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Spatially Averaging Electrodes Disha B. Sheth, Richard Diefes, and Miklo ´ s Gratzl* Department of Biomedical Engineering, Case Western Reserve University, Cleveland, Ohio 44106 Both potentiometric and voltammetric measurements report on the concentration of the analyte in the closest layer of the sample solution that is unperturbed by the measurement. Besides this local concentration, the solution composition within the thin boundary layer adjacent to the electrode|solution interface is influenced also by local mass transport and other electrochemical processes necessary for signal transduction. This local perturbation of concentrations is typically corrected for by calibration so that the ultimate output of the measurement is the local concentration, at a distance of a few micrometers to about 100 µm from the electrode surface. In contrast to many optical techniques, the electrochemical approach is therefore only capable of measuring local concentrations but cannot be used to assess three-dimensional (3D)-averaged bulk concentrations of inhomogeneous samples. This may pose a problem in very small samples where homogenization by stirring is difficult. We present here the concept of spatially averaging electrodes that can, due to their special design, report 3D spatially averaged bulk concentration of inhomogeneous samples that have some type of symmetry. Within a given type of symmetry an infinite variation of concentration distribution may exist. We illustrate the concept of spatially averaging electrodes with results obtained in microliter-sized hemispherical samples with a source in the center of the drop. Optical measurement techniques use light to probe the sample solution. As light can pass through the entire width of the sample before being detected, under proper circumstances optical techniques can report the average concentration of an inhomogeneous sample. This is not the case for electrochemical measurements as an electrode can probe the sample only in its vicinity. Therefore, homogeneous samples obtained after some type of stirring are traditionally measured with electrochemical techniques. Here we present the concept of spatially averaging electrodes that, due to their designed geometry, can report three-dimensional (3D)averaged bulk concentration of an inhomogeneous sample. Typically, both potentiometric and voltammetric measurements perturb the solution composition within a thin boundary layer adjacent to the electrode. We call here the concentration of the closest unperturbed layer of the sample solution local concentration. In addition to information about this local concentration, the direct output of the measurement also contains information on the processes at the electrode|solution interface and associated local mass transport that are necessary for transduction. This * To whom correspondence should be addressed. E-mail: miklos.gratzl@ case.edu. Fax: 216-368-4969. 10.1021/ac802288g CCC: $40.75 2009 American Chemical Society Published on Web 02/13/2009
interference that is due to the measurement itself, however, typically can be corrected for with calibration. Thus, the concentration that is assessed by an electrochemical measurement is the local concentration that is present in the sample at a distance of a few micrometers to about 100 µm from the electrode surface. Inhomogeneous samples may exist in a number of contexts, such as in microfluidic systems at nodes where reagents are mixed and in laboratory-on-a-chip platforms. Another application where this problem may arise is the diffusional microtitration of microlitersized hemispherical sample drops where continuous dosing of a reagent into the sample via a miniature diffusion port in its center is effected.1-4 The center can also be a diffusional sink. The hemispherical drop may be used for absorption of an analyte gas from the environment. In all these cases there is radial symmetry within the drop. Diffusional acid|base titrations involve the movement of H+|OH- whose diffusion is fast, the diffusion coefficient, D, being in the order of 10-4 cm2/s. Therefore a 1-5 µL sample is nearly homogeneous even without stirring during the entire titration process. However, in samples of larger volume, and/ or in other types of titration where the reagent and/or analyte are heavier ions, significant inhomogeneity within the sample may arise during reagent dosing. This problem does not necessarily cause a deterioration of the performance of the technique as long as optical detection is used that is capable of reporting average bulk concentrations.5,6 Electrochemical detection, however, becomes problematic. We present here the concept of spatially averaging electrodes, SAE, that can, due to their special design, report 3D spatially averaged bulk concentrations of inhomogeneous samples. The local concentrations along the SAE are inhomogeneous in such cases. We show here that the SAE is capable of indicating the 3D-averaged bulk concentration despite that these inhomogeneous local concentrations are measured directly. The condition for this approach to work is that the concentration distribution in the sample exhibits a given type of symmetry. Within a specific type of symmetry, however, an infinite variation in the type of concentration distribution may exist. We illustrate here the concept of SAE with results obtained in microliter-sized hemispherical sample drops having a source in the center of the drop, which gives rise to nearly radial symmetry in the evolving concentration distributions. The center of the drop in the actual experiments was a diffusional port through which (1) (2) (3) (4) (5) (6)
Gratzl, M. Anal. Chem. 1988, 60, 2147–2152. Gratzl, M. Anal. Chem. 1988, 60, 484–488. Xie, H. J.; Gratzl, M. Anal. Chem. 1996, 68, 3665–3669. Kao, L. T.-H.; Hsu, H.-Y.; Gratzl, M. Anal. Chem. 2008, 80, 4065–4069. Gratzl, M.; Yi, C. Anal. Chem. 1993, 65, 2085–2088. Hui, K. Y.; Gratzl, M. Anal. Chem. 1997, 69, 695–698.
