Anal. Chem. 1997, 69, 4482-4489
Improved Ruggedness for Membrane-Based Amperometric Sensors Using a Pulsed Amperometric Method Xiaomei Wang and Harry L. Pardue*
Department of Chemistry, 1393 BRWN Building, Purdue University, West Lafayette, Indiana 47907-1393
This paper introduces a new approach to the use of membrane-based amperometric sensors which is expected to improve the ruggedness of these sensors significantly relative to the steady-state method in common use. In this new method, the fixed-voltage source used with the conventional steady-state method is replaced by a pulsed-voltage source. Unlike the fixed-source approach, which yields steady-state currents corresponding to large differences between analyte concentrations inside and outside the isolating membrane, the pulsed-source approach permits measurement of currents corresponding to near-equilibrium conditions between solutions inside and outside the membrane. Because the measured currents correspond to near-equilibrium conditions, results are expected to be virtually independent of variables that affect rates of mass transport to and across the membrane. The new approach is evaluated using the “oxygen electrode” as a model system. Results obtained using the new method are compared with results obtained using the conventional steady-state option as well as a coulometric approach described recently. The reproducibility of the pulsed amperometric approach and the scatter of data about least-squares calibration lines are an order of magnitude or more better than for the conventional steady-state option. As expected, the pulsed amperometric method is 40-100-fold less dependent on changes in membrane thickness, stirring rate, and temperature than the conventional steady-state option. The purpose of this study is to develop a measurement and data-processing method to improve the ruggedness of membranebased amperometric sensors such as the “oxygen electrode”.1,2 The oxygen electrode is one of several types of devices that use selective membranes and amperometric electrodes to isolate and detect a variety of species.3 Most applications of these devices are based on measurements of steady-state currents with a constant applied potential. The steady-state currents result from a balance between two or more competing rate processes, including the rate of mass transport from the bulk solution to the electrode surface and the rate of the electrochemical reaction at the electrode. Accordingly, results based on steady-state currents usually have large dependencies on variables such as stirring rate and membrane properties, which influence rates of mass transport, and temperatures, which influence rates of mass transfer and (1) Hitchman, H. L. Measurement of Dissolved Oxygen; Wiley: New York, 1978. (2) Wang, H. Y.; Li, X. Biosensor 1989, 4, 273-285. (3) Cao, Z.; Buttner, W. J.; Stetter, J. R. Electroanalysis 1992, 4, 253.
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electrochemical processes. The poor ruggedness of the steadystate option makes it necessary to control a variety of variables within narrow tolerances to obtain reliable results.1,4,5 Earlier studies in this6,7 and other4,5 laboratories have shown that alternative measurement/data-processing approaches can be used to improve the ruggedness of the membrane-based oxygen sensor significantly relative to the more conventional steady-state option. All of these alternative options can be grouped into one general category, identified herein as a “pseudoequilibrium” approach. In all these pseudoequilibrium options described to date,4-7 the solution inside the membrane must be equilibrated with the sample solution prior to application of the polarizing voltage. After the polarizing voltage is applied to the preequilibrated sensor, some feature of the resulting current is measured and related to analyte concentration. In the earliest study,4 the measurement objective was the rate of decrease in current immediately after application of the polarizing voltage. In a subsequent study,5 the measurement objective was the Faradaic current following application of a voltage pulse of a few milliseconds, with an off-time between pulses to permit restoration of oxygen depleted by the previous pulse. In a later study, the measurement objective was the Faradaic component of the current at the instant voltage was applied after preequilibration of the oxygen electrode with the sample solution; the initial current was obtained by nonlinear extrapolation of current vs time data to the point at which polarizing voltage was applied.6 In the most recent study,7 the measurement objective was the charge corresponding to electrolysis of all the oxygen in a fixed volume of solution inside the membrane, which was obtained by nonlinear extrapolation of charge vs time data to t ) 0 (the point at which polarizing voltage was applied) and t f ∞ (the point at which electrolysis current would reach zero in the absence of mass transport from the sample solution). All of these options had desirable performance characteristics, including linear dependence on analyte concentration and reduced dependence on variables that affect rates of mass transport. However, all the approaches have one or more limitations, including the need to equilibrate the electrode and sample prior to application of the polarizing voltage,4-7 potential interference from charging currents,4-7 and/or inability to compensate for variables that affect electrode response.4-6 (4) Mancy, K. H.; Okun, D. A.; Reilley, C. N. J. Electroanal. Chem. 1962, 4, 65-92. (5) Mancy, K. H. In Chemistry and Physics of Aqueous Gas Solutions; Adams, W. A., Ed.; Electrochemistry Society: Princeton, NJ, 1969; pp 281-289. (6) Uhegbu, C. E. and Pardue, H. L. Anal. Chem. 1992, 64, 2378-2382. (7) Williams, S.; Pardue, H. L.; Uhegbu, C. E.; Smith, A. M.; Studley, J. Talanta 1996, 43, 1379-1385. S0003-2700(97)00106-6 CCC: $14.00
