Anal. Chem. 1980, 58, 2481-2485
method and those obtained from the titrimetric procedure (7) are in agreement within 0.004 O/U units. The overall precision achieved by the proposed spectrophotometric method is &0.002 O/U units. In the case of highly sintered UOz+,, dissolution at high pressure becomes essential.
.
CALCULATIONS The observed value of the slope at 420 nm for U(V1) is 0.051 89. For U(1V) it is 0.024 18 a t 420,0.05156 a t 544, and 0.21871 a t 665 nm. Suppose absorbance at 665 nm is AI, a t 544 nm is A2, and a t 420 nm is A3 Cu(Iv) (mg/mL) = A,/0.21871 = A2/0.05156
A, - (A,/0.21871)0.02418 CU(V1)
(mg/mL) =
-
0.05189 A3 - (A2/0.05156)0.02418 0.05189
Then
ACKNOWLEDGMENT The authors are thankful t o C . K. Mathews, Head, Radiochemistry Programme, IGCAR, Kalpakkam, for his keen
2481
interest and help extended during the preparation of the paper. Thanks are also due to S. K. Nayak and R. B. Yadav for useful discussions. Registry No. U308, 1344-59-8;U,7440-61-1; 02,7782-44-7.
LITERATURE CITED (1) Metz, C. F.; Dahlby, J. W.; Waterbury, G. R. In Proceedings of a Symposium on Analytical Methods in The Nuclear Fuel Cycle; 1972; IAEASM-149/33, pp 35-44. (2) Buldini, P. L.; Ferri, D.: Paluzzi, E.; Zambianchi, M. Anawst (London) 1984, 109, 225-227. (3) Kuvik, V.; Krtli, J.; Moravec, A. Radiochem. Radioanal. Lett. 1982, 5 4 , 209-220. (4) Yadav, R. B.; Ahmed, M. K.; Nagarajan, K.; Kaliappan, I.;Rao, P. R. V. I n Proceedings of The Nuclear Chemistry and Radiochemistry Symposium, Nov 1981; pp 537-539. (5) Khatoon. F.: Rao. C. S. Radiochem. Radioanal. Len. 1985. 95, 241-246. (6) Kuhn, E.; Baumgartei, G.; Schmieder, H.; Gorgenyi, T. 2.Anal. Chem. 1973, 267, 103-105. (7) Nagarajan, K.; Saha, R.; Yadav, R. B.; Rajagopalan, S.; Kutty, K. V. G.; Saibaba, M.; Rao, P. R. V.; Mathews. C. K.; J . Nucl. Mater. 1985, 130, 242-249. (8) Tolk, A.; Lingerak, W. A. Proceedings of a Panel on Analytical Chemistry of Nuclear Fuel; IAEA: Vienna, 1972; STI/PUB/337, pp 51-58. (9) John, M.; Vaidyanathan, S.; Venkataramana, P.; Natarajan, P. R. Convention of Chemists; Indian Chemical Society, 1976; Anal-58. ( I O ) McGlynn, S. P.; Smith, J. K. J . Mol. Spectrosc. 1961, 6 , 104-187. (11) Ryan, J. L. "Absorption Spectra of Actinide Compounds" in MTP I n ternational Review of Science : Inorganic Chemistry, Series One ; Bagnall, K. W., Ed.: Butterwotths: London, 1972; Vol. 7, pp 323-367.
RECEIVEDfor review February 28, 1986. Accepted May 23, 1986.
Optimization of Microelectrode Array Geometry in a Rectangular Flow Channel Detector Lawrence E. Fosdick and James L. Anderson*
Department of Chemistry, The University of Georgia, Athens, Georgia 30602
The optlmum geometry was investlgated theoretically for a mlcroelectrode array flow detector operated at a constant applied potentlal on one wall of a rectangular channel under andmOns of laminar Row. Concentration proflles and current were calculated by uslng the backward Impllclt flnlte dlffere w e nunerical procedure for electrode arrays In whlch either the spacing between actlve electrode elements was systematically varied from one end of the array to the other while malntalnlng a constant electrode length parallel to flow or the lengths of individual electrode elements In the array were systematically varied whlk malntalnlng a constant gap length. Both studies indlcate that the optlmum electrode response Is obtained wlih a mloroelectrode array having active sites of constant length, separated by uniform gaps. Experimental data for an electrode wlth uniform electrode and gap lengths and an electrode with uniform electrode and varying gap lengths support the theoretlcal predictions.