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Figure 1. Experimental setup: (A) top view schematic, (B) side view schematic, and (C) picture of the MEMS-fabricated SAE with and without a 1 µL sample drop on top of it. The basic design criterion for an SAE for hemispherical samples is also shown here. The sample is divided into small hemispherical shells (light gray band in panels A and B); if the thickness of the shell is small the length of the spiral electrode in a particular shell is proportional to the square of the average radius of that shell (in panel A). The spiral electrode is fabricated on a transparent Pyrex substrate surrounded by a hydrophobic ring of 1.56 mm i.d. The spiral electrode with 0.34 mm2 active surface area starts adjacent to the junction hole and ends at the hydrophobic ring. The junction of 0.1 mm diameter in the center of the spiral electrode is plugged with agarose gel saturated with 50 mM K3Fe(CN)6 + 0.2 M KCl. The sample is placed on top of the junction and the spiral electrode inside the hydrophobic ring to form a hemispherical drop. The cylindrical electrode with 0.15 mm2 active area and the microdisc electrode with 0.0046 mm2 active area are inserted radially into the hemispherical sample at the depth of 0.6 and 0.2 mm, respectively. The air-jet parallel to the Pyrex substrate can be used to stir the sample. The reference and the counter electrodes were positioned in the reagent reservoir underneath the Pyrex substrate.
ferricyanide ions were delivered into buffer sample drops. The diffusion coefficient of ferricyanide ion (D ) 6.3 × 10-6 cm2/s) being almost 2 orders of magnitude lower than that of H+ or OH-, diffusion cannot effectively homogenize even a 1 µL drop during the delivery process. Yet, the voltammetric current measured by a spatially averaging platinum line electrode deposited onto a two-dimensional (2D) substrate obtained in an unstirred solution is essentially identical with that in a stirred solution. In contrast, a cylindrical or a disk-shaped platinum electrode in the same sample reported very different values in stirred versus unstirred solutions. EXPERIMENTAL SECTION Apparatus. Instrumentation. The experimental setup (Figure 1) consists of a spatially averaging Pt spiral, Pt microdisc, or a cylindrical Pt working electrode incorporated in the diffusional microtitration platform.3,6-9 The Pt spiral electrode is fabricated on transparent Pyrex substrate which also serves as the supporting base on which the sample rests. The microdisc and the cylindrical (7) Diefes, R. S.; Hui, K. Y.; Dudik, L.; Liu, C. C.; Gratzl, M. Sens. Actuators, B 1996, 30, 133–136. (8) Cserey, A.; Gratzl, M. Anal. Chem. 1997, 69, 3687–3692. (9) Shetty, G. N.; Syed, N.; Tohda, K.; Gratzl, M. Anal. Sci. 2005, 21, 1155– 1160.
