Anal. Chem. 1989, 6 1 , 1305-1308 1
.’
Wavelength
.
\~
. . .- .-.
Position
Figure 4. smoothed response surface for Mg in a dc plasma. One dimension represents wavelength, and the other dimension represents vertical height in the center of the dc plasma.
dimensions, 25 points wide in the X dimension and 5 points wide in the 2 dimension. When designing a two-dimensional smoothing procedure, one must choose a smoothing width for each dimension. There have been a number of discussions of how to optimize the smoothing width in linear applications (IO,11). The choice of widths involves the trade-off of two factors: First, any smoothing procedure will generate some distortion of the peak, which increases with smoothing width; however, the noise is increasingly attenuated as the width increases. The position of the intersection of these two factors depends on peak shape and noise level in addition to the order of the smoothing integers. For each of the smoothing equations, the surface was smoothed for each combination of smoothing widths from 1 to 31 in the X dimension and from 1to 19 in the 2 dimension. The mean square error was computed for each case, summing the squares of the difference between the smoothed surface and the noise-free surface; the noise was accumulated only over the portion of the surface in which the magnitude of the noise-free surface exceeds 0.1% of the peak maximum. The smoothing results are shown as a series of contour plots in Figure 3. While this demonstration does not purport to provide a rule for optimum smoothing, it does show that a nonsymmetrical smoothing procedure is required. The most obvious and significant point is that the minima occur at
1305
parameters that are far from symmetric. As a final “real-world” example, the smoothing operation was applied to a surface acquired with a two-dimensional charge-coupled device array. The image, shown in Figure 4, results from the excitation of Mg in a dc plasma, with spatial resolution in one dimension and spectral resohtion in the other. In the spectral dimension the emission peaks are very narrow while in the spatial dimension, there exist broad regions of both atomic and continuum emission. Consequently, the smoothing width is small in the wavelength dimension and large in the spatial dimension. The resultant surface clearly and rapidly conveys an optimal position for the measurement of Mg emission. CONCLUSIONS Two-dimensional smoothing integers for PLSC are readily computed from the formulas provided. These formulas make possible the selection of any width in either dimension according to the requirements of the data set. With these tools, smoothing procedures for surfaces can be optimized separately in each dimension. ACKNOWLEDGMENT We express our appreciation to Rafi Jalkian, Jeff Kolczynski, Skip Pomeroy, and Bonner Denton of the University of Arizona for providing images acquired by charge-coupled device imaging technology, including the data on which Figure 4 is based. LITERATURE CITED (1) Madden, H. H. Anal. Chem. 1978, 50, 1383. (2) htzlafi, K. L. Intrcductbn to Computerdsslsted Experirnentatlon; Wiley-Interscience: New York, 1987. (3) Savitzky, A.; Goiay, M. J. E. Anal. Chem. 1964, 36, 1627. (4) Steinier, J.; Termonia, Y.; Deltour, J. Anal. Chem. 1972, 4 4 , 1906. (5) Rossi, T. M.; Warner, I. M. Appl. Spectrosc. 1984, 38,422. (6) Massart, D. L.; Vendeglnste, 8. G. M.; Demlng, S. N.; Michotte, Y.; Keutman, L. ChemomeMcs: A Textbodc;Elsevier: New York. 1988. (7) Kolczynski, J. D.; Pomeroy, R. S.; Jalkian, R. D.; Denton, M. B., to be submitted for publication. (8) Edwards, T. R. Anal. Chem. 1982, 5 4 , 1519. (9) Bevington. P. R . Data Redwtlon and Error Anaksis for the phvsical Sciences; McGraw-Hill: New York, 1969. (10) Enke, C. G.; Neman, T. A. Anal. Chem. 1976, 48, 705A. (11) Edwards, T. H.; Willson, P. D. Appl. Spechosc. 1974, 28, 541.
RECEIVED for review December 20,1988. Accepted March 1, 1989.