The amperometric response of a microelectrode array in a flow-through channel is a subject of interest due to the ever increasing role of electrochemical detectors in chromatography and continuous flow analysis (1-4). Several studies indicate that microelectrode arrays operating a t a single, constant applied potential offer significant advantages over solid 0003-2700/86/0358-2481$01.50/0
electrodes of the same geometric area in regard to signal/noise enhancement, thus improving analytical detection limits and sensitivities per unit active area (2-5). Previous studies for characterization of microelectrode arrays have focused primarily on regularly spaced arrays (1, 5 ) ,although preliminary consideration has also been given to nonuniform arrays (5). However, techniques such as photolithography, which show promise as methods of microelectrode array fabrication, are capable of producing electrodes of a wide variety of geometries (6-8). Also, any microelectrode array fabrication technique is subject to some degree of variance in the size of the active sites and/or the size of the gaps between active sites. An understanding of the effects of fabrication tolerances is necessary to reconcile properly any theoretical treatment of microelectrode array response to experimentally acquired results. This paper will examine the effect of varying electrode size or interelectrode gap size on the response to a microelectrode array, using the backward implicit finite difference method. This study assumes that the microelectrode array consists of strip electrodes on one wall of a rectangular channel, oriented perpendicular to the direction of flow, with the electrode width equal to the channel width. Mass transfer limited response is assumed under laminar flow conditions, neglecting migration and diffusion parallel to flow. The validity and limitations of these assumptions for very fine array elements are presented elsewhere (5). 0 1966 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 12, OCTOBER 1986
EXPERIMENTAL SECTION Microelectrode arrays used to test the effect of electrode spacing consisted of 600 nm thick platinum deposited over 27 nm of chromium on a glass substrate. The microelectrode arrays, generous gifts from D. Engelhaupt and T. Li of Martin Marietta Aerospace Corp., were fabricated using microlithography and ion milling, using methodology similar to that reported by Murray and ceworkers (7). Two microelectrode arrays, each with 10 active sites and a fractional inactive area, 8, of 0.98, were produced to test the effect of varying the interelement spacing. A microelectrode array consisting of 10 active elements 0.0008 cm in length parallel to flow, separated by inactive zones 0.050 cm in length, was used to test the uniformly spaced case, where the spacing control exponent a , is equal t o 0 (see Discussion). A second microelectrode array, in which the interelement spacing, or gap lengths, increased from the leading edge of the array toward the trailing edge, was produced from a microelectrode array, originally consisting of 49 active elements 0.0008 cm in length, separated by 0.0102-cm inactive zones, and connected at one end to a common bus bar. This microelectrode array was modified by cutting the connections between elements and the connecting bus bar to achieve a linear increase in interelement gaps from one end to the other of the array. The gap length between the first two elements was 0.0102 cm, with each succeeding gap length increasing by 0.0110 cm. Thus the second microelectrode array approximates the case where the electrode spacing exponent, a , is greater than 0, with a value near 0.5. The current response of these electrodes was tested by using flow injection techniques and an electrochemical cell described elsewhere (8). The mobile phase contained 0.1 M, pH 7.1 phosphate buffer in distilled/deionized water. The electroactive species was 5.93 X lo4 M ferrocenecarboxylic acid (FCA) (Aldrich, used as received) in 0.1 M phosphate buffer. The injection volume was 1.1 mL with a dead volume between injector and detector of 0.09 mL, to ensure a steady-state concentration, equal to the injection concentration, in the cell.
