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Anal. Chem. 1981, 53, 202-206
Spectroelectrochemical Observation of Diffusion Profiles by the Parallel Absorption Method Rlchard Pruiksma and Richard L. McCreery” DepaHment of Chemistty, The Ohio Sfate University, Columbus, Ohio 43210
Improvements in a spectroelectrochemicaltechnique whlch Involves absorption of light passing parallel to an electrode surface are discussed, A laser beam passing near an electrode is absorbed by electrogenerated chromophore, and the region being monitored Is deflned by a slit whlch can be posltloned at various dlstances from the electrode. The technique Is slgniflcantly more sensitive than most spectroelectrochemical methods and does not depend on the optlcal propertles of the electrode surface. The technique allowed spatlal resolution of the diffuslon layer, with concentration vs. distance profiles belng obtained for distances down to less than 20 pm from the surface. The resolution of the method was found to be llmlted by diffractlon of the llght by the electrode, with the 3-pm silt appearing to sample approxlmately 25 pm of the diffusion layer.
Several geometries for spectroelectrochemical experiments, including internal and external reflection and transmission through a transparent electrode, have found wide use in optically monitoring and characterizing species generated at an electrode surface. The nature and merits of those approaches have been reviewed and will not be repeated here (1-5). A much less common geometry occurs when a light beam passes by an electrode parallel to the surface. This configuration was used for interferometry with some success, but the basic quantity being measured was refractive index rather than absorptivity (6, 7). While interferometry can provide information about the spatial distribution of chromophore, concentrated solutions (0.1 M) of single components (e.g., CuS04) are required, and light-deflection effects limit the techniques to distances greater than about 100 pm from the surface. Applications to the dilute solutions usually encountered in electrochemistry are seriously restricted by these complications. A “focal cylinder” method has been described, where a laser beam is focused down to a beam waist and the electrode is moved relative to the beam waist (8). Measurement of absorbance of the beam by an electrogenerated material permits the spatial distribution of chromophore within the diffusion layer to be observed. However, the resolution of this technique is limited to the width of the beam waist (-50 pm), which is a significant fraction of the diffusion layer thickness (-200-400 pm a t most). In addition, regions closer to the electrode than -100 pm could not be sampled due to obstruction of the conical beam by the electrode. A previous report discussed the use of a narrow slit to intercept a portion of a laser beam passing parallel to a planar electrode (9). By movement of the slit relative to the electrode, different portions of the diffusion layer were sampled, and absorption measurements using light from the slit were used to construct concentration vs. distance profiles. In addition to providing spatial resolution, the technique is quite sensitive, since the path length (determined by the electrode length along the optical axis) is long compared to most spectroelectrochemical techniques. However, the initial results using this technique suffered from significant quantitative error 0003-2700/81/0353-0202$01 .OO/O
(-15-20%) for distances greater than 60 pm from the electrode, and absorbance values were unpredictable at shorter distances. In addition, the slit used in the initial work was reported to be 10 pm wide (later discovered to be 18pm wide), limiting the resolution to that value. A subsequent report from a different laboratory (10) discusses an experiment with similar beam geometry but without a slit. A beam from a conventional spectrometer traveling very close to a planar electrode was used to monitor electrogenerated chromophores. The sampled region of the beam was not defined by a slit, rather the whole beam was analyzed for absorbance. While the technique showed high sensitivity, the response could not be quantitatively predicted from theory because of uncertainties in the shape and position of the beam relative to the electrode. In addition, the time response was slow due to the time required for diffusion to “fill” the beam with chromophore. The use of a slit in the present work allows more accurate definition of the region being analyzed, therefore permitting theoretical comparison and spatial resolution. The present paper describes significant improvementa over the original work from this laboratory which permit higher resolution and observation of the region within 60 pm of the electrode. In addition, the fundamental limitations of the technique are discussed.