© 1997 American Chemical Society
The present study was undertaken to develop a method that would eliminate the need for the preequiiibrium step and would reduce or eliminate effects of charging current. The new option involves the use of a pulsed-voltage source combined with curvefitting methods to extrapolate transient responses to equilibrium values. The primary purpose of the pulsed source is to facilitate the measurement of currents corresponding to a near-equilibrium condition between oxygen concentrations in the solutions inside and outside the membrane. Ideally, it would be desirable to use a very narrow pulse width to minimize the amount of oxygen electrolyzed inside the membrane. However, in practice it is necessary to use a pulse width of sufficient duration (e.g., 20 ms8) to permit charging currents to decay to negligible values before Faradaic currents are measured near the end of each pulse.8 Accordingly, the time interval between pulses (e.g., 6 s) is selected to be long enough to permit most of the oxygen depleted by electrolysis to be replaced by diffusion from the sample solution. If these conditions can be met, it should be possible to obtain a current corresponding to an oxygen concentration inside the membrane approximately the same as that in the sample solution. Currents measured in this way should be virtually independent of variables that affect rates of mass transfer of oxygen to and through the membrane. Two options are used to obtain the pseudoequilibrium current. In one option, identified herein as the direct measurement option, the measurement process described in the previous paragraph is continued until the system reaches a steady state, and the nearequilibrium current is measured and related to concentration. In the other option, identified herein as the extrapolation option, the current is measured at several points in time as the response approaches the pseudoequilibrium condition, and a curve-fitting method9,10 is used to extrapolate the time-dependent currents to the pseudoequilibrium values. Results reported herein compare performance characteristics of this pulsed-voltage amperometric (PVA) option with results obtained using both the conventional fixed-voltage steady-state (SS)3 and fixed-voltage coulometric (FVC)7 options. Primary emphasis is on ruggedness to changes in selected variables including membrane thickness, stirring rate, and temperature. However, performance characteristics discussed also include reproducibility and linearity. EXPERIMENTAL SECTION Instrumentation. Commercially available oxygen-selective electrodes (Model 5331, Yellow Springs Instrument, Yellow Springs, OH) and electrochemical instrumentation (Model CV50W voltammetric analyzer, Bioanalytical Systems, Inc., West Lafayette, IN) were used throughout this study. Current vs time data were collected and stored temporarily using software with the electrochemical instrumentation. The current/time data were subsequently transferred to a laboratory computer (Model PMV1448NI, Gateway 2000, North Sioux City, SD) for final processing. Solutions. All solutions were prepared in doubly distilled water. (8) Voltammetry: Stationary solution techniques. Heineman, W. R. In Chemical Instrumentation: A Systematic Approach; Strobel, H. A., Heineman, W. R., Eds.; Wiley: New York, 1989; pp 1126-1132. (9) Mieling, G. E.; Pardue, H. L. Anal. Chem. 1978, 50, 1611-1618. (10) Mieling, G. E.; Pardue, H. L.; Thompson, J. E.; Smith, R. A. Clin. Chem. (Winston-Salem, NC) 1979, 25, 1581-1590.
Cleaning Solution. A solution containing 3% by weight of ammonia was used to clean the electrode surface between experiments and before storage. Probe Solution. The solution inside the membrane contained 2.2 mol/L potassium chloride plus 0.0625 mL of a surfactant (Photo-Fro, Eastman Kodak Co., Rochester, NY) per 32 mL of solution. Samples. An air-saturated solution was used to test effects of membrane thickness, stirring rate, and temperature on different measurement options. Serial dilutions of air-saturated solutions were used to test linearity. Samples used to test linearity were prepared by diluting measured aliquots of air-saturated water with deaerated water. Air-saturated water was prepared by exposing doubly distilled water to the atmosphere for at least 24 h. Oxygen-free water was prepared by deaerating water with oxygen-free nitrogen for at least 30 min prior to use. One potential problem in handling the solutions used to test linearity is loss (or gain) of oxygen to (or from) the atmosphere surrounding the solutions. To minimize this problem, the tip of the oxygen electrode, inserted through a rubber stopper that sealed the top of the temperature-controlled sample compartment (a 25 mL water-jacketed beaker), was first immersed in a measured volume (usually 10 mL) of oxygen-free water with a stream of nitrogen flowing over the surface of the oxygen-free water. Then, after the solution was equilibrated to the desired temperature (26 °C unless stated otherwise), a measured aliquot of air-saturated water (5-10 mL) at the same temperature was added to the oxygen-free water through a hole in the rubber stopper. Measurements were started immediately after addition of the aliquot of air-saturated water and were completed within 60-90 s after addition of the aliquot of sample. Total sample volumes were usually in the range of 