Anal. Chem. 1989, 61, 295-302 (14) Itaya, K.; Uchlde, I.; Neff, V. D. Acc. Chem. Res. 1986, 19, 162-168. (15) Itaya, K.; Ataka, T.; Toshima, S. J . Am. Chem. SOC. 1982, 104, 4767-4772. (16) Deekin, M. R.; Melroy, 0.J . E/ectroanal. Chem. Inferfackl €kcfrochem. 1988, 239, 321-331. (17) Deakln. M. R.; Melroy, 0. R. J . Nechochem. Soc.,in press. (18) Ellis, D.; Eckhoff, M.; Neff, v. D. J . phvs. Chem. ~ 8 1 8, 5 , 1225-1231. (19) Reichenberg, D. In Ion Exchange; Marinski, J. A., Ed.; Marcel Dekker: New York, 1966; Vol. 1, Chapter 7. (20) Sauerbrey, 0. 2. 2.fhys. 1959, 155, 206.
,
295
(21) Itaya, K.; Shoji, N.; UchMa, I. J . Am. Chem. Soc. 1984, 106, 3423-3429.
RECEIVEDfor review September 2,1988. Accepted November 21, 1988. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this work. A summer salary award from the Council on Research and Creativity at Florida State University also is acknowledged.
Signal-to-Noise Ratio in Microelectrode-Array-Based Electrochemical Detectors Stephen G . Weber Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
The signal-to-nolse ratlo at an electrode array depends on the electrode area, the perimeter-to-area ratlo of the electroactlve portion of the surface, the mass transfer coefficient of the analyte-eiectrode comblnation, the measurement bandwidth, and the sources and magnttudes of the nolses. Simple models for chronoamperometry with an array In quiescent solution and for hydrodynamlc current at an array In one wail of a rectangular condutt through which analyte-containing solution is following are glven. Noises from seven sources, includlng environmental noises, are considered In a noise model. The signal and noise models are combined to yield a model for slgnal-tmoke ratlo at array-based electrochemical detectors. There exists an optimum array density for a given area that depends on the noise power, noise reslstance, the current density at a sparse array, and the current density at a solid electrode of the same area. Approximations that lead to Simple expresshs for the optimum eiectroacthm area fractlon and noise resistance lead to results that are in good agreement with more complex and less approximate calculations. Electrodes of millimeter dimenslons consisting of about 1% active surface with electroactlve “pieces” of micrometer dimenslons are anticipated to yield detection limits of about 1 fmoi injected Into a typlcai packed-column liquid chromatograph. Thls corresponds to about lo-’’ M anaiyte In the detector and about an order of magnitude improvement over solid electrodes.
INTRODUCTION Electrochemical detectors have proven to be useful owing to their selectivity and low detection limits (Id). Notwithstanding the success the detector has enjoyed, there are still electroactive compounds such as tyrosine containing neuroactive peptides (6)that are present in concentrations below the detection limit of electrochemicaldetectors. It is possible to decrease detection limits considerably by using microbore liquid chromatography (7,8). In this analytical system there are two means by which detection limits can be improved over those found for large column liquid chromatography. The concentration of the mass injected is larger because the peak volume is smaller, and the flow rate and dimensions are smaller so the electrochemical detector efficiency is higher. 0003-2700/89/0361-0295$01.50/0
Caliguri and Mefford realized an improvement of a factor of 20 from the former effect and 2.5 from the latter effect. Note that the major improvement in the mass detection limit of the system is because of the peak volume effect. The efficiency increase only provides a factor of 2.5. We are interested in understanding ways to improve the concentration detection limits of electrochemical detectors (9,lO). From our signal and noise studies we have predicted that there is an optimum area for electrochemical detectors. Wightman (11)has pointed out that the area dependence on sensitivity depends on the electrode shape. We have therefore developed a simple computational approach to the signalto-noise ratio (snr) problem in order to understand how one should, in principle, construct the best detector. Our approach is based on array electrodes (12-36). Both experimental and theoretical work have shown that such electrodes, in which many electrodes are used to collect current, are more efficient than a solid electrode with the same electrode area. This efficiency arises from the diffusive or hydrodynamic