EXPERIMENTAL SECTION The overall experimental configuration is the same as reported earlier (9) but with significant changes in the cell and mechanical components. An argon ion laser with up to 2.4 W of output at 515 nm was used instead of the previous He/Ne laser to provide more power through the smaller slit and to accommodate a different chemical test system. The laser output was controlled at 1 W, of which -100 mW reached the cell after attenuation by neutral density filters. The spatial filter used in the initial work was found to have no effect on the results and was not used with the argon laser. The electrode and slit arrangement, shown in Figure 1, was designed so the slit was as close as possible to the trailing edge of the working electrode. A luggin capillary was attached to the reference electrode and positioned near the working electrode out of the beam. The working electrode was cut from an optical flat ( h / 4 ) and coated with either Hanovia liquid bright gold or 70 8, of chromium before vapor deposition of 3000 8, of gold. The flatness of the electrode was checked interferometrically after coating and was found to be as good as the original optical flat. The length of the working electrode along the optical axis was 0.19 cm. Electrical contact with the active surface was made with a gold strip clamped to the coated side of the working electrode. The electrode was aligned parallel to the beam by the method described previously (11). The slit consisted of a razor blade scratch on a gold-coated optical flat. The Hanovia liquid gold coating produced good slita with this technique. The slit width was determined from its diffraction pattern by use of standard equations for single slit diffraction and had a value of 3 pm. The slit was moved relative to the electrode by an “inchworm” piezoelectric translator (Burleigh Instruments, Fishers, NY) with an accuracy of fl pm. Unlike a micrometer, this translator has no tendency to rotate so slit motion is achieved with a minimum of mechanical disturbance to the electrode and slit alignment. The slit was aligned parallel to the trailing edge of the electrode by visually observing 0 1981 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 53, NO. 2, FEBRUARY 1981 Inchworm Translator
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Figure I, Cell and electrode configuration, showing rerlationship of SIR and electrode. Beam enters from right, and light passing through the slit is collected by lens at left. Electrode holder could be moved In the directions indicated to orient the electrode surface relative to the beam. The cell length was 25 mm along the optical axis. the output of the slit while it was scanned along the axis perpendicular to the electrode plane. The slit was aligned until the transition into the shadow region of the electrode occurred at the same time for all points along the width of the beam. The transition from light to dark at the electrode edge was sufficiently sharp that this procedure was accurate to -2 pm. This alignment procedure also established the approximate zero point for the distance from the slit to the electrode plane, but a more accurate determination was made by using the properties of diffraction at an edge. At the geometric shadow of an edge, the intensity of the diffracted light is one-fourth the incident intensity (12). Thus the slit was considered to be at x = 0 relative to the electrode when the intensity of light passing through the slit was one-fourth that of its constant value at distances far away (>lo0 pm) from the electrode. A well-understood test system with sufficient absorptivity at 515 nm is the two-electron oxidation of o-dianisidine in 1 M sulfuric acid at 0.9 V v5. SCE. The literature value (13) of the diffusion coefficient (4.38 X lo4 cm2/s) was verified by use of chronoamperometry. The diffusion coefficient of the oxidized form was determined by chronoamperometric reduction of a solution oxidized with a stoichiometric amount of Ce4+. The diffusion coefficients of the oxidized and reduced forms were equal within experimental error ((5%). The molar absorptivity was determined by conventional spectrophotometry of the oxidized form (21600 at 515 nm) and by external reflection spectroelectrochemistry (22700 at 515 nm). The lower of these two values is less reliable, given the slow degradation of the oxidized form, and the higher value agrees well with the literature value (23000 at 510 nm) (14). Thus the literature value will be used in all calculations below. Theoretical absorption vs. time curves were calculated from the molar absorptivity (230001, electrode length (0.19 cm), and a concentration averaged over the width of the slit. For example, the average concentration for a 3-pm slit positioned at 25 pm from the electrode is the average of digital simulation (15) results for the distances from 24 to 27 pm. Thus the finite width of the slit is taken into account in the absorbance vs. time plots. The theoretical concentration vs. distance plots presented result directly from digital simulation with no averaging of concentrations. RESULTS Figure 2 shows several absorption vs. time transients for the generation of oxidized o-dianisidine at a diffusion controlled rate. After 5 s of oxidation, the potential was returned to 0.10 V to carry out a diffusion-controlled reduction. The indicated distances are the distance of the slit away from the electrode, as determined by the piezoelectric translator. While these plots are similar to those observed previously, much closer distances to the electrode were used, with good response being observed at distances of 20 gm or less. The theoretical line calculated from the literature values for e and D is in-