15-20 mL, so that the surface-to-volume ratio was not unusually large. These precautions were expected to minimize the amount of oxygen transferred between aqueous samples and the surrounding atmosphere before and during the measurement step. For temperature dependence studies, special precautions, including a large volume-to-surface area ratio and the shortest possible measurement times, were taken to reduce problems from changes in oxygen solubility with temperature. For these studies, the sample compartment was a 250 mL bottle with a neck opening just large enough for insertion of the oxygen electrode. The bottle was equilibrated in a thermostated water bath at the desired temperature, and sufficient air-saturated water at room temperature was added to fill the bottle to the narrow neck with the electrode inserted far enough into the water sample to minimize surface effects. The electrode was inserted immediately, and water in the bottle was stirred with a magnetic stirring bar to speed temperature equilibration. Measurements were completed within 5-6 min after adding water to the bottle. Procedures. All voltages specified throughout are relative to the silver/silver chloride electrode without any applied offset voltage. Preparation of Sensor. Before each set of experiments, the surfaces of the platinum and silver electrodes were cleaned with a cotton swab dipped in 3% ammonium hydroxide solution and rinsed thoroughly with doubly distilled water. The bare probe was then wetted with the probe solution, and a piece of membrane was fastened onto the probe surface using an O-ring. The probe Analytical Chemistry, Vol. 69, No. 21, November 1, 1997
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was tested at the beginning of each day using an air-saturated solution. The probe was considered to be in good condition if the steady-state current was within 1% of the average value obtained over several days using an air-saturated solution. Sample Treatment. Samples were handled as described above. The sample compartment was a water-jacketed 25 mL beaker, around which water was circulated from a temperature-controlled water bath. Except for the temperature dependence study, measurements were made at 26 °C. Pulsed-Voltage Amperometric Option. For some of the pulsedsource studies, the oxygen content of the solution inside the membrane was first depleted by application of a fixed polarizing voltage (-0.8 V vs Ag/AgCI); in other pulsed-source studies, the oxygen concentration inside the membrane was not depleted between samples. In all the pulsed-source studies, voltage pulses with a 20 ms pulse width were used, and current was sampled during the final 3 ms of each pulse. In studies designed to establish optimum conditions, different pulse amplitudes, numbers of pulses, and periods were studied. For the final studies of linearity, imprecision, and variable dependencies, we used a -0.8 V pulse amplitude, a 20 ms pulse width with current sampled during the final 3 ms of each pulse, and a 6 s interval between pulses. Results show that these conditions satisfy the competing criteria of acceptable sensitivity, short measurement time, and limited depletion of oxygen inside the membrane by the measurement process. Data Processing. Data for current vs time were processed in two ways. In one option, identified herein as the extrapolation option, a fit of a first-order model to current vs time data was used to extrapolate the data to t ) 0 and t f ∞. The difference in the two extrapolated values was used to determine concentration. In the other option, identified herein as the direct measurement option, the average of the last five data points was computed and used as the measured value of the pseudoequilibrium signal, I∞. Fixed-Voltage Coulometric Option. A second general approach,7 identified herein as a coulometric option, was also evaluated. In this option, the oxygen electrode was first equilibrated with each sample, after which a fixed polarizing voltage (-0.8 V) was applied, and current vs time data were digitized and stored. The current vs time data were then integrated to obtain charge vs time, and a combined zero-order/first-order fit of the data was used to obtain the first-order component of the charge. The maximum value of the first-order component of the charge is expected to correspond to electrolysis of all the oxygen in a fixed volume of solution inside the membrane. This computed value of the first-order component of the charge, Q∞1, was used to quantify oxygen concentration. MATHEMATICAL TREATMENT In all of the discussion below, it is assumed that measurements are made after charging currents have decayed to zero. Any uncompensated charging current will likely causes a non-zero offset in calibration plots. It has been suggested that the current/time behavior of membrane-based amperometric sensors is described by the Cottrell equation,3,5 which suggests that the transient current should vary inversely with the square root of time. However, because the experimental conditions used in this study do not satisfy the assumptions inherent in the derivation of the Cottrell equation (fixed applied voltage and diffusion control of mass transport), the Cottrell equation was not expected to fit timedependent responses. As will be shown later, this expectation 4484