RESULTS AND DISCUSSION Effect of Electrode Spacing on Current. To study the effect of electrode spacing on current, a model spacing function was developed to provide an orderly variation of interelectrode gap spacing over a wide range. For this study a spacing function was selected in which electrodes would be spaced either with uniform space between them or with separation between electrodes varying smoothly from a small gap between electrodes at the leading edge of the electrode array to an increasingly larger gap as the total distance along the array increased. The rate of gap variation with distance was variable over a wide range. For completeness, the model was also required to reverse the spacing function such that electrodes at the leading edge of the array would be relatively far apart, becoming closer together toward the trailing edge of the array. The signals of different electrodes or arrays can most usefully be compared a t constant geometric area, since the maximum signal will be achieved with a solid electrode (no gaps) ( I , 2,s). The noise levels can most usefully be compared at constant active area, since noise is proportional to area for conventionally sized electrodes (9). Therefore, the model was constrained to maintain constant active and geometric areas, to ensure that changes in the current, and hence the signal/noise ratio, would be attributable only to the electrode spacing. The electrode arrays for this study were defined by the constraint iv
N- 1
W L = CW, + CG,
(1)
W = (1- B)WL/N
(2)
J=1
J=1
and
where W , is the length of the j t h electrode element parallel
to flow, Gj is the length of the j t h gap, WL is the total length of the array, 6 is the fraction of the area that is inactive (i.e., gaps), and N is the number of active electrodes. In this portion of the study, all active electrode elements of the array were maintained at a constant length, W . Combining eq 1 and 2 yields N-l
W L = N W + CC, j=1
(3)
The model selected to calculate the j t h gap length was defined as
(4) where i- 1
L, = jw+ CC, i=l
(5)
Here, L, is the cumulative array length up to the leading edge of the j t h gap, G, is the length of the j t h gap, a is an exponent that is varied to control electrode spacing, and kN is a scaling constant required to maintain the total electrode array length constant. The value of ItN varied with the number of electrodes, N. Terms involving G, and L, were normalized by the total electrode length, W, in eq 5,so that kN is dimensionless. The spacing function defined by eq 4 results in equal gap lengths when a = 0 and gap lengths that decrease toward the trailing edge of the electrode array when a < 0 and increase toward the trailing edge when a > 0. Figure 1 shows the electrode spacing for electrode arrays of ten elements for a values of +1.0 and 0.0. The array spacing when a is -1.0 is the mirror image of the +1.0 case. Due to the nature of the spacing function, the value of k N was calculated iteratively using a binary chop technique for given values of a , 0, N , and W,. The current at each active element was calculated by using the backward implicit finite difference method (5, IO),assuming conditions of steady-state, fully developed laminar flow in a rectangular cell. Figure 2 shows the electrode response relative to a uniformly spaced electrode array, whose response is defined as I", as a function of a and 0 for an electrode array containing 20 active elements. Here, a was varied from -1.0 to +1.0, for 0 values ranging from 0.5 to 0.98, corresponding to fractional active areas between 50% and 2%, respectively. As shown by Figure 2, the optimum current was obtained at an a value of zero, corresponding to equally spaced electrode elements. Response decreased symmetrically as a deviated from zero, with no difference between the extremes where electrodes are very close together at the leading edge or a t the trailing edge. The trend in response relative to a was qualitatively consistent within the limits used for this study, for 0.50 5 0 5 0.98, and 5 2 N 5 80. The calculated current did not deviate noticeably from that for an equally spaced array for values of a close to zero. Table I shows the a values at which a 1% deviation in current occurs, as well as the decrease in current at the limits tested in the model. The effect of electrode spacing on response increases as the fractional inactive area increases, because the electrode length is constrained to a constant value. However, the dependence of electrode response on geometry is not strong, yielding at most a 13% diminution of response at the extremes considered, confirming our earlier results for a more limited set of conditions (5). It is notable that the effect is of greatest importance for the sparsest electrode arrays. The optimum response for a uniform spacing was somewhat surprising, since the increasing spacing model ( a > 0) provided increasingly greater time for the diffusional replenishment of the electroactive species at increasingly larger gaps between successive electrodes. It was anticipated that the response for an a value of -1 would be significantly worse
ANALYTICAL CHEMISTRY, VOL. 58, NO. 12, OCTOBER 1986
2483
'"I
E l e c t r o d e No. Figure 3. Response of the Individual electrode elements normalized relative to corresponding elements of an equally spaced electrode array: (a) a = +1.0; (b) a = -1.0. Soli line is at +1.0, corresponding to the normalized response of an equally spaced array. In all cases = 0.90, N = 20.