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electrodes are fixed to an XYZ micromanipulator (World Precision Instruments Inc., Sarasota, FL) and inserted in the sample from the top. A CHI 1030 potentiostat (CH Instruments, Austin, TX) controls the cell consisting of the working Pt electrode, Ag|AgCl reference, and stainless steel counter electrodes. A 0.5 mm diameter capillary air-jet was used to stir the sample with the output air pressure adjusted to ∼15 psi when results obtained with stirring were to be compared to those in unstirred solution. Electrodes. The spiral electrodes are fabricated on transparent Pyrex substrates (Dow Corning, Midland, MI) using microelectromechanical system (MEMS) fabrication techniques in the microfabrication laboratory at Case Western Reserve University (CWRU) as described in previous work.7 These electrodes are 3.4 mm in arc length, 0.1 mm wide, and cover 2π radians in angular distance as shown in Figure 1. The microdisc and cylindrical electrodes are made by pulling a 1 mm diameter capillary (Fisher Scientific) containing a 76 µm diameter Pt wire (Sigmund Cohn Corp., Mt. Vernon, NY). The capillary is pulled using a Narishige microelectrode puller (Narishige, Japan). The pulled capillary with the Pt wire is sealed by exposing the glass|Pt junction of the electrode to a Bunsen burner flame for a fraction of a second. The microdisc electrode is then polished on Grit 600 SiC abrasive paper (Buehler, Lake Bluff, IL) such that only the cross section of the Pt wire is exposed. The cylindrical electrode is made by cutting the excess Pt wire at the pulled end to expose only the required length of the wire. Commercially bought Ag|AgCl electrode (Bioanalytical Systems, Inc., West Lafayette, IN) is used as reference, and a stainless steel wire is the counter electrode. Minimizing Evaporation of the Sample. The sample is surrounded by a reservoir of DI water both of which are covered by a plastic lid to maintain humid atmosphere around and thus minimize evaporation. The air used to stir the sample in some experiments is first humidified by passing it through a water bath for the same purpose. Materials. All chemicals used in these experiments are from Sigma (Sigma Chemical Co., St. Louis, MO) and Fisher (Fisher Scientific, Pittsburgh, PA). All aqueous solutions are prepared using 18.2 MΩ · cm deionized water from Millipore Milli-QUV plus (Billerica, MA). Electrochemical Cell. A 0.2 M KCl solution is used as supporting electrolyte for all the solutions. The junction hole in the center of the Pyrex chip (Figure 1, top view) is plugged with 1% w/w type-I agarose prepared in 50 mM K3Fe(CN)6 + 0.2 M KCl solution. The junction is in direct contact with 50 mM K3Fe(CN)6 + 0.2 M KCl solution stored underneath the Pyrex chip in the reagent reservoir. The reference and the counter electrodes are positioned inside this reservoir to complete the cell. Ferricyanide is delivered into the sample through this junction via diffusion. A hydrophobic ring of silicone elastomer (Dow Corning, Midland, MI) of 1.56 mm i.d. confining the hemispherical sample is thick-film printed on the Pyrex chip. A hydrophobic band of silicone elastomer (Dow Corning, Midland, MI) is applied on the glass electrode shaft of the microdisc and cylindrical electrodes at the distance of insertion from the active electrode end. All the solutions and agar gel are freshly made and used within 1 day of their preparation.
Procedures. Reconditioning Procedure. A 1 µL drop of 50 mM K3Fe(CN)6 is placed on top of the Pyrex chip with a 2.5 µL Eppendorf pipet inside the hydrophobic ring atop the junction for 15 min before every delivery experiment, to allow the junction to equilibrate and thus reproduce the same initial condition for each experiment. The respective working electrode (spiral, microdisc, or cylinder) is also preconditioned in the mean time by applying 100 cyclic voltammograms (CVs) from 0.6 to -0.2 V at the scan rate of 0.5 V/s. For experiments with the microdisc and cylindrical electrode the electrodes are inserted radially into the sample from the top using the XYZ micromanipulator. The microdisc is inserted to a depth of 0.2 mm, whereas the cylindrical electrode is inserted to a depth of 0.6 mm. The drop is then removed with lint free wipe, and a 1 µL drop of 0.2 M KCl without K3Fe(CN)6, used as the test sample, is placed to start the delivery experiment. CVs are taken during the delivery as explained below. Experimental Protocol. At time zero the test sample is placed atop the Pyrex chip, and after waiting for 5 min current measurements are started. Five CVs from 0.6 to -0.2 V at 0.5 V/s scan rate are performed following the waiting period (during which the cell is open circuited) of 40 s. The last step is repeated five times in total. The sample is not stirred between the CVs for the first four sets, but is stirred for 25 s during the last 40 s waiting period before running the final five CVs. The data were collected at a sampling rate of 50 data points/s. Obtaining the Ox-Red Peak Current per Unit Area. The data are post processed to obtain the ox-red peak current which is defined as the difference between the reduction and the oxidation peak currents for the last (fifth) CV of every run to obtain the ox-red peak current at the time of that CV. The first four CVs for all the runs are used only for electrode reconditioning. This ox-red peak current is normalized for the active electrode area for each of the electrodes. Statistical Considerations and Control Experiments. Each of the delivery experiments with the spiral, the microdisc, and the cylindrical electrodes are performed four times for statistics. Control experiments without stirring and for quantification of evaporation of the drop are also performed with each of the electrodes. THEORY By dividing the total volume of an