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was confirmed experimentally. Accordingly, an alternative mathematical treatment is described here. As will be shown later, equations resulting from this treatment fit time-dependent responses for current and charge well, permit extrapolation of current vs time data to pseudoequilibrium values, and permit resolution of the components of charge corresponding to regions before and after steady-state conditions are reached. Current-Time Relationships. The analyte concentration inside the membrane will be controlled by two competing processes, namely a decrease due to electrolysis at the electrode surface and an increase due to mass transfer from the sample solution to the electrode surface. This is represented mathematically by
-(dCt/dt) ) (ke/nFV)Ct - km(Cs - Ct)
(1a)
in which ke is a proportionality constant between electrolysis current, It, and concentration (It ) keCt), Ct is the time-dependent analyte concentration inside the membrane, Cs is the analyte concentration in the sample, n is the number of electrons involved in the electrochemical reaction, F is the Faraday constant, V is the volume of fluid inside the membrane, and km is a mass transfer constant. Integrating eq 1a to obtain an expression for Ct and using the relationship It ) ke Ct, we obtain
It ) (kekm/γ)(1 - exp(-γt))Cs + ke[exp(-γt)]C0
(2a)
in which C0 is the analyte concentration inside the membrane at t ) 0 (the point at which the polarizing potential is applied), Cs is the analyte concentration in the sample, γ ) (ke + kmnFV)/nFV, and other symbols are as defined above. This equation predicts an exponential change toward a steadystate current, with the direction of the change depending on the relative values of C0 and Cs. Pulsed-Amperometric Option. If measurement conditions can be designed so that the rate of depletion of the net oxygen concentration inside the membrane by the electrochemical process is negligible, then diffusion would be the rate-limiting process. Our approach to establishing this condition in this study was to use a voltage amplitude (0 to -0.8 V), peak width (20 ms), and time interval between peaks (6 s) such that most of the oxygen consumed by each pulse would be restored by diffusion between pulses. Assuming that these conditions are such that diffusion is the rate-limiting process, then the first term in eq 1a, which corresponds to the rate of the electrochemical process, can be ignored, and the simplified form of eq 1a is
(dCt/dt) = km(Cs - Ct)
(1b)
Integrating this equation and following procedures described for development of eq 2a, we obtain
It = keCs + ke[exp(-kmt)](C0 - Cs)
(2b)
If the initial oxygen concentration inside the membrane is greater than the oxygen concentration in the sample (C0 > Cs), then, according to eq 2b, the initial current will be greater than the final current, and the time-dependent current will decrease toward
the equilibrium value. On the other hand, if the initial oxygen concentration inside the membrane is less than the oxygen concentration in the sample (C0 < Cs), then the initial current will be less than the final current, and the time-dependent current will increase toward the equilibrium value. At long times (f ∞), the last term approaches zero, and eq 2b simplifies to
I∞ = keCs
(2c)
Whatever the initial conditions, for situations in which it is possible to measure the electrolysis current without depletion of the net oxygen concentration inside the membrane, then, according to eq 2c, the equilibrium current, I∞, should be proportional to the oxygen concentration in the sample. Moreover, because this equilibrium current should correspond to equal oxygen concentrations inside and outside the membrane, results obtained in this way should be independent of rates of mass transport processes, and the resulting methodology should be very rugged to variables such as stirring rate and membrane properties that influence rates of mass transport. On the other hand, results obtained in this way would depend on the kinetics of the electrochemical processes. For situations in which the oxygen electrode is preequilibrated with sample prior to application of polarizing voltage, then the current measured as described above should represent sample concentration without any influence from mass transport processes. Alternatively, if the electrode is not preequilibrated with the sample prior to application of polarizing voltage, then the current vs time response can be monitored to equilibrium or, if eq 2b is valid, then a portion of the current vs time response can be extrapolated to equilibrium using a first-order model.9,10 Fixed-Voltage Coulometric Approach. As indicated above, the pulsed-source amperometric approach is expected to compensate for changes in variables that affect mass transport processes but is not expected to compensate for variables that affect the kinetics of the electrochemical processes at the electrode surface. However, it is expected that it should be possible to use a charge-based (coulometric) approach to compensate for both groups of variables. Noting that the rate of change of charge with time is equal to the current (dQ/dt ) I), equating eq 2 to the rate of change of charge and integrating, we obtain
Qt ) (kekm/γ)Cst + (kekm/γ2)(exp(-γt) -1)Cs + (ke/γ)[1 - exp(-γt)]C0 (3a)
in which all symbols are as described above. This equation predicts combined first-order/zero-order behavior for charge vs time. In the conventional applications of the oxygen electrode, in which a fixed voltage is applied to the electrodes, we have observed experimentally that the rate of electrochemical reaction inside the membrane is approximately 10-fold faster than diffusion of oxygen across a single layer of the membrane. For example, in an early study,11 the pseudo-first-order rate constant for equilibration of oxygen across a single layer of the isolation (11) Uhegbu, C. E.; Pardue, H. L. Anal. Chim. Acta 1990, 237, 413-420.