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Figure 2. Absorbance vs. time profile for double step oxidation of 0.123 mM odianisidlne In 1 M H2S0,: (A) 25 pm from electrode; (B) 52 pm from electrode; (C) 90 p from electrode. Solid lines are experimental curves; dashed lines were calculated from theory with e = 23000 and D = 4.38 X lo-'. cluded for comparison. From a series of absorbance vs. time profiles at various distances, it is possible to construct concentration vs. distance plots at various times, as described previously (9). Several such diffusion profiles are shown in Figure 3 for the forward step of the double step experiment and in Figure 4 for the reverse step. The irregularities in the curves at -40 pm were observed in all cases regardless of time, and all experimental curves were consistently lower than theory, although the error decreased with time and distance. It is possible that localized heating caused by light absorption can lead to refractive index gradients which scatter the light (thermal lens effect) (16). To check that this effect was absesnt and that localized heating was not causing convection, we varied the input laser power over an order of magnitude. No change in the absorbance vs. time response was observed, except for increased noise at low power levels. In order to estimate the effect of diffraction by the electrode, we calculated theoretical diffraction patterns for conditions appropriate to this experiment. For the purpose of this calculation, the electrode was assumed to have an infinitely short dimension along the optical axis. The theory of diffraction from a thin edge has been examined in detail (12) and can be modified to assess the effect of an electrogenerated chromophore on the diffraction pattern. For accomplishment of this task, the distance away from the electrode edge was divided into short increments, and each increment was assigned a concentration calculated from linear diffusion equations. Each increment attenuates the incoming light by
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an amount related to the concentration, with increments near the electrode appearing “darker” during the forward step. The path length used to determine the attenuation was that of the real electrode, even though the diffraction calculation assumed a thin edge. Once the increments were established, the contribution of light from each increment was summed at the slit, taking interference of light from different increments into account. This calculation is a straightforward application of Huygens principle ( l a ,where each light s o m e to be summed is given an amplitude related to the chromophore concentration. Finally, the log of the ratio of light reaching the slit without chromophore to that with chromophore was calculated, yielding an absorbance value with diffraction taken into account. The only approximation in this calculation are that the electrode diffracts light like a thin edge and that all of the absorbance occurs over the thin dimension of this edge. The results of the diffraction calculations are shown in Figure 5. An ideal profile is shown which neglects any diffraction effects. The two curves which take diffraction into account differ only in the distance along the optical axis between the electrode edge and the slit (2). In the real experiment this distance corresponds approximately to the distance between the leading edge of the electrode and the slit. Notice that as 2 increases the oscillations caused by diffraction increase in period and amplitude and that the diffraction correction becomes negligible as 2 approaches zero.
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DISCUSSION As noted in the previous paper, a light beam traveling parallel to an electrode provides a very sensitive probe of chromophore generation. In the present work sensitivity is improved by moving closer to the electrode, yielding, for example, 0.3 absorbance units after 5 s of o-dianisidine oxidation at 25 Fm. A comparable experiment using an optically
ANALYTICAL CHEMISTRY, VOL. 53, NO. 2, FEBRUARY 1981
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Figure 6. Effect of slit width on response for distances close to the electrode. Solid curve is experimentalresult for a 3-pm slit poskioned at the electrode surface ( x = 0 f 2 pm). Dotted curve is theoretical absorbance averaged over the distances from 0 to 3 pm. Dashed curve is theoretical absorbance averaged over distances from 0 to 25 m .