transport of electroactive material from regions of the array surface that are insulating to regions that are electroactive. Composite electrodes or random microelectrode arrays such as Kel-Graf (25), which consists of Kel-F and graphite, have shown higher snr in hydrodynamic systems than solid electrodes. Optimum performance seems to be at around 15% carbon (26). Regular microelectrode arrays (24,27,29, 31-35) have been studied as well. The agreement between calculated and predicted current is excellent (34). For low flow rates, arrays are less flow rate sensitive than solid electrodes (27) because at low flow rates the dominant mass transport path is edge diffusion. This work is an attempt to formulate a theory for array electrodes that is at the same time simple and useful. Signal production in chronoamperometry will be discussed in some detail, as will steady-state signal production in flowing streams. The latter is relevant to amperometric detectors, in which the lowering of detection limits is of interest. Thus, after a brief overview of the noise model, snr calculations illustrating several main points are given. It is concluded that an optimum electrochemically active area exists that depends upon the background current, electrode capacitance, the current-tovoltage converter noise power and noise resistance, the importance of environmental contributions to noise, the bandwidth of the measurement, and the perimeter-to-area ratio 0 1989 American Chemical Society
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of the electrochemically active area.
ss2
m
THEORY Signal Production at Arrays. Vetter, in 1952, drew the analogy between the steady state at a partially blocked electrode and the analogous electrical problem (12). Both solutions result from solutions of Laplace's equation. For small concentration perturbations, the relationship is
CT
1
7
I\ Flgure 1. Equivalent circuit for chronoamperometry at an array. The impedances are as follows: Q, uncompensated resistance; W 1,
where Rd is the diffusional resistance, R, T, n, and F have their usual meanings, Cois the bulk concentration,D is the diffusion coefficient, u is the specific conductance of the medium, and Rn is the ohmic resistance. The analogous case for ac current was studied as well. In the case where the surface is mostly inactive, the individual active regions can be thought of as hemispheres. The resulting solution for ac current is
? = (1- O)nFDC,([
($>"'+ t]
cos (ut) -
Warburg impedance to actual electroactive area; SS1, radial diffusion Warburg impedance to whok array; SS2, radal diffusion to whole array: CT, charge transfer; C, double layer capacitance.
to edges; W2,
Table I. Values of Mass Transfer Coefficients for the Impedances in Figure 1' mass transfer coefficient/cm s-l
areab/&-:
w1
( D w / ~ ) (1 ' / ~- j ) (j= (-l)'/')
SS1 w2
4D/irrl
1-0 1-0 0 1
process
SS2
CT where t is the current density, 8 is the inactive fraction of the surface, w is the radial frequency of the potential, t is time, and ro is the radius of the hemisphere. The result is obviously the sum of a Warburg term (dominant a t high frequencies) and a spherical diffusion term. The same result is obtained if one represents the diffusional impedance as the sum of a Warburg impedance in parallel with a spherical diffusion resistance. Landsberg and Thiele (13), using an electrical result of Smyth for the resistance between two disk electrodes of different radii separated by a distance z , established approximate relationships for hydrodynamic current at a rotating disk array. The result is shown as
nFDCo
-
i=
6
+ f(r2,r1,6)
(3)
'The individual microelectrode radius is rl. width is rs.
(44
*ne
A-B
More recent work has been reviewed by Wipf and Wightman (37). An array of microelectrodes has a well understood, if difficult to calculate, behavior. In a chronoamperometric experiment the short time behavior is that of a planar electrode with the same electroactive area as the array. The Cottrell equation holds with an area of (1- @ A ,A being the whole array area (blocked plus unblocked). A t intermediate times diffusion from the insulating regions to the electroactive regions is seen as an approach to steady state. As the insulating regions become depleted, mass transfer from the bulk becomes the dominant effect. Note that the process of mass transfer from the bulk to the depleted regions is in series with the process of mass transfer from the depleted regions to the electrode.