membrane was determined to be 0.2 s-1, corresponding to a halflife of about 3.5 s. In a subsequent study,6 it was determined that the pseudo-first-order rate constant for electrochemical depletion of oxygen inside the isolation membrane is 2.3 s-1, corresponding to a half-life of about 0.3 s. Accordingly, for the usual configuration of the oxygen electrode, it follows that the rate of electrochemical depletion of oxygen inside the membrane is approximately 10fold faster than the rate of mass transport of oxygen across a single layer of the isolation membrane. According to eq 3a, the charge vs time relationship for the oxygen electrode operated in the conventional mode (fixed potential) should be described by a combined first-order/zeroorder process. The zero-order component of the charge vs time response corresponds to integration of the steady-state current that results when the rate of mass transport is equal to the rate of the electrochemical reaction inside the membrane. The firstorder component corresponds to the part of the process during which the oxygen concentration inside the membrane is changing with time. Accordingly, because the rate of the electrochemical process inside the membrane is at least 10-fold faster than the rate of mass transport of oxygen across the membrane, it follows that the first-order component of the charge vs time data corresponds primarily to the electrochemical process inside the membrane. By fitting a model for parallel first-order and zero-order processes to charge vs time data, it should be possible to resolve the first-order component of the charge from the zero-order component. As explained in the next paragraph, the maximum value of the first-order component, Q∞,1 obtained from such a deconvolution process should be proportional to oxygen concentration and independent of variables that affect rates of mass transport as well as the rate of the electrochemical reaction. Because the layer of solution between the membrane and the platinum electrode is very thin (a few micrometers), it is reasonable to assume that the total charge will correspond to electrolysis of all the oxygen in the fixed volume of solution between the membrane and the platinum electrode. Finally, if the solution inside the membrane is equilibrated with the sample solution before the potential is applied to the electrodes, then the total charge should be equivalent to depletion of oxygen in a small fixed volume of the sample solution. Accordingly, in the absence of charging current, the maximum value of the first-order component of the charge should be proportional to oxygen concentration and independent of variables that affect rates of mass transport as well as those that affect the rate of electrochemical reaction at the electrode. These expectations can be supported mathematically by assuming that the rate of mass transport is slow compared to the rate of the electrochemical process. For example, assuming that km(Cs - Ct) , (ke/nFV)Ct, eq 1a can be simplified to
-(dCt/dt) = (ke/nFV)Ct
(1c)
Following procedures used to obtain eq 3a from eq 1a, it is easily shown that the first-order component of the charge for the circumstance described above should be given by
Qt,1 = nFV[1 - exp(-δt)]Cs
(3b)
in which δ ) ke/nFV, and all symbols are defined above. Analytical Chemistry, Vol. 69, No. 21, November 1, 1997
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Alternatively, eq 3b can be obtained from eq 3a by assuming that kmV , ke, a reasonable assumption given that km < ke and the volume between the platinum electrode and the membrane is very small, and noting that C0 ) Cs because the solutions inside and outside the membrane are to be equilibrated before voltage is applied across the electrodes. As the first-order process approaches steady state (when f ∞), the exponential term in eq 3b approaches zero (exp(-δt) f 0 when t f ∞). Accordingly, the maximum value of the firstorder component of the charge, Q∞,1 is given by
Q∞,1 = nFVCs
(3c)
This expression predicts a proportional relationship between the maximum value of the first-order component of charge and concentration and involves neither mass transport nor electrochemical rate parameters. Accordingly, the coulometric approach applied to a situation in which an oxygen electrode is preequilibrated with sample solution prior to application of a polarizing voltage has the potential to compensate for variables that affect both mass transport and electrochemical processes. RESULTS AND DISCUSSION Unless stated otherwise, all random uncertainties are cited at the level of one standard deviation unit ((1 SD). Pulsed-Voltage Amperometric Option. Response Curves. Assumptions leading to eqs 2b and 2c require that currents be measured without disturbing the net oxygen concentration inside the membrane. In the absence of charging current, one could use a very narrow pulse to minimize the amount of oxygen consumed during each pulse. However, to allow time for charging currents to decay, we chose a pulse width of 20 ms.8 Because it was expected that this pulse width would cause depletion of oxygen inside the membrane, it was necessary to consider different periods between pulses to allow time for oxygen to be restored by diffusion. Accordingly, experiments were done using a 20 ms pulse width and different periods between pulses. Some typical response curves illustrating effects of the interval between pulses as well as other features are illustrated in Figure 1. The solid points in each frame represent experimental data; the solid, dashed, and dotted lines represent fits of various models to the data. Figure 1A,B represents response curves for situations in which the oxygen electrode was immersed in an oxygen-free solution and potential was applied until the current decayed to zero before an aliquot of an oxygen-containing sample was added to the solution surrounding the electrode. Data in Figure 1A represent a situation with a short (2 s) time period between pulses. The plot curves gradually toward a nearlinear region at longer times, obviously not satisfying the firstorder relationship predicted by eq 2b. Our interpretation of this behavior is that the time interval between pulses is not sufficient for all the oxygen depleted by each pulse to be restored by diffusion across the membrane between pulses. Data in Figure 1B represent the situation for a 6 s time interval between pulses. The plot approaches a steady-state value after about 60 s and, as discussed below, is fit very well by a first-order model, as predicted by eq 2b. Virtually identical behavior was observed for delay intervals of 15 and 30 s between pulses. Our interpretation of this behavior is that the time interval between 4486 Analytical Chemistry, Vol. 69, No. 21, November 1, 1997
Figure 1. Data for current vs time for a membrane-based oxygen sensor. Fitting model, all frames: (a, ‚‚‚) Cottrell equation; (b, s) firstorder/zero-order; (c, - - -) first-order. (A) Oxygen inside membrane depleted before sample (5 mL, air-saturated) was added to 10 mL of oxygen-free solution; pulse width, 20 ms; pulse period, 2 s. (B) Same as (A), except 3 mL of oxygen-saturated solution added to 10 mL of oxygen-free solution; pulse period, 6 s. (C) Oxygen inside membrane not depleted between samples. Conditions: 5 mL of sample added to 10 mL of oxygen-free solution; pulse width, 20 ms; pulse period, 6 s. (a-f) 0, 2.36 × 10-3, 4.33 × 10-3, 6.0 × 10-3, 7.43 × 10-3, and 8.66 × 10-3 mmol/L.