flow
e
Table 11. Effect of Varying Electrode Spacing on Current Response
type of no. of spacing electrodes uniform varying
+low
Figure 1. Ten-element electrode array produced by eq 4: (a) a = +1.0, 8 = 0.90, N = 10;(b) a = 0,8 = 0.90,N = 10.
l.OOOEt00 +
H
+
\ H
,
1.400E-01 -8.000E-01
-4.000E-01
0.000E-01
a
4.000E-01
0.000E-01
Flgure 2. Relative response of 20-element electrode arrays normalized to the equally spaced array (a = 0), (a) 6 = 0.50, (b) 8 = 0.70, (c) e = 0.90, (d) 6 = 0.98. I o is the response of an equally spaced array for a given value of 8 .
Table I. Effect of Varying Electrode Spacine a for
decrease (%) for
6
1%decrease
a = fl
0.50 0.70
0.70 0.50 0.30 0.35
0.90 0.98
2.20 4.98 10.2 12.7
Calculated current decrease as a function of model exponent a, for 20-element electrode arrays with constant electrode length and variable gap lengths. 0 is fraction of surface that is inactive. a
than expected for a = 0 and a = +1, rather than identical with the response a t a = +l. A plot of current at each active element normalized to the current at the corresponding element of an equally spaced
10 10
0
current, nA exptl predicted
0.98 90.6 f 3 0.98 86.2 2
*
93.8 90.9
no. of
measurements 6 7
array ( I O ) (Figure 3) shows that although the increased electrode spacing does indeed increase the efficiency of electrodes toward the trailing edge of the array, this benefit is balanced by decreased efficiency near the leading edge, where the active elements are closer than in the equally spaced array, resulting in a slight overall decrease in efficiency. Conversely, the efficiency of active elements near the leading edge is greater for the array having a 0 than for the equally spaced array ( a = 0), but increasingly worse near the trailing edge of the array. Efficiency for an asymmetrically spaced array (a 0) would be greater than for an equally spaced array (a = 0) if the smallest gap spacing of the asymmetric array were equal to the gap spacing of the equally spaced array, but this condition violates the constraint of constant geometric area and is therefore not considered here. The currents measured for the microelectrode arrays used in this study are compared to theoretical predictions in Table 11. As predicted, a microelectrode array whose electrode spacing varies linearly across the array (approximately equal 0.5) shows a slight (4.4%) decrease in to the case for a current relative to a uniformly spaced array with the same geometric and fractional active areas and same number of electrodes. The ratio of response of the two microelectrode arrays is within 2% of the ratio predicted by theory, although both electrodes show currents ca. 3% lower than theoretically predicted. The latter deviation may reflect partial blockage of the electrode surfaces by contaminants, though the difference is not statistically significant when the standard deviations in the current measurements are considered. The magnitude of the observed currents at the test electrodes provides insight into two additional parameters of practical interest. Firstly, since responses are comparable to or slightly lower than theoretical predictions, it appears that any turbulence due to the 600-nm thickness of each platinum electrode element above the substrate is not significant enough to cause any positive deviation from theoretical predictions assuming completely laminar flow at the experimental flow rate of 1 mL/min. Secondly, it is apparent that the effects of longitudinal diffusion parallel to flow are still of little consequence for element lengths of 8 wm, since longitudinal diffusion would tend to enhance response. These results are consistent with an approximate treatment estimating less than
2484
ANALYTICAL CHEMISTRY, VOL. 58, NO. 12, OCTOBER 1986
5% enhancement for 10-pm element length ( 5 ) and a treatment assuming inviscid flow (11). The results obtained here for a uniformly spaced and a progressively spaced electrode are also supported by earlier results for composite electrodes with randomly spaced active elements (2). Experiments with Kelgraf electrodes for varying percent graphite showed experimental responses within 1170 of theoretical predictions for a uniformly spaced electrode for all but one composite electrode out of nine studied, with 0.75 5 8 5 0.95 and N N 31. A detailed comparison of theoretical predictions to experimental results for uniformly spaced gold microelectrode arrays at a single potential has been presented elsewhere (8). The decrease in response for the asymmetric electrode is not accompanied by a decrease in noise, since the total electroactive area remains constant, resulting in a slight decrease in signal/noise compared to the uniformly spaced microelectrode array. These experimental results support the theoretical prediction that the optimum microelectrode array geometry is uniformly spaced, though response depends weakly on spacing. Other electrode geometries show greater promise than an asymmetric electrode spacing pattern at a single potential for enhanced sensitivity and signal/noise, such as the interdigitated electrode array described earlier (12). Effect of Varying Electrode Lengths on Current. This portion of the study examined the effect of varying the lengths parallel to flow of the active elements in the electrode array while maintaining a constant gap length. An electrode array having elements of equal lengths and constant spacing was used for comparison. The criteria used for the interelement spacing study were also used in this portion of the study, where arrays were modeled, with electrode elements either of constant length or of varying length from shorter to longer, or longer to shorter, in length, from the leading to the trailing edge of the array. The electrode arrays for this study were generated from eq 1,where a constant gap length, G, was defined by the relation
G = BW,/(N- 1)
(6)
where 8, W,, and N are as defined previously. Combination of eq 1 and 6 yields
CWj
j= 1
(7)
where 1-1
+ CW,
1.y
k
\ t-
..
