correlation between χ and uc/(4Dr0) is shown in Figure S-9 in the Supporting Information for four randomly selected systems. By combining eqs 6 and 8, we obtain

χ=

KFR Kenh 4DσKET 2

ΔTel,ss2

(ε − ε∞)ω2τD = 4r0 σ + 4ε0r0 s R total 1 + ω2τD2 ⎡ ω2τD ⎤ ⎥ = 4r0⎢σ + ε0(εs − ε∞) ⎢⎣ 1 + ω2τD2 ⎥⎦ 1

(9)

The term is square brackets can be called the effective conductance of the solution:

According to our simulations, for a platinum electrode in 0.5 M KCl, the proportionality coefficient between χ and ΔT2 should be about 0.005 K−2, the experimental value based on our previous paper7 is about 0.006 K−2; therefore, the agreement is quite good, but it may result from a lucky cancelation of errors. Effect of Dielectric Relaxation. According to electrostatics, a metal sphere placed in a dielectric develops an electrical capacitance given by Cs = 4πεr0 (where ε is a product of the dielectric permittivity of space and relative permittivity of the medium, and r0 is the sphere radius). Similarly, following Newman’s approach,18 we can assign electrical capacitance to an inlaid disk electrode of radius r0:

Cs = 4εr0

σeff = σ + ε0(εs − ε∞)

This equation accounts only for the capacitance which develops across the solution (not across the body of the electrode), and this is not the double layer capacitance. This capacitance is parallel to the solution resistance and we will call it the solution capacitance. For microelectrodes, Cs is very tiny; for a 5 μm electrode, Cs = 1.4 × 10−14 F and the double layer capacitance is about 1.6 × 10−11 F. Nevertheless, at high frequencies a very substantial current passes through Cs. This passage of current through solution capacitance is accounted for in eq 3, which describes the electrothermal force; however, at high frequencies, the passage of current through Cs may also contribute to the heating of the electrode. Normally, capacitors do not produce heat, but according to the Debye theory the relative permittivity of a medium has real (ε′) and imaginary (ε″) components:

ε″ =

ΔT 2 ω Δχ

(15)

KFR (ω)σ KFR (0)σeff

=

KFR (ω) KFR (0)

σ ω2τ

D σ + ε0(εs − ε∞) 1 + ω2τD2

(16)

The relationship predicted by eq 16 for a Pt electrode is shown in Figure 3 as a dashed line; it matches experimental data very well. This shows that the effect of the dielectric relaxation of water can easily be observed under ultrahigh frequency voltammetric conditions. At this point we should also discuss another related issue, dielectric relaxation in the double layer. Unfortunately, very little is known about relaxation processes in the double layer. In general, it is believed that water in the double layer is more sluggish than in the bulk. Bockris et al.21 suggested that the relaxation time of water in the double layer can be around 10−6 s. Many years ago we suggested the double layer relaxation time can be close to 10−7 s,15 but later we obtained some evidence which indicates (although not with certainty) that the viscosity of water in the double layer (at the zero charge potential) is only 1.7 times higher than the bulk viscosity.16 Since the double layer is just at the electrode surface, any heat generated in this layer may substantially affect the electrode temperature. Our preliminary estimates suggest that the double layer heating may account for a few (or even more than 10) percent change in the electrode temperature, and it can be evaluated experimentally. We are planning to perform such experiments in the near future.

(11)

(12)

(εs − ε∞)ωτD 1 + ω2τD2

=

ΔT 2 0

where εs is the static (low frequency) permittivity, ε∞ is the limiting high frequency permittivity, τD is the Debye relaxation time, and ω is the angular frequency. For water at 25 °C, εs = 78.36, ε∞ = 5.2, and τD = 8.27 × 10−12 s. Consequently, the imaginary component of the solution capacitance for a disk electrode is given by

Cs″ = 4ε0r0

1 + ω2τD2

Δχ

( ) ( )