pulses is sufficient for most of the oxygen depleted by each pulse to be restored between pulses. Accordingly, the steady-state response in this situation represents a pseudoequilibrium condition between solutions inside and outside the membrane.
Fitting Models. Three models (the Cottrell equation, a combined first-order/zero-order model, and a first-order model) were evaluated for fitting current/time responses such as those in Figure 1A,B. The dotted lines in Figure 1A,B represent the Cottrell model. The solid line in Figure 1A represents a fit of a combined first-order/zero-order model to the data; the dashed line in Figure 1B represents a fit of a first-order model to the data. A fit of the first-order/zero-order model to data in Figure 1B was virtually superimposed with the fit of the first-order model and is not included. As expected, the Cottrell equation did not fit any of our data well; on the other hand, the combined first-order/zero order model fit all the data very well. More significantly, for longer intervals between pulses, the current/time responses follow first-order behavior as predicted by eq 2b, supporting the validity of assumptions used to develop this equation. Other implications of this behavior are discussed below. The dashed line in Figure 1A illustrates how the first-order component of a combined first-order/zero-order response can be resolved. This is most relevant to the coulometric option discussed below. Measurement Options. Two measurement options were evaluated in this study. In the option illustrated by data in Figure 1B, current vs time data were obtained for a situations in which the oxygen concentration inside the membrane was depleted prior to addition of the sample and application of the pulsed polarizing voltage. These data were particularly useful in evaluating different curve-fitting models, as illustrated above. In the other option, samples were added without depletion of oxygen inside the membrane, and the resulting current/time responses were monitored to equilibrium. Some results obtained in this way are illustrated in Figure 1C for situations in which pulsed-voltage measurements were made immediately after addition of sample and without prior depletion of oxygen inside the membrane. In each case, the response approaches a pseudoequilibrium steadystate value representative of the oxygen concentration in each sample. The smaller changes relative to data in Figure 1A,B result from the fact that differences between initial concentrations inside and outside the membrane are smaller in the latter case (Figure 1C) than in the former. Data-Processing Options. According to eq 2c, in the absence of charging current, the equilibrium current, I∞, obtained for situations such as that in Figure 1B should be proportional to oxygen concentration and independent of variables that affect mass transport across the membrane. As noted earlier, two dataprocessing options, identified herein as the extrapolation option and the direct measurement option, were evaluated for the pulsedvoltage approach. In the extrapolation option, a first-order model9,10 is used to extrapolate current-time data from regions before equilibrium to values at equilibrium (see Figure 1B). In the direct measurement option, the current-time response is monitored to equilibrium and several data points in the equilibrium region are averaged to obtain a direct measure of the equilibrium current. Linearity Data. Figure 2 is a plot of intercept-corrected current vs concentration (5.1 × 10-3-1.3 × 10-1 mmol/L) obtained using the direct measurement, pulsed-amperometric option; similar results were obtained with the extrapolation option. On the one hand, the plot is observed to vary linearly with concentration for smaller concentrations, which is consistent with
Figure 2. Intercept-corrected current vs concentration for pulsed amperometric option. Correction, 7.85 µA; (- - -) linear least-squares line for first seven data points.