inhomogeneous sample into small-volume compartments, each of homogeneous concentration, the average concentration of the entire sample can be calculated by weighting each compartment’s concentration with its volume, summing up these weighted concentrations, and dividing the sum with the total volume. This procedure assumes that the sample can be divided into parts of a simple and repetitive shape that are individually homogeneous. This is generally the case when some type of symmetry in the actual concentration distribution exists. To obtain the average bulk concentration in such a sample electrochemically, the volumes for each compartment need to be translated into a corresponding fraction of active electrode surface area that measures the concentration in that compartment. On the basis of this concept SAE can be designed for different symmetries. Here we derive the theory for a line SAE deposited onto a 2D substrate that can be used to measure 3D-averaged bulk concen-
tration in a hemispherical sample having arbitrary, but radially symmetric, concentration distributions. The volume compartments in this case are hemispherical shells, one of them conceptually shown in Figure 1 with a gray band between dashed lines. A corresponding spatially averaging line electrode is shown with a spiral-shaped solid line in the top view. The requirement for this electrode to work properly is that its length within any shell be proportional to the volume of that shell. We derive here the equation defining the required shape for a voltammetric SAE, but the result will be approximately valid also for a potentiometric (such as redox) electrode.10 The electrode current measured in any volume fraction is directly proportional to the local concentration of the analyte present in the vicinity of the electrode in that volume fraction. Assuming that the electrode is a line of negligible width, the total current measured is i)s
∫
L
0
c(l) dl + b
∫
L
0
dl ) Sc¯ + B
where s is the slope of calibration per unit length, b is background current per unit length, l is running length of the electrode with l ) 0 where the electrode begins, c(l) is concentration of the measured species at electrode length l, and L is the total length of the electrode. If this electrode is an SAE then the total current will directly reflect, as shown in the equation on the right, the spatially averaged bulk concentration of the entire sample, ¯c, where S and B are the cumulative slope and background, respectively, for the entire length of the electrode. Assuming that the electrode begins (i.e., l ) 0) at a distance R0 from the center of symmetry (e.g., the source or sink of material), the first integral term can be expressed as a function of radius, r, as follows:
∫
L
0
c(l ) dl )
∫
R
R0
c(r)w(r) dr ) Lc¯
(1)
where w(r) ) (dl)/(dr) is a weighting function accounting for the change in integration variable from arc length to radius. The average concentration in a hemispherical solution domain is ¯c )
3 (R - R03) 3
∫
R
R0
c(r)r 2 dr
(2)
From eqs 1 and 2 we get w(r) )
3L dl ) r2 ) Rr2 dr R3 - R 3 0
where R is a geometry related constant: R ) (3L)/(R3 - R03). The arc length of the SAE as a function of radius will then be l(r) )
R 3 (r - R03) 3
(10) Diefes, R. S. Diffusional Microtitration System for the Analysis of Biomedical Samples. Master’s Thesis, Case Western Reserve University: Cleaveland, OH, 1993.
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The arc length versus the angle in polar coordinates is then described by the following differential equation: dl ) dr
1 + r ( dθdr ) 2
2
(3)
Solving eq 3 for θ θ)
1( 2 4 √R r - 1 - arctan √R2r4 - 1) 2
The value of θ is nonimaginary when R2r4 > 1. Assuming R2r4 . 1,
r)
2θR
(4)
This equation defines a special nonlinear spiral, called Fermat’s spiral.11 This is the shape of an SAE for a hemispherical sample with radially symmetric concentration distributions. As a comparison, an SAE that would measure 3D-averaged concentration in hypothetical samples of cylindrical symmetry would be an Archimedean spiral: θ r) β
(5)
where β is a geometry constant. The length increments of a linear spiral in concentric cylindrical shells increase linearly with radius similar to the volume of the individual shells themselves. In contrast, the volume of concentric hemispherical shells increases quadratically with the radius. This requires a quadratically increasing arc length for a unit increase in radius. Equation 4 shows that the angle and, thus, the arc length indeed increase quadratically with the radius. RESULTS AND DISCUSSION To demonstrate the concept of SAE we fabricated a thin (width 100 µm and height 5000 Å) Pt line electrode deposited on top of an inert substrate in the shape of a Fermat’s spiral designed according to eq 4, as shown in Figure 1A, top view. The images in Figure 1C show an actual device made to perform measurements in 1 µL hemispherical sample drops, shown without and with a sample drop deposited on it. Cyclic voltammograms of K3Fe(CN)6 were recorded using this Pt spiral electrode in stationary, unmixed buffer drops while ferricyanide was entering the sample from the junction in the center. Both the dosing of ferricyanide into the drop through the junction and its spreading in the drop occur via diffusion alone. This leads to approximately hemispherical symmetry of the evolving concentration distributions in the drop. To monitor the processes of diffusional dosing and spreading, cyclic voltammetry was performed at regular intervals in the sample drop. The difference between the currents at the reduction and oxidation peak (ox-red differential current) divided by the electrode area was plotted against time as shown in Figure 2C. The red circles (O) are from an experiment conducted without (11) Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed.; CRC Press: Boca Raton, FL, 1997.