transparent electrode would yield 0.019 absorbance units under the same conditions. Obviously larger absorbances could be achieved with longer electrodes, although the spatial resolution would be sacrificed (see below). Observation of very weak electrogenerated chromophores should be possible, provided they last long enough to diffuse into the sampled region of the beam (a few seconds or less). An additional attractive aspect of the technique is the lack of constraints on the optical properties of the electrode. While it should be flat, it need not be transparent or reflective, and changes in these properties with time or potential are unimportant. Such optical changes have been observed with transparent (18)or reflective electrodes (29). A significant improvement with the present technique is observation of distances much closer to the electrode surface. Several attempts to resolve events closer than 60 pm to the surface failed (6,8,9),yet distances of 20 pm or less were used here. While agreement with theory is not outstanding at short distances, the results were within about 20% of predicted values and qualitatively correct results were observed at distances of 0-3 pm from the surface (Figure 6). Other approaches were unable to provide spatially meaningful results in this region. The diffusion profiles presented here were obtained over a wider range of distances and at higher resolution (3 pm) than the previous work (18 pm). The diffusion profiles and absorbance vs. time curves from which they were derived agree qualitatively with the expected results. At greater distances from the electrode, the concentration changes more slowly in response to potential changes at the electrode, as expected for a diffusing chromophore. However there is not good agreement between experiment and theory particularly a t short times when the diffusion layer is thin. The error ranges from about 10% at 5 s to about 20% at 1.0 s. The best agreement obtained was for the reverse portion of double step experiments where the error was less than 10% over all distances greater than 20 pm. The source of the consistently low experimental results is unlikely to be the apparatus itself, since errors in alignment and positioning were reduced to values low enough to be neglected. The error in the alignment of beam and electrode was about 0.8 mrad with the technique desoribed, implying that a portion of the beam will cut through a 2-rm segment of the diffusion layer over a 0.19-cm electrode length. This error in distance from the electrode is small compared to the resolution element (3 pm). The position of the slit relative to the electrode surface was based on the shadow of the electrode, which had a transition from light to dark covering a total of about 10 pm. While the error caused by diffraction
205
cannot be rigorously assessed because of the thickness of the electrode edge, it is likely to be significantly less than 10 pm. Furthermore, absorbance measurements made at distances with apparently negative x coordinate show no increase in magnitude, as would be expected if x = 0 were not correctly assigned. Thus the error caused by inaccurate positioning of the slit is estimated at 3 pm, and certainly less than the error between theory and experiment. An additional check on the optical system was made by a long term (-5 min) electrolysis which “filled up” the electrode region with chromophore. The absorbance value reached by this electrolysis had the expected Beer’s law value. Finally, a different chemical test system was tried (trianisylamine) in a different solvent (acetonitrile) and wavelength (632.8 nm) with virtually identical results. Therefore it seems unlikely that the quantitative error results from problems with alignment or the chemical system. The diffraction effect depicted in Figure 5 shows that within the constraints of the calculation, diffraction should not cause an overall negative deviation from theory. It is worthwhile to note that the first minimum for a 0.19-cm electrode occurs a t about 40 pm, the same distance as the reproducible disturbance in the experimental points. However this calculation is not sufficient to explain the consistent negative deviation of experiment from theory. The most likely reason for this error is the finite thickness of the electrode, a factor which was not included in the calculation. A beam of light traveling parallel to a surface of finite length will experience three major phenomena: absorption by chromophore, a phase shift caused by refractive index changes, and diffraction by the electrode surface. On the basis of interferometric data (20), the refractive index gradients expected in the diffusion layer for the concentrations used here are much too small (-lod units across the diffusion layer) to cause an observable deviation of the beam from a parallel path. On the other hand, diffraction by the electrode will scatter the light as it passes by the electrode, resulting in a distortion of the wavefronts as they traverse the diffusion layer. Roughly speaking, light entering the diffusion layer at one distance may leave at a different distance, resulting in a loss of resolution. This effect was not assessed by the diffraction calculation, since the program assumed a thin edge, the only case for which a solution is available. An additional factor contributing to this effect is the phase shift caused by the refractive index changes upon oxidation of the substrate. While this shift is not large enough to cause an overall deflection of the direction of propagation, it will affect the diffraction process. When a phase shift correction is included in the diffraction calculation, the error between theory and experiment is decreased. However the phase shift required to make theory and experiment agree is large (21),requiring a change in refractive index of 0.4 unitsM-’of electrogenerated species, a value much larger than that expected (0.03 M-l) (20). One concludes that refractive index changes may contribute somewhat to the diffraction calculation but that the more serious effect is the thick electrode edge. In any case, diffraction with or without a phase shift will cause an effective decrease in resolution, and the slit will be sampling a wider than expected portion of the diffusion layer. While the magnitude of the resolution degradation caused by diffraction is difficult to assess, it can be estimated by theoretically increasing the width of the slit. Figure 6 shows theoretical absorbance vs. time curves in which the absorbance was averaged over a 3- or 25-pm window. Note that the 25-pm slit agrees with experiment better than a 3-pm slit, even though the experimental data were obtained with a 3-pm slit. This observation implies that the 3-pm slit was in fact sampling light which had been scattered through approximately 25 pm of the diffusion layer. A lower effective resolution would
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explain the sluggish response near the electrode, since concentrations averaged over 25 pm change more slowly than those within 3 pm of the electrode as shown in Figure 6. A reasonable conclusion is that the wave properties of the light prevent a resolution of 3 pm because the light diffracts near the electrode surface, allowing light from a wider region than 3 pm to enter the slit. This effect should be less severe for shorter wavelengths, but apparently the difference was not observable for the two wavelengths (515 and 633 nm) used here. One would predict that a shorter electrode would yield better results, since the diffracted light would travel a shorter distance and the slit would sample a smaller increment of the diffusion layer. It was observed experimentally (21)that a shorter electrode gave better results, but the short (0.04 cm) electrode was exceedingly difficult to align, and the small scale made mechanical operations very difficult. However, it can be concluded that the wave properties of the light ultimately degrade resolution when attempting to monitor events on a scale of only a few tens of wavelengths. The same effect also limits the response time of the absorbance, since a wider effective sampling window leads to sluggish response. In addition, there is a trade-off between path length and spatial resolution, since longer electrodes worsen the diffraction problem. Even without spatial resolution, however, a longer electrode could be used to enhance sensitivity and lower detection limits. In conclusion, a spectroelectrochemical experiment using light parallel to the electrode represents a geometry with some attractive features. High sensitivity results from long optical path length, and regions within -20 pm of the electrode surface can be monitored. Concentration vs. distance profiles can be constructed with diffraction of the light being the
ultimate limit on spatial resolution and response time. Applications of the technique to problems involving reactions of electrogenerated species and more complex mass transport phenomena are in progress.