(9)
i=1
Here, Lj is the cumulative electrode array length up to the leading edge of the j t h active element and other terms are as described earlier. The pattern of element length variation is analogous to that of Figure l a for variable gap length, but with electrodes and gaps interchanged and the first gap omitted. The first element length calculation had to include the first gap length, since the array has a zero length a t the leading edge of the first electrode element, resulting in a W1 length of zero if the first gap length was not included. Figure 4 shows the electrode response relative to a uniformly spaced electrode array as a function of a and 8 for an electrode array containing 20 active elements. Here, a was varied from -1.0 to +1.0 for 8 values between 0.50 and 0.98. Results were calculated by using the backward implicit finite difference method as described earlier. The model exponent used to control electrode element lengths, a , was varied between -1.0
3
Electrode No.
I
Figure 5. Response of the individual electrode elements normalized relative to corresponding elements of an electrode array having elements of constant length: (a) a = +1.0; (b) a = -1.0. Solid line is at 1.O, corresponding to the normalized response of an array in which all elements are of equal length. I n all cases 8 = 0.90 and N = 20.
+
Table 111. Effect of Varying Electrode Lengths" 8
0.70 0.90
0.98
The model used to calculate the individual active element lengths is
Lj = j G
Figure 4. Response of 20-element electrode arrays normalized relative to the array having elements of equal lengths (a = 0): (a) 8 = 0.50, (b) 8 = 0.70, (c) 8 = 0.90, (d) 8 = 0.98. I o is the response of an equally spaced array for given value of 8.
0.50
N
W,=(N-l)G+
t
1 OOOEtOO
CY for 1% decrease
0.50 0.40 0.35 0.35
decrease (%) for N
= fl
2.91
3.88 4.53 4.53
'Calculated current decrease as a function of model exponent CY, for 20-element electrode arrays with constant gap length and variable electrode length. 6 is fraction of surface that is inactive. and +LO. Again the optimum current response of the electrode arrays occurs a t an a value of zero, decreasing symmetrically as a deviates from zero. The rate of decrease of response for CY values near zero was more rapid than in the previous study where gap lengths were varied. However, the overall decrease in response at the extreme limits tested is noticeably smaller, less than 5% at the extremes used in this study. Table I11 shows the change in a required for a 1% decrease in total current as well as the percent decrease in current at the limits tested. A plot of the current response a t each active element, normalized to the corresponding element of a uniformly spaced electrode, I", for an electrode array containing 20 active elements (Figure 5) shows similar trends to those where electrode spacing was varied. When the element lengths decrease toward the trailing edge of the array, the first set (half) of the elements produces more current than the corresponding elements of the equally spaced model, while the second set (half) of the elements produces less current. The converse is true in the case where elements increase in length toward the
ANALYTICAL CHEMISTRY, VOL. 58, NO. 12, OCTOBER 1986
trailing edge of the array. The net result is similar to the fist portion of this study where gap lengths were varied, in that the direction of variation in electrode spacing, Le., the sign on the exponent CY,makes no difference in the magnitude of the total current for any tested values of CY, 8, and N . This case was not experimentally tested, since the experimental test for variable gap length showed acceptable agreement with theory, and since the significantly smaller differences predicted here would be difficult to measure reliably, and thus do not warrant an experimental test.
CONCLUSIONS Noise is directly proportional to the active area of an electrode over a wide range of practical experimental conditions, so that the signal to noise ratio, SIN, depends only on sensitivity, i.e., electrode response per unit concentration, if the total active area is constant (5, 9). The constraint of a constant total geometric area used in this study, combined with comparison of electrode arrays of equal fractional inactive area, 0, allows the direct comparison of S / N and response. The optimum fraction, 8, of inactive surface area, is a compromise in S/N ratios. Noise is directly proportional to total active area (5,9),until electrodes are quite small, while the signal from a sparse array is dependent on geometric electrode area, 8, and the number of electrodes in the array (I, 5). However, sparse arrays will yield smaller currents than a solid electrode of the same geometric area, and as a result, other sources of noise, e.g., instrumental noise, will become important in very sparse electrode arrays (5, 9). Optimum values of 0 and/or number of electrodes for a given array geometry can be estimated from measured noise parameters (5). In addition, diffusion parallel to flow will become important when electrode lengths are sufficiently small relative to channel height for sufficiently slow flow rates ( 4 , 5 ) . Approximations for estimating the importance of parallel diffusion are presented elsewhere (5). Variations in the interelement spacing or the element lengths themselves across the length of an array result in no inherent benefit, but actually a slight decrease of the overall
2485
signal to noise ratio for an electrode array. Experiments confirm this theoretical prediction. However, the effects of fairly large variations in interelement gap lengths or element lengths do not appear to alter the response of the complete electrode array greatly. The optimum electrode spacing and sizing appears to be obtained for the simplest geometry, where all element lengths are equal and alf interelement gap lengths are equal. An analogous situation applies for the dimensions of adjacent elements of interdigitated miceoelectrode arrays held at alternating potentials to enable successive oxidation and reduction of an analyte, as shown elsewhere (12). These results should greatly simplify the design and fabrication of practical microelectrode array flow sensors.
ACKNOWLEDGMENT The authors are grateful to Dare11 Engelhaupt and Tom Li of Martin Marietta Aerospace Corp., Orlando, FL, for their generous gift of the microelectrode arrays used in this project. LITERATURE CITED (1) Fliinovsky, V. Yu. Nectrochim. Acta 1980, 25, 309-314. (2) Anderson, J. L.; Whiten, K. K.; Brewster, J. L.; Ou, T. Y.; Nonidez, W. K. Anal. Chem. 1985, 57, 1366-1373. (3) Taiiman, D. E.: Wsisshaar, D. E. J. Lis. Chromatoor. 1983, 6 , 2157-2172. (4) Caudill, W. L.; Howell, J. 0.; Wlghtman, R. M. Anal. Chem. 1982, 54, 2532-2535. (5) Moldoveanu, S.; Anderson, J. L. J. Elecfroanal. Chem. 1985, 785, 239-252 - - . -- -. (6) Thormann, W.; van den Bosch, P.; Bond, A. M. Anal. Chem. 1985, 57,2764-2770. (7) Chdsey, C. E.; Feldman, E. J.; Lundgren, C.; Murray, R. W. Anal. Chem. 1986. 58. 601-607. (8) Fosdlck, L. E.; Anderson, J. L.; Baginski, T. A,; Jaeger, R. C. Anal. Chem., in press. (9) Morgan, D. M.; Weber, S. G. Anal Chem. 1984, 56, 2560-2567. (10) Anderson, J. L.; Moldoveanu, S . J. Nectroanal. Chem. 1984, 179, 107-117. (11) Cope, D. K.; Tallman, D. E. J. Nectroanal. Chem. 1986, 188, 21-31. (12) Anderson, J. L.; Ou, T. Y.; Moldoveanu, S . J. Elecfroanal. Chem. 1985, 196, 213-226.
-
RECENEDfor review January 21,1986. Accepted June 9,1986. This work was supported in part by a grant from the U S . Geological Service, Water Resources Research Program.