behavior expected from eq 2c. On the other hand, the uncorrected data have nonzero intercept and curve toward the concentration axis at higher concentrations. The nonzero intercept most likely corresponds to uncompensated charging current. The nonlinearity at higher concentrations may result from loss of oxygen from samples during the measurement. It was difficult to determine if steady-state currents behaved similarly because the scatter of those data was such that such curvature would have been obscured such. We did not pursue this issue further because our primary focus was on the improved ruggedness expected for this approach. Data for linear least-squares fits given below are limited to the concentration range from 5.1 × 10-3 to 8.7 × 10-2 mmol/L. Variable Dependencies. Three variables, membrane thickness, stirring rate, and temperature, were used to test the ruggedness of the pulsed-source amperometric option. The effects of each variable were studied using air-saturated solutions to minimize potential problems associated with loss or gain of oxygen to or from the atmosphere. To facilitate direct comparisons of the dependencies with other options discussed later, measured signals for different values of each variable are normalized to the signal for the smallest value of each variable. It can be shown that the slope of a plot of normalized responses vs values of the variable of interest is proportional to the relative error coefficient (REC) for that variable; i.e., the slope is proportional to the percentage error in concentration per unit change in the variable of interest. Effects of membrane thickness and stirring rate on results obtained using the direct measurement and extrapolation approaches to the pulsed-voltage amperometric (PVA) option are represented by the open circles and diamonds in Figures 3 and 4. As predicted by eq 2c, the equilibrium currents obtained using this approach are virtually independent of changes in these variables. The temperature dependence of the PVA option is represented by open circles in Figure 5. As expected from the treatment leading to eq 3c, this option is not as effective in reducing effects of temperature as in reducing effects of membrane thickness and stirring rate, which affect rates of mass transport. These results tend to support the expectations based on the development of eqs 2b and 2c. Analytical Chemistry, Vol. 69, No. 21, November 1, 1997
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Figure 3. Effects of membrane thickness on different measurement options. Ordinate is the ratio of currents at other membrane layers to that at one layer (25 µm). Reference values: SS (0) 0.228 µA; PVA extrapolation (O), 14.94 µA; PVA direct measurement (]), 14.92 µA.
Figure 5. Effects of temperature on different measurement options. Ordinate is the ratio of signals obtained at other temperatures to signal obtained at the lowest temperature for each set of experiment. Reference values: SS (0), 0.28 µA; FVC (4), 2.04 µC; SS (9), 0.28 µA; PVA extrapolation (O), 23.3 µA. Table 1. Linear Least-Squares Statistics for Current vs Oxygen Concentration for Different Options slope (µA/ (mmol L-1)) intercept/slope pooled (µmol/L) RSD SEE/slope corr RSDa (%) option value (%) value SD (µmol/L) coeff (r) SSb pvAc PVAd PVAe FVCf
1.43 16.2 13.5 13.6 12.8l
4.2 4.4 1.7 1.5 2.4
24 220 540 530 5.5
9.6 8.5 50 47 4.0
5.2 4.7 4.4 4.5 0.011
0.991 0.992 0.993 0.991 0.99
2.299g 0.73h 0.81i 0.73i 0.4k
a
Six groups of three runs each. b Fixed-voltage steady state. Pulsed-voltage amperometric methods: cwithout depleting oxygen between samples; dextrapolation; edirect measurement. f Fixed-voltage coulometric method. g-k Average signals: g0.089 µA; h4.6 µA; i7.90 µA; j7.88 µA; k1.38 µC. l In µC/(mmol L-1). c-e
Figure 4. Effects of stirring rate (arbitrary scale) on different measurement options. Ordinate is the ratio of currents at other stirring rates to that at speed 0. Reference values: SS (0) 0.179 µA; PVA extrapolation (O), 13.38 µA; PVA direct measurement (]), 13.35 µA. The stirring rate is in arbitrary units.
Fixed-Voltage Coulometric Option. Response curves for the fixed-voltage coulometric option have been presented earlier (Figure 2 in ref 6). As expected from eq 3a, a model representing parallel first-order/zero-order processes fits the data very well. Moreover, using the combined first-order/zero-order model, it was possible to resolve the first-order component, of the signal from the zero-order component, as is illustrated for current/time data in Figure 1A (plot c). As predicted by eqs 3b and 3c, the maximum value of the first-order component, Q∞,1, resolved as mentioned above, was proportional to oxygen concentration. As predicted by eq 3c, results were independent of membrane thickness and stirring rate. Results discussed below show that the first-order component of the charge is also less dependent on temperature than either the PVA or steady-state (SS) options, as also predicted by eq 3c. Comparisons of Options. To permit comparisons of performance characteristics, several features of the conventional steadystate and the recently described fixed-voltage coulometric7 approaches were also quantified. Least-Squares Statistics/Imprecision. Results for linear leastsquares fits of current or charge vs concentration and relative 4488 Analytical Chemistry, Vol. 69, No. 21, November 1, 1997
pooled standard deviations (RPSDs) are summarized in Table 1. To facilitate direct comparison of results for the different options, intercepts and standard errors of the estimates (SEEs) are divided by the slopes to convert all parameters to concentration units. The most apparent differences among the data are the larger values of standard errors of the estimates and pooled standard deviations for the steady-state option. The smaller intercept for the steady-state option reflects the fact that steady-state currents are much smaller than pseudoequilibrium currents measured using the pulsed amperometric approach. Variable Dependencies. Figures 3-5 include data which permit visual comparisons of effects of different variables on results by the different options. In every case, the ruggedness of the pulsedvoltage amperometric option is much better than that for the steady-state option. As expected (eqs 2c and 3c), the ruggedness of the fixed-voltage coulometric option to changes in temperature is slightly better than that of the pulsed-voltage amperometric option. The variable dependencies, quantified as relative error coefficients, are included and compared in Table 2. The pulsed-voltage amperometric option is 40-, 100-, and 10-fold less dependent on membrane thickness, stirring rate, and temperature, respectively, than the steady-state option. The fixed-voltage coulometric option is 60- and 6-fold less dependent on temperature at 25 °C than the
Table 2. Relative Error Coefficients (RECs) for Different Measurement/Data Processing Options membrane thicknessa
option SS PVA FVC a
REC (mmol L-1 µm-1)
ratiob
0.0097 0.00025
40 1
stirring ratec
temperatured
REC REC (mmol L-1 (mmol L-1 au-1) ratiob °C-1) ratiob 0.011 0.00011
b
100 1
0.03 0.003 0.0005
60 6 1
c
25-50 µm. Ratios of relative error coefficients. 0-2 arbitrary units (au). d At 25 °C.