1-0
The array half-
A simple model for mass transfer to an array can be inferred from these studies. Figure 1shows a model for the interfacial impedance that includes the following impedances: CT, charge transfer; W1, Warburg impedance at short times; SS1, steady state term for the diffusion of material from the insulating region to the electroactive region; W2, Warburg impedance to the insulated fraction of the array; SS2, steady-state term for the whole array; Q, ohmic resistance; CDL, double layer capacitance. The mass transfer coefficients and associated areas of each term are shown in Table I. Impedances are calculated by using the following relationships: i = E / Z = nFACoh
Z = E/nFACoh
where 6 is the Levich diffusion layer thickness and f is a complex function of 6 and the radii of the two disks. In this problem one views the smaller disk of radius rl as corresponding to a circular electroactive region and the larger disk radius r2 as the half-distance between microelectrodes (center to center). The form of eq 3 allows one to draw an analogy between blocked electrodes and CE behavior (14-17). One may visualize the region of solution above an insulating space as being a source of electroactive analyte, as 2 is in the CE mechanism shown below.
Z-A
(DW/2)1/' (1- j ) 4D/irr3 k
(5)
where h is a mass transfer coefficient. Letting E be RTInF, one arrives at an expression for the ith impedance
Z, = RT/n2FACohi
(6)
We must now insert an impedance representing hydrodynamic mass transfer into the scheme. One way to do this is to put a hydrodynamic resistance in parallel with each Warburg impedance. Another way is to use the kinetic analogy and the resulting frequency-dependent impedance of Gueshi (14-16). The former is a bit contrived since one has to have two impedances for the hydrodynamic resistance to mass transfer, one in parallel with each Warburg term, yet there is only one hydrodynamic process. In neither case is the rather important influence of depletion layer recharge taken into account. Thus we have chosen an alternative. The current due to radial diffusion to individual portions of the electrode is added to the current due to hydrodynamic mass transfer (see Figure 2). This is used to determine the current to a sparse array. A correction term is calculated by using Filinovsky's procedure (24). Finally we apply the empirically determined (34) values of certain numerical coefficients to force the approximate model to fit numerical computations. One needs to define a physical system before the mass transfer coefficients in eq 6 can be calculated. We have chosen to use the thin-layer cell because it is well understood (38,39), and most experimental work on arrays has been done with this geometry. Solutions for the problem of an array of rec-
ANALYTICAL CHEMISTRY, VOL. 61, NO. 4, FEBRUARY 15, 1989
297
Table I1 source of noise
magnitude of noise power amplification
1. voltage noise in current to voltage converter input (instrument side of electrode interface) 1. voltage noise on solution side of electrode interface 3. Johnson noise in Re(Zb) 4. Johnson noise in Re(Zr) 5. Current noise in current to voltage converter 6. shot noise in background current (iB is background current) 7 . impedance noise
Allf A2If 4kTRe(Zh) 4k T R e ( Z f ) Adf 2eiB
(AdD(id2
%le
Flgure 2. Equivalent circuit for the hydrodynamic system. The boxed combination of SS1, radial diffusion to the electroactive portion of the electrode, and H, the hydrodynamic mass transfer term, governs the steady-state behavior, the Warburg impedance is only important in voltammetry.