LITERATURE CITED Winograd, N.; Kuwana, T. In “ElectroanalyticalChemistry”; Bard, A. J., Ed.; Marcel Dekker: New York, 1974; Vol. 4. Kuwana, T. Ber. Bunsenges. Phys. Chem. 1973, 77, 858. Heineman, W. R. Anal. Chem. 1978, 50, 390A. Winograd, N.; Kuwana, T. J. Elecfroanal. Chem. 1976, 23, 333. McCreery, R. L.; Prulksma, R.; Fagan. R. Anal. Chem. 1079, 57, 749. Muller, R. H. Adv. Electrochem. Ehsctrochem. Eng. 1073, 9 , 281. Srinivasan, V. S. A&. Electrochem. Elecfrochem. Eng. 1973, 9 , 389. IkeshoJi,T.; Seklne, T. Denkl Kagaku 1977, 45, 575. Pruiksma, R.; McCreery, R. L. Anal. Chem. 1979, 51, 2253. Tyson, J. F.; West, T. S. Talanta 1980, 27, 335. Referemce 9, p 2254. Hecht, E. “Schaum Outline Serles, Optics”; McGraw-HIII: New York, 1975; p 197. Adams, R. N. “Electrochemlstryat Solid Electrodes”;Marcel Dekker: New York, 1989; p 218. Bishop, E.; Hartshorn, L. Q. Analyst (London) 1971, 96, 26. Feldberg. S. W. In “Electroanalytlcal Chemistry”; Bard, A. J., Ed.; Marcel Dekker: New York, 1970; Vol. 3. Dovlchl, N. J.; Harrls, J. M. Anal. Chem. 1978, 57, 728. Reference 12, pp 159-181. Mackev. L. Ph.D. Thesis. The Ohlo State Universltv, Columbus. OH. 1975, p 135. McIntyre, J. D. E. A&. Elecfrochem. Electrochem. Eng. 1073, 9 , 61. Srlnivasan, V. S. A&. Elecfrochem, Electrochem. Eng. 1973, 8, 331. Pruiksma, R. Ph.D. Dissertation, The Ohlo State University, Columbus, OH, 1980.
RJXEIVFDfor review September 10,1980. Accepted November 18,1980. This work was supported by The National Science Foundation (CHE 7828068) and The National Institute of Mental Health (MH-28412).
Linearization Methods for First-Order Kinetic Analysis Lowell M. Schwartz Department of Chemistry, Unlversity of Massachusetts- Boston, Boston, Massachusetts 02 125
Three linearization methods are presented for determlnlng the parameters of a first-order klnetlc decay superposed on a flnlte sum of power terms. Although primary consideration is glven to the estimation of the rate constant and Its statistical uncertalnty, the preexponential factor and the coefflcients In the polynomial are discussed as well. The three methods depend upon the freedom of the analyst to preprogram the tlme schedule of recording data polnts, and recording at equally spaced tlme Intervals Is a convenient and suitable program. Special cases of these three general methods have been previously suggested by Guggenhelm, ref 1, by Gutfreund and Sturtevant, ref 7, by Kezdy, Jaz, and