steady-state and pulsed-voltage amperometric options, respectively. All these findings are consistent with predictions based on eqs 2c and 3c. Despite our efforts to reduce effects of temperature on the solubility of oxygen, it is probable that the problem was not eliminated completely. For example, all the plots exhibit apparent decreases in response at higher temperatures, probably resulting from losses of oxygen from the test solutions. However, it is important to note that all the measurement options were evaluated using as nearly as possible the same conditions and procedures. Accordingly, although the results for temperature dependence may include some solubility effects, it is apparent from the data that the pulsed amperometric and coulometric approaches have much smaller temperature dependencies than the steady-state option. Initially, we had expected that the pulsed-voltage amperometric option would have a positive temperature coefficient. However, repeated experiments confirmed that the small negative slope of this option was reproducible. A possible explanation for this behavior is that the rate of mass transport from the bulk solution to the solution inside the membrane is too slow to compensate for the increased rate of electrolysis at higher temperatures. Thus, the steady-state concentration of oxygen inside the membrane is lower at higher temperatures, than at lower temperatures, and the steady-state current measured using the pulsed approach deviates more from the true equilibrium value at higher temperatures than at lower temperatures. Effects of stirring rate, membrane thickness, and temperature on steady-state results found in this study are consistent with observations made in previous studies.4-7,12 Although previous authors have not quantified relative error coefficients, improvements in the effects of stirring rate and membrane thickness achieved here agree qualitatively with results reported by others.4,5 (12) Falck, D. Curr. Sep. 1997, 16 (1), 19-22.
Although effects of temperature on membrane permeability and, consequently, on steady-state currents have been discussed previously,4 no results showing reduced temperature dependence were reported. The present study shows that both the pulsed amperometric and coulometric approaches can reduce temperature coefficients significantly relative to the steady-state option still in common use.12 CONCLUSIONS The present study is similar to that reported by Mancy5 in the sense that a pulsed-voltage source is used. It differs from Mancy’s work in that it includes a more accurate mathematical description of the transient response; it uses curve-fitting methods9,10 to compute the pseudoequilibrium signal from transient data, it quantifies and compares the variable dependencies in terms of relative error coefficients, and it evaluates effects of temperature on this and the coulometric approach.7 Most features (sensitivity, scatter about least-squares lines, reproducibility, and ruggedness) of the pseudoequilibrium options are better than the same features for the steady-state option found in this study and reported elsewhere.4-7 Of the three pseudoequilibrium options evaluated here, the coulometric approach is the most rugged. However, because of the preequilibration requirement of the coulometric option, it is the least convenient of the three options. Because the pulsed-voltage amperometric option does not require any preliminary adjustment of conditions, it is more convenient than the coulometric option. Moreover, the ruggedness of the pulsed-voltage amperometric option is orders of magnitude better than that of the steady-state option and similar to that of the coulometric option. Accordingly, given the greater simplicity and similar ruggedness of the pulsed-voltage amperometric option relative to the coulometric option, it would appear that the PVA option could be the one of choice for situations in which the highest degrees of ruggedness are not needed. As noted earlier, the “oxygen electrode” is just one of many types of membrane-based amperometric sensors. Accordingly, although this study focused on the oxygen electrode, conclusions from this study should apply to other analogous sensors. ACKNOWLEDGMENT This study was supported in part by Grant No. GM 13326-26 from the National Institutes of Health. Received for review January 28, 1997. Accepted July 21, 1997.X AC970106L X
Abstract published in Advance ACS Abstracts, October 1, 1997.
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