tangular electrodes in a thin layer flow cell have been obtained by Filinovsky (24),Anderson (31,33,34),and Tallman (29, 35). Quantitative arguments based on the parallel processes (27) of hydrodynamic and radial diffusive mass transport have been given by Caudill et al. (27). We have used the equation of Anderson (34) to calculate the effect of depletion layer recharge. This equation, based on the original work of Filinovksy, has empirical parameters the values of which were determined by Anderson by comparison to their backward implicit finite difference calculation. The computations of Cope and Tallman are more accurate, yet are more difficult to do. The major differences between the present treatment and Filinovsky’s are the inclusion of radial diffusion and the use of a two-dimensionalm a y of square electrodes rather than linear array of band-shaped electrodes. Computation of the Current to an Array. Let there be a two-dimensional array of N electrodes arranged in K files and M ranks. Each individual electrode is of width and length 2rl. The array width is 2r3 and the array length is 2r4. The hydrodynamic current to a single electrode can be written (34)
(7)
where rLis 4r1r@/ Ub, D is the molecular diffusion coefficient of the solute, rSr is the volume flow rate, b is the spacer thickness, and y is defined implicitly. The current arising from radial diffusion is 4312
ie = -nFCDrl *1/2
= arl
(8)
where the assumption has been made that the current to a square is the same as the current to a disk of the same area. The current at a single rank (across the flow direction) consisting of K electrodes is then
irmk= K(7-
+ arl)
(zf/z,)2 1 Z? Z? Z?
where N is the number of (solid) ranks. Clearly M replaces Filinovhy’s N. The fraction complementary to 8, electroactive arealtotal area, has been called x (12). The electroactive fraction along the direction of flow over a file of electrodes is
CDL
ih = 1.467nFOCrL2I3
1 + (Z,/Zd2 (Zf/Zd2
(9)
If the current at one rank had no influence on downstream ranks, then the total current would be A4-irank. A correction to this must be made to account for depletion of the electroactive species. We use the same procedure for the depletion as Filinovsky (24). In Filinovsky’s notation this is
= Mr1/r4
(11)
The electroactive fraction across the width has already been used above (eq 7), and it is &ank
= Krl/r3
(12)
The product of the two is the total electroactive arealtotal area X
=
XfileX,,k
= MKrI2/r3r,
(13)
We must replace Filinovsky‘s 1 - 8 with xfile. Thus, the current to a sparse array is
The ratio of this current to that at a solid electrode of the same dimensions is
(15) In the last expression k has been substituted for the value 113. The reason for this will become evident. Filinovsky’s equation for nearly solid electrodes (dense arrays) is used directly with the replacement of 1 - 8 by zfde. The two current ratios are combined in the following way:
I=
Idenselsparse Idense
+ Isparse
(16)
Finally, the corrections of Fosdick et al. (34) were made. They recognized the approximate nature of the treatment but appreciated the ease of use of the equation of Filinovksy. Thus, they obtained the values of the parameters a and k , in eq 15, by fitting eq 16 to more exact backward implicite finite difference calculations. In practice eq 16 was used to obtain a mass transfer coefficient, then an impedance was calculated (eq 5 and 6). For voltammetric detection, a Warburg impedance, W1, is needed, as shown in Figure 2. In the current work presented here, only the steady-state component was considered. Noise. The noise model is the same as that of Morgan and Weber (9) with one change. Operational amplifier voltage noise and external voltage noise (e.g. from the auxiliary electrode output amplifier) are treated slightly differently. The noises are shown in Table 11. There are several parameters of the instrument and environment that need to be specified. The input voltage noise in the current voltage converter is given by AJf. We have used l l f noise for all of the calculations since, for two widely
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ANALYTICAL CHEMISTRY, VOL. 81, NO. 4, FEBRUARY 15, 1989