Experimental Determination of the Coefficient In the Steady State Current Equation for Spherical Segment Microelectrodes Zbigniew Stojek
Department of Chemistry, Warsaw University, ul. Pasteura 1, 02-093 Warsaw, Poland J a n e t Osteryoung*
Department of Chemistry, SUNY University of Buffalo, Buffalo, New York 14214 A variety of shapes of microelectrodes (i.e., electrode with at least one dimension in the micrometer range) have been examined and characterized as reported in the literature so far. This includes disks, rings, cylinders, lines, and hemispheres (I). Usually microelectrodes are made from metals or graphitic materials. However, mercury microelectrodes are also needed. Several such microelectrodes have been prepared and applied (2-13). Application of mercury microelectrodes for voltammetric purposes requires that the mercury surface is coherent and does not consist of isolated droplets. It has been shown that 0003-2700/89/038 1-1305$01.50/0
hemispherical mercury electrodes which work well can be prepared by electrodeposition of the predetermined amount of mercury on a platinum microdisk in a separate plating step (4,8,10,13). Iridium has also been proposed as a substrate (9). However, iridium wires thinner than 127 pm are not available commercially. Recently a dropping mercury microelectrode has been constructed. The operation of the device is based on heating a mercury reservoir which drives the droplets through a capillary (14). In many of these cases the geometry is not well-defined. For example, Wehmeyer and Wightman (4)assume that 0 1989 American Chemical Society
1306
ANALYTICAL CHEMISTRY, VOL. 61, NO. 11, JUNE 1, 1989
mercury deposited on platinum substrates is hemispherical. However the deposition current at substrates of different sizes (all other conditions being identical) is about the same, and the radii of the “hemispheres” determined experimentally are in general greater (as much as 24 times) than the radius of the substrate. For this case apparently the mercury most closely resembles a shielded sphere with a point of attachment, and the deposition current increases with time as t1i2,as expected for growth of a single nucleus. We are concerned here with a different geometry, that of the spherical segment. This model applies when the metallic substrate is uniformly wet by mercury, and deposition of mercury increases the height of the segment, but the base (the metallic substrate) remains fixed. The diffusion problem for the intermediate geometry of the spherical segment is complex and has not been solved for the general case. However, there is considerable interest in use of such geometries experimentally. Therefore, the aim of the present paper is to determine experimentally the dependence of the steady-state current on the height of the spherical segment. This has been accomplished by analysis of the chronoamperometric deposition of mercury on an easily wettable microdisk electrode. The steady-state current thus measured a t a given time was directly related to the height of the spherical segment through Faraday’s law, assuming that the volume of mercury is always in the shape of a spherical segment. A coherent mercury microelectrode obtained by deposition of mercury on a metallic disk substrate (of radius r ) has the shape of a segment of a sphere (of radius R ) with altitude h and one base of radius r. The volume, V, and lateral surface area, A , of such an electrode are (15) V = xh(3r2 h 2 ) / 6 = xh2[3R - h ] / 3 (1)
+
+
A = x ( r 2 h2) = 2xRh (2) respectively. The radius of curvature of the lateral surface is R = (h r 2 / h ) / 2 (3)
+
The steady-state current at such an electrode can be described by I,, = knFCDr (4) where C denotes bulk concentration of the depolarizer and D is its diffusion coefficient. For a microdisk electrode ( h = 0), k = 4 (16),and for an ideal hemisphere ( h = r ) , k = 2 x . The latter can be obtained by applying considerations of symmetry to the well-known equation for a spherical electrode. In some cases it is more convenient to express the steady-state current in terms of electrode area. Then I,, = k‘nFADC/r, where k ’ = 4 / x for a disk and k ’ = 1 for a hemisphere. The ~]. quantity k’is related to k (eq 4 ) by k ’ = k / x [ l + ( / ~ / r ) The situation where h is not equal to r complicates the analysis of expermental results, since then the coefficient for the steady state current equation, k, is not known. However, it is much more convenient to make an electrode of known but arbitrary h than to be constrained to the hemispherical geometry. The case h = 0 is not a practical alternative. EXPERIMENTAL SECTION Chronoamperometric experiments involving electrodeposition of mercury on microdisk electrodes were done on a Micro-g antivibration table produced by Technical Manufacturing Corp. The instrumental setup consisted of an EG&G PARC Model 273 potentiostat in the high-speed mode controlled by a MC 68020 Masscomp computer. The cell together with the electrometer probe was kept in a Faraday cage. Platinum and silver microdisk electrodes of radius 12.5pm were employed as the working electrodes. The outer diameter of the insulator was about 1 mm. This ratio of diameters of about 400 should ensure conformity with the model of an electrode embedded in an infinite insulating plane. A saturated calomel
1 0
I ~
0 0
I
ANALYTICAL CHEMISTRY, VOL. 61, 0
hlr
k
hlr
k
0.128 0.2112 0.2528 0.2804 0.3148 0.349 0.383 0.417 0.450 0.4836 0.549 0.6136 0.677 0.739 0.800 0.859 0.917 0.974 1.0
4.0 4.03 4.087 4.172 4.215 4.257 4.342 4.413 4.484 4.541 4.626 4.797 4.981 5.166 5.378 5.591 5.776 6.003 6.192 6.2832
1.0838 1.1375 1.2415 1.2919 1.3411 1.4147 1.439 1.533 1.6237 1.712 1.797 1.880 1.961 2.040 2.116
6.670 6.897 7.365 7.55 7.82 8.174 8.316 8.770 9.224 9.650 10.10 10.53 10.94 11.34 11.75
0 . 3
.
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Table I. Coefficient k,in the Steady-State Current Equation (eq 4) for Spherical Segment Microelectrodes"
6
\
NO. 11, JUNE 1, 1989
c
-C 5
0 3
-0
0 1 E
/
I V
-0 3
-0 5
-0
Figure 2. Cyclic staircase curve obtained with a 12.5 p m radius silver disk electrode in 0.25 M HCIO,. Step height = 10 mV, step width = 0.1 S, = 30 OC.
r
than 1 nA. Still, to enhance the rate of wetting, it was necessary to apply for 0.1 s a prepulse to +0.48 V, where silver is moderately oxidized to Ag'. A curve obtained with a silver microdisk electrode pretreated in this manner is shown in Figure 1. The curve is monotonically increasing with time after the first few seconds and does not display the sharp decreases in current characteristic of coalescence of droplets. The decrease in current in the first few seconds is the usual transient response to a potential step. The subsequent continuing increase is due to increase in electrode area as mercury is deposited. The resulting minimum current, here I = 2.84 pA a t t = 11 s, should be close to the steady-state current. At short times, when the amount of mercury deposited is small, we use the equation for the time-dependent current at a disk
Iexp = 1 8 , f(.)
h / r = height of spherical segment divided by radius of circular substrate.
10
(5)
where f(7)= 1 + 0.71835~-'/~ + 0 . 0 5 6 2 6 ~ --~ 0.0064558~"/~, /~ T = 4Dt/r2,and t is time (16). After 11 s f ( ~=) 1.041, which gives for I , the value 2.73 WAand for the diffusion coefficient 1.13 X cm2/s. Diffusion coefficients reported in the literature for HgZ2+in HC104 at 30 O C are 1.05, 1.08, 1.25, and 1.21 X cm2/s in 0.1,0.5,1, and 2 M HC104, respectively, with an error estimated by the authors as 6% (20). Our value fits well in the range of these data. In fact, at t = 11 s, the amount of charge is 3.13 pC, which corresponds to a height of 1.947 Km. Thus the value of the diffusion coefficient should be adjusted downward to 1.12 X cm2/s, still well within the range of expected values. While time in the chronoamperometric experiment elapses, the value of f(7)decreases, reaching the values of 1.029 and 1.0155 a t 22 and 77.33 s, respectively (assuming constant radius of 12.5 pm). During this time period the current actually increases, however, due to the increase in surface area. The mercury volume and area of an ideal hemisphere on a 12.5 pm radius silver disk were reached at 77.33 s, for which time f ( ~ )calculated for a disk electrode of the same area as the resulting hemisphere is 1.022. This deviation from the steady-state current a t a hemispherical electrode can be also calculated from the well-known equation for a sphere We assume that the equation for f ( ~ )for the hemisphere is the same, and thus f(7)= 1.024 for t = 77.33 s and r = 12.5 Ctm. The ratio of HgZ2+reduction currents after 77.33 and 11 s (ideal hemisphere and disk, respectively) is 0.4429/0.284 = 1.559. Applying eq 5 with the values of f(7)above, we obtain the ratio 1.580, whereas the theoretical ratio is 2x/4 = 1.5708. The difference of only 0.6% is a good test for correctness of the experiment. We have assumed that the current measured
0 0 5
0 0
1.5
1 0
2.0
I
2.5
hr"
Figure 3. Coefficient k plotted vs h l r
ratio (see
Table I).
a t 77.33 is more reliable as a reference point than that at 11 s and have corrected the latter value so that the ratio is 1.5708. The final results presented in Figure 3 and Table I give the coefficient k as a function of the ratio h/r. We were not able to find a simple function which yields these values. Because the formula for f(7)is uncertain, the results of Table I should not be applied for situations where f ( ~is)much different from unity. For 0 5 h 5 r, the accuracy appears to be within about 2%. For h > r one is extrapolating rather than interpolating, so the treatment is less certain. From eq 3, for h = 2r, R is only 5r/4 r, so the shape still resembles a hemisphere on an insulating plane more than a shielded sphere with a point of attachment. Also the regular behavior of the current-time transient in this region ( t > 77 s) suggests that the extrapolation is reasonable. Special caution should be taken, however, with double potential step experiments for h > r, because products formed on the forward step might be trapped in the region between the mercury and the insulating substrate. It should be emphasized that the main difficulty is to achieve experimentally the smooth curve of Figure lb. The smooth curves themselves are quite reproducible. These results should prove useful to the experimentalist and also should guide attempts to solve the corresponding diffusion problem.
-
LITERATURE CITED ( 1 ) Ultramicroelectrodes : Fleischmann, M., Pons, S., Rolison. D., Schmidt, P. P.,Eds.; Datatech Science: Morgantown, NC, 1987. (2) Cushman, M. R.; Anderson, C . W. Anal. Chim. Acta 1981, 130, 323.
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Anal. Chern. 1989, 67, 1308-1310
(3) Schulze, G.; Frenzel, W. Anal. Chim. Acta 1984, 759,95. Wehmeyer, K. R.; Wightman, R. M. Anal. Chem. 1985, 5 7 , 1989. Ciszkowska, M.; Stojek, 2. J . Nectroanal. Chem. 1985, 191, 101. Baranski, A. S.;Quon, H. Anal. Chem. 1986, 58, 407. Golas, J.; Osteryoung, J. Anal. Chim. Acta 1988, 787, 211. Howell, J. 0.: Kuhr, W. G.; Ensman, R . E.; Wightman, R. M. J . Electroanal. Chem. 1988, 209, 77. (9) Golas, J.: Galus, Z.; Osteryoung, J. Anal. Chem. 1987, 59,389. (10) Baranski, A. S.Anal. Chem. 1987, 59,662. (11) Sottery, J. P.; Anderson, C. W. Anal. Chem. 1987, 59, 140. (12) Li. L. J.; Fleischmann. M.; Peter, L. M. Nectrochim. Acta 1987, 32, (4) (5) (6) (7) (8)
1585. (13) Stojek, 2.; Osteryoung, J. Anal. Chem. 1988, 6 0 , 131. (14) Pons, J. W.; Daschbach, J.; Pons, S.; Fieischmann. M. J . Nectroanal, Chem. 1980, 239, 427. (15) Lange’s Handbook oichemisfry, 13th ed.; Dean, J. A , , Ed.;McGrawHill: New York, 1987; pp 1-13.
(16) Aoki. K.; Osteryoung, J. J . Electroanal. Chem. 1984, 760. 335. (17) Hassan, M. Z.; Unterecker, D. F.; Bruckenstein. S. J . Electroanal. Chem. 1973, 4 2 , 161. (18) Metals in Mercury; Solubility Data Series; Hirayama, C., Gaius, Z.,Guminski, C., Eds.; Pergamon Press: London, 1987. (19) Stojek, 2.; Zublik, Z. J . Nectroanal. Chem. 1975, 6 0 , 349. (20) Levart, E.; d’Ange d’Orsay, E. P. J . Nectroanal. Chem. 1966, 12, 277.