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Figure 2. Ox-red peak currents (explained in the Experimental Section) from one experiment out of four parallels calculated using (A) microdisc, (B) cylindrical, and (C) spiral electrodes during the course of K3Fe(CN)6 delivery into a 1 µL sample of 0.2 M KCl with and without stirring. The ox-red peak currents were obtained from the recorded CVs (0.6 to -0.2 V at the scan rate of 0.5 V/s). Maximum change in ox-red peak current after stirring the sample is observed for the microdisc electrode (A), 11.2%; followed by the cylindrical electrode (B), 7.9%; the minimum change in ox-red peak current after stirring the sample is observed for the SAE spiral electrode (C), 0.2%.
stirring. The black dots indicate ox-red currents obtained in an identical experiment except that the sample drop was homogenized by stirring before the last set of CVs was recorded. For comparison, the same experiments were performed using a Pt microdisc (Figure 2A) and a cylindrical Pt electrode (Figure 2B). These electrodes are schematically shown in Figure 1, side view. The data shown in Figure 2 are representative of four separate experiments for each electrode. An increase in current with time is observed in the experiments performed with each of the three electrodes. To verify that this increase was caused mainly by the continuous delivery of K3Fe(CN)6 into the sample through the junction in the center and its diffusional spreading, two control experiments were done. The first experiment was to confirm that evaporation has a minimal effect on sample concentration during the time course of a typical experiment. A sample drop was placed inside the hydrophobic ring on Pyrex substrate that did not have a junction hole in the center. To provide for a reference an Ag|AgCl electrode was inserted directly into the sample from the side, instead of using an agar junction as in the delivery experiments. When a 1 µL buffer drop containing 50 mM ferricyanide was deposited on this setup and the ox-red peak current per unit area was measured using a microdisc electrode, a current increase of 9% was measured during the time of the actual experiments. The increase measured by the same microdisc electrode during an
actual delivery experiment when the sample was not stirred was 81% (data not shown). In the second control experiment a 50 mM ferricyanide sample was deposited on top of an actual device with a junction in the center that was saturated with 50 mM K3Fe(CN)6. The normalized ox-red peak current obtained from this experiment showed that there was only 7% increase in the measured current at the end of the experiment (data not shown). These control experiments confirm that the change in current due to sample evaporation or possible other effects in the course of the experiment is small, and thus, the current increase measured during the actual delivery experiments is due primarily to diffusional dosing of K3Fe(CN)6 via the junction and its spreading in the drop. In actual delivery experiments the absolute ox-red peak current was found to be largest for the Pt spiral electrode, followed by the cylindrical and then the microdisc electrode, reflecting the decreasing active surface area of these electrodes. To better compare the change in ox-red peak current across the electrodes, the current values were normalized for the electrode area as explained in the Experimental Section. These normalized currents are shown in Figure 2 indicating that the highest normalized ox-red current is obtained with the microdisc electrode. This is because of efficient, nearly spherical mass transport expected at a microdisc electrode. Cylindrical mass transport is less efficient which results in comparatively smaller current per area for the Pt wire electrode. The lowest normalized currents were measured at the Pt spiral since mass transport at the flat spiral electrode is approximately hemicylindrical. The rates of change of the curves in Figure 2 that correspond to the individual sensitivities follow this trend. The normalized ox-red peak currents obtained at 9 min after the beginning of delivery without stirring, and immediately after stirring of the drop, can now be analyzed using the data shown in Figure 2. The current for the microdisc electrode obtained in the experiment after stirring is 49 µA/mm2 (Figure 2A). This is 11% larger than the current value of 44 µA/mm2 obtained without stirring at the same time point. The same increase caused by stirring for the cylindrical electrode is 8%, from 37 to 40 µA/ mm2 (Figure 2B). The change found in an identical experiment performed with the Pt spiral electrode is, in contrast, negligible at 0.2% from 39.5 and 39.6 µA/mm2 (Figure 2C). The figure shows the results of one experiment for each electrode type out of four parallels. The current change between unstirred and stirred drops from the four repeats done with the Pt spiral electrode is 0.8% ± 0.9% which is significantly smaller than the average change obtained for the cylindrical electrode, 6.3% ± 1.4%, and the same for the microdisc electrode, 10.8% ± 1.3%. These results indicate that the fabricated nonlinear spiral Pt electrode deposited on a 2D substrate indeed measures the true 3D average concentration of inhomogeneous but radially symmetrical samples. These results lend support to the validity of the concept of SAE. Finally, a theoretical observation can be made by comparing the linear spiral and the here-derived square-root-type spiral electrodes (Figure 3). One might think that the best performance for spatial averaging could be obtained by using a solid circular