(εs − ε∞)ωτD 1 + ω2τD2

ω2τD

It is clear that the effect of the dielectric relaxation of the solution can be, under some conditions, substantial. In 1 M KCl below 200 MHz, σeff/σ is practically equal to 1, but this ratio increases to 1.075 at 2000 MHz; in 0.1 M KCl σeff/σ = 1.0066 at 200 MHz and 1.65 at 2000 MHz. Most of our experimental results presented in the previous paper7 were obtained around 200 MHz so they are practically unaffected by the dielectric relaxation of water; however, for the experimental results presented in Figure 3 the effect is significant. Experimental data in this case are obtained as normalized ratios ((Δχ)/(ΔT2)) (i.e., slopes of the limiting current enhancement factors plotted versus ΔT2). The relationship can be described by eq 6 in which σ is substituted with σeff given by eq 15:

(10)

ε −ε ε′ = ε∞ + s 2 ∞ 2 1 + ω τD

([14])

(13)

An imaginary capacitance has real impedance (1/Cs″ω); therefore, this is a resistive element which produces heat upon the passage of current. Since Cs″ is in parallel with the solution resistance, the overall conductance of the electrode is given by 1358

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CONCLUSIONS Our simulations correctly describe all experimentally observed trends in the behavior of hot microelectrodes: proportionality of the electrode temperature toVRs2, proportionality of the limiting current enhancement factor to VRs4 or ΔT2, and inverse proportionality of the limiting current enhancement factor to the solution conductivity. Numerical simulations for platinum microelectrodes predict that the direction of the solution flow around the electrode should be independent of frequency, and this is in agreement with previously published7 experimental observations. Furthermore, numerical simulation of the transmittance of light through the solution around a microelectrode in the presence of convection (Figure S-12 in the Supporting Information) agrees with the observed distribution of the reduction product of methyl viologen at hot Pt and Bi electrodes (Figure S-10 in the Supporting Information). Simulations of the temperature distribution around hot microelectrodes seem to be reasonably accurate, because simulation results for a hypothetical thermally nonconductive electrode agree (within 2%) with the previously published6 analytical solution. Yet, there are some indications that our simulations underestimate the electrode surface temperature by, perhaps, 16−18%. This conclusion is derived from the fact that the calculated proportionality coefficient between the electrode temperatures and VRs2 is lower than experimentally observed. We speculated that this discrepancy may be due to additional heating provided by double layer relaxation processes. If this is true, then detailed studies of ac heated microelectrodes, in combination with previously described studies of faradaic rectification,8 may provide a new insight into the dynamics of double layer processes. Ultrahigh frequency voltammetry can also be employed in studies of other fast processes, which are not accessible or barely accessible to assessment by other electrochemical techniques. For example, one can use this method to study very fast electron transfer processes, very fast adsorption processes, the Debye− Falkenhagen effect22 (if it really exists) and the dielectric relaxation of solvents and solutes. However, the use of our method in indirect high-frequency impedance measurements is only one of several possible applications. The essence of UHF voltammetry is in carrying out voltammetric measurements in the presence of a huge and nonuniform alternating electrical field around the electrode (as shown in Figures S-1 and S-2 in the Supporting Information) and in the presence of a large temperature gradient (as shown in Figure 1 and Figure S-3 in the Supporting Information). This generates electrophoretic and electrothermal effects which have been already successfully exploited in analysis, in microfluidic systems.11 Our method has a significant advantage in such applications, because, in our case, both effects are generated at the same microelectrode at which an electrochemical detection is carried out. Our method can also be easily implemented in scanning electrochemical microscopy to provide localized analysis. Specific applications of the method will be described in subsequent publications.

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convection at hot Pt and Bi microelectrodes. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Phone: (306) 9664701. Fax: (306) 9664730. E-mail: andrzej. [email protected]. Present Address †

University of Texas at Austin, Chemistry and Biochemistry Dept., 1 University Station A5300, Austin, TX 78712-0165.

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ACKNOWLEDGMENTS The financial support of the Natural Sciences and Engineering Research Council, Canada (NSERC), through an Individual Discovery Grant is gratefully acknowledged.

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REFERENCES

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ASSOCIATED CONTENT

* Supporting Information S

Twelve additional figures illustrating properties of ac heated disk microelectrodes, discussion of the direction of the electrothermal force, brief discussion of the heating rate and the buoyancy effect, a list of equations used in numerical calculations, and three videos illustrating the electrothermal 1359

dx.doi.org/10.1021/ac202234v | Anal. Chem. 2012, 84, 1353−1359