RECEIVED for review October 17, 1988. Accepted March 10, 1989. This work was supported in part by the U S . Office of Naval Research and by the Grant 01.17.04.01 from the Polish government.
Generalized Digital Smoothing Filters Made Easy by Matrix Calculations Stephen E. Bialkowski Department of Chemistry and Biochemistry, U t a h State University, Logan, U t a h 84322-0300 Reported herein is a simple procedure for the determination of the impulse-response filter function used for data smoothing by the convolution process ( I ) . This procedure is based on the use on matrix techniques used in regression analysis and included in many software packages or algorithms found in many texts on numerical analysis (2-4). It is a general technique and so may be used to determine the filter function coefficients for Savitsky-Golay type power series polynomials ( 5 ) ,finite Fourier series, and almost any other basis functions. The essence for the least-squares smoothing filters made popular by the work of Savitsky and Golay is regression of the raw data within a finite and “moving” interval, with some function or set of functions which the user chooses as being appropriate for modeling the particular data ( 1 , 5 , 6). In the analysis by Savits!cy and Golay, a power series in the abscissa dimension is used as the model function and the regression analysis is performed analytically in order to obtain the filter coefficients. These filter coefficients make up what is known as the impulse response of the filter ( I ) . The latter is convoluted with the raw data to obtain the smoothed data estimate at the middle point of the sample interval. In addition to the Savitsky-Golay middle point estimators, there have been reports of filters that may be used to estimate other than the central point. In particular, Proctor and Sherwood (8) used orthogonal polynomial filters which allowed estimation of any data within the sample interval, and Leach et al. used the least-squares method to determine the coefficients for an initial point filter (9). As a consequence of the rather tedious task of obtaining the filter coefficients in analytical form, one normally determines only those coefficients that are the estimators for either the middle or end points of the odd smoothing interval. Subsequently, filter smoothing of those points that are close to either the start or the end of the raw data sequence has historically presented a problem. Solutions to this end point problem have ranged from their elimination from the smoothed data set to partial smoothing by simple averaging (6, 7 ) . This loss or partial filtering of data is not necessary. In general, regression analysis can be used to determine the coefficients of the basis set functions which best fit, in the least-squares sense, the raw data over the entire smoothing interval. As an example, consider the power series polynomial given by x [ k ] = a. + a,h a2k2 ... + a,km (la)
+
+
0003-2700/89/0361-1308$01 50/0
or equivalently m ,,.
x [ k ] = CUjk’ j=O
where x [ k ] is the raw data sequence, k is the sequence index, and the a, are the coefficients of the mth order power series to be determined by regression. Equation 1 is actually a single element of a more general problem of estimating each x[k] within the current data interval. The least-squares solution for the coefficients which describe the x [ k ] data over this interval is well-known (2, 4). Cast in the form of a matrix equation, the least-squares solution for raw data with constant variance may be written (4) minllx - Valla
(24
with the simple solution
x = Va
(2b)
where x is a column vector of length n, a is the length m + 1 column vector of coefficients, and V is the n by m + 1matrix of basis functions arranged in columns. With the power series basis set, V is the Vandermonde matrix with elements ui, = i1-l. The minimum p2-norm expression in eq 2a is equivalent to finding those coefficients that minimize the errors in the least-squares sense. Although there are fast algorithms for determining the coefficients of Vandermonde systems (4),there is information in the end result of using the pseudoinverse solution that may be used to determine relative variances. The solution for the coefficients vector in eq 2 obtained by using the pseudoinverse of V is V+x = a (3) where V+ = (VTV)-’VT,and superscript T indicates the transpose. The determination of a describes the data in terms of the power series coefficients, but it does not provide the data estimate, that is the smoothed data. To obtain the estimate, the coefficients are used to reconstruct the data over the interval by multiplication by the basis set matrix f = (VV+)x
(4) where 2 is the estimate of x within the span of the basis set and is the same vector length as the raw data vector. We herein call VV+ an estimation matrix since it estimates each point within the sample interval. 6 1989 American Chemlcal Society