used potentiostats, namely the PAR 174A and the BAS LC-4, the noise was found to be of this form (9),although from a Kiethley 427 picoammeter it is white (IO). Values of A, for the two instruments mentioned are in the range (0.05-2) X V2 at 1 Hz. We have not measured A2, but have estimated it to be of the same order of magnitude as A,. The value for A3 is also instrument dependent. We have measured a value of A2 at 1Hz. Environmental conditions govern A4. Values on the order of lo4 are reasonable (9,IO).In other words, the root mean squared (rms) noise in the background current is about 0.1% at 1 Hz. The bandwidth is an important parameter in determining the noise in the measured signal. In all calculations a characteristic time, T, is used to compute a frequency band such that 1/(5.2r~)< f < 5 / ( 2 a ~ ) .If a concentration detection limit is desired, then 7 is chosen arbitrarily. If a mass detection limit is desired, T is governed by the chromatographic peak width. The latter is calculated from the column parameters, the Knox equation, and the linear velocity of the fluid. Computational Details. The computations were performed in ASYST on an IBM-PC. The general flow of the programs is as follows. Input and calculation of lumped parameters are followed by a calculation of electrode area. Values for r1 and r3 (r4is always taken as equal to r3 and M equal to K) are used with the number of electrodes, N , to compute x . At each value of N the faradaic impedance and background current (9)are calculated. The background current yields an interfacial resistance. This and double-layer capacitance are combined in parallel and then added to an uncompensated resistance to yield an interfacial impedance in the absence of analyte. It is this impedance that is used for Zi, in the calculation of noise (see Table 11). These calculations are carried out in the specified frequency band, and the rms noise in the band is calculated. From Table I1 it can be seen that Zf plays a major role in determining several of the noises. Thus, one is not free to choose the value of Zfarbitrarily. It is important that the value of the feedback resistance be a realistic one. When using the impedance formalism for obtaining a current, one must specify a voltage. Of course, a small voltage (> R, at the detection limit. In other words, a t the detection limit, the signal is much smaller than the background current. In this case the noise can be approximated as
"in
In eq 20 (re,: is the variance in the voltage measured at the amplifier output, the u? term represents terms 1 , 2 , and 7 in Table 11, and the u t term represents term 5 in Table 11. The impedance noise, term 7, is included in the left term in the right-hand side of eq 20 in its equivalent voltage noise form
In this equation, E is the voltage applied, RTInF, and uz2/Zh2 is A41f. The value of Zh is the parallel combination of the double layer capacitance and a leakage resistance. Its magnitude is inversely proportional to the electrode area. The leakage resistance decreases exponentially (9,25) as the magnitude of the actual applied potential increases
=xN,, .
(:;:;: ) + m. l
y
Let us abbreviate eq 23 as Is- = mx, then, rephrasing eq 16, with the approximation that Id,- = ImEd
I=
mxIsolid
(24)
mx + Isolid
We ask that the signal out of the current to voltage converter be a constrained Eout,thus, ignoring signs Rf = Eout/I
= Eout(Imlid +
A)
One can now write an approximate expression for the noise
Equation 26 shows that the noise depends not only on the system noise sources but also on the signal. We now describe the noise by two parameters, the noise power and the noise resistance
Equation 26 becomes
This equation can be used to determine the optimum value of the fractional coverage x . Differentiation of eq 29 with respect to x at constant total area (leaving Im&-J, r3, and r4 constant) leads to the value of x at minimum noise
'Opt
=
(
ZO2Isolid
113
(r3r4I2mRn2)
This value can be substituted back into eq 29, and the result differentiated with respect to R , to determine the optimum noise resistance. First we make the substitution shown in eq 31. 2 , is evidently the interfacial impedance that would exist
if the solid electrode had the current density of the sparse array and if the signal and background current were strictly proportional. Then
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 4, FEBRUARY 15, 1989
Differentiating with respect to R,, one arrives at the optimum value of R,, given that x has been optimized. The results are shown in eq 33 and 34
Further insight is gained by substitution into eq 34 (using eq 7 and 22) r. '0
0.2
where