Figure 3. Comparison of two possible 2D SAE designs: (A) line electrode on a 2D substrate for cylindrical symmetry; (B) same for hemispherical symmetry. Uniform roughness corresponds to a hypothetical circular solid electrode for cylindrical symmetry (C). Roughness increasing with radius illustrates the surface of a solid circular electrode for radial symmetry. (D) The density of the ripples illustrates the required specific surface area as a function of radius.
electrode as the base of the sample. This is, interestingly, not the case, however. A solid 2D circular electrode would be identical with a linear spiral with infinite wounds (β f ∞ in eq 5). Such a solid disk would therefore be suited as an SAE for a hypothetical cylindrical diffusion field, but not for a hemispherical concentration field. The analogous solid disk electrode for a radially symmetrical system would correspond to Fermat’s spiral with infinite wounds (eq 4 when R/2 f ∞). This would be a disk whose effective area above the same 2D domain would increase at increasing distances from the center. In other words a solid electrode for the spatial averaging of radially symmetric concentration distributions would have to have fractal dimensions (2 + f)D where f is the additional fractal dimension that increases with the radius. A possible realization of such a hypothetical disk is a solid electrode that is flat (2D) at the center but becomes increasingly rough with increasing distance from the center (see Figure 3D for illustration). We note that one could consider just an arc segment (similar to a pizza slice) of a solid disk electrode (analogous here to a whole pizza) with modified radial edges. These two edges should be curved outward so that the area of the “pizza slice” increases quadratically with the radius to obtain 3D spatial averaging for a hemispherical droplet. To reassemble a whole “pizza” from such slices would lead to increasing overlap between slices farther from the center. This clearly shows the fractal nature of the SAE Analytical Chemistry, Vol. 81, No. 6, March 15, 2009
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problem: apart from the center, increasingly larger areas are needed than what is available in a flat surface. The 2D “pizza slice” and 1D “spiral” SAE design are just two of the many possible SAE designs governed by the same principles of averaging in an inhomogeneous hemispherical droplet. A zerodimensional version of the same is a radial array of microdisc electrodes with decreasing spacing between these electrodes, positioned at the intersection points of a radius of the droplet and the Fermat’s spiral, as one travels radially outward from the center of the hemispherical drop. Within the scope of these generic designs many different variants are possible by modifying the geometric constant R (specific for hemispherical sample) which influences the density of the nonlinear spiral. We chose to explore the 1D line electrode because of two practical considerations: (1) to keep bulk depletion of the electroactive species during the measurement minimal, and (2) to incorporate redundancies in the measurement to correct for eventual minor deviations from radially symmetric concentration distributions.
and made if the type of symmetry of the possible concentration distributions is known. This concept may find useful applications in the future for electrochemical measurements in small domains that are difficult or undesirable to mechanically stir. Potential application areas may be in diffusional microtitrations for quantitative analysis of biomedical, environmental, and industrial samples with limited available volume. Laboratory-on-a-chip and microfluidic applications are other areas where the SAE concept could be implemented and used. Gas absorption into small drops for preconcentration and following measurement would also produce radially symmetric concentration distributions.
CONCLUSIONS In this work we introduced the concept of SAE and proved on a specific example that such electrodes can indeed be designed
Received for review October 30, 2008. Accepted January 20, 2009.
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ACKNOWLEDGMENT We thank Professor C. C. Liu and Laurie Dudik, managing director, CWRU for fabrication of the sensors.
AC802288G