Several important points can be gained from consideration of eq 35. Of the two terms in the denominator, the one on the right represents the effect of "edge diffusion". The one on the left arises from convective diffusion. If r1 is relatively large, there is no influence of edge diffusion and the optimum value of x is then unaffected by changes in 0 or b. The optimum value of x decreases as rl decreases. In all cases the optimum value of x decreases as the electrode length increases. This reflects the fact that the total electrode area is important as well as the distribution of electroactive surface. When the radius rl is smaller, the optimum value of x becomes smaller and more sensitive to the conditions of the cell. Increasing the velocity of fluid (increase u,decrease b, decrease r3 and the cell width simultaneously) decreases the denominator and increases the value of xopV Once xopthas been determined, one can further optimize the system. The choice of electronics and environmental conditions will lead to a range of possible noise powers, p, and noise resistance R,. As eq 35 shows, the snr is greater the lower the noise power in the system. Thus one ought to opt for the lowest noise power available. Recall that besides the characteristics of the current to voltage converter (terms 1 and 5 in Table 11), the power includes terms 2 and 7 which have their origin in environmental effects and the output from the auxiliary electrode driver. With a quiet, two-electrode system at low applied potential, the former two may dominate. When this is the case, the operational amplifier used in the current to voltage converter is the dominant source of noise. Amplifiers with a given noise power are available with a range of values of R,. This is because amplifier design allows one to decrease u,2 at the expense of ur and vice versa. Amplifiers with low p are always preferred. One must choose the amplifier with the lowest available p that has an R, given by eq 33. However, under many circumstances items 2 and 7 are important, especially for high values of x . This complicates the choice of amplifier. If one has significant contributions from terms 2 and 7, and they cannot be reduced in magnitude, then one must find a current to voltage converter amplifier with a low noise current to obtain a low noise power with an environmentally determined noise voltage. The auxiliary electrode driver must be chosen for low a, (low R,) while the current to voltage converter must be chosen for low ui (high R,). The use of the same amplifier type for both jobs is obviously inappropriate. Let us now consider some examples of the noise in array detectors. We must return to the issue of stray capacitance and bandwidth. The calculations have been done with
0.4
0.6
0.8
1.0
X Figure 5. Signal-to-noiseratio versus active fraction. The following were used in the calculation: rl, 50 pm; r 3 and r,, 3200 pm; b, 0.0125 cm; D , 5.0 X 10" crn2s-'; k,, 100 cm s-I;Cm, 20 pF/cm2;R,, 100 0;C ,30pF, €-, 1.O V; frequency band, 0.01-0.25 k;0,1.05 rnL rnin-I . Noises at 1 Hr were as follows: A,, 1.0 X V2; A,, 1.0 X
V2; A,, 1.0 X
A'; A, (0.003)'.
chromatography in mind, and a bandwidth calculated from chromatographic peak width. The noise in the frequency band is calculated by numerical integration. The signal is taken as the steady-state signal. The Warburg impedance, which will play an important role in future calculations on voltammetry a t arrays in flow streams, is assumed infinite. The frequency band chosen would be difficult to realize in practice; thus the noise magnitude is somewhat underestimated. The influence of the stray capacitance will be seen when the -3 dB frequency, 1/(27rRJska,.), becomes as low as the upper frequency governed by the chromatography. A t this point, peak skewing would be evident. We have flagged such conditions in our calculations. Weisshaar et al. published data on a Kelgraf composite electrode (26). The carbon particle size used in the fabrication was small, but the conducting segments of the material were of the order of 50 pm. We have done as well as we could in simulating the dimensions and electronics of their apparatus. A plot of snr for a 1pM solution continuously being pumped through the system is shown in Figure 5. The data of Weisshaar et al. show a maximum at 0.10 < x < 0.15, whereas the maximum in the simulationsis at x 0.02. No reasonable changes in conditions force the maximum to the right. An important inference is that composite electrodes based on the mixing of conducting and nonconducting phases may never reach their theoretical maximum in performance because the method of conduction is by a percolation process, and percolation is highly composition sensitive. In fact, polymercarbon composites display a dramatic reduction in resistivity in the 10-20% carbon range, although if the arrangement of the carbon within the composite is not random, the transition from conductor to resistor may occur at other values of carbon content (40). Caudill et al. studied arrays of 100 5pm-radius carbon disks in a flow stream. They found that the detector was virtually flow rate independent when a thick spacer (0.12 cm) and low flow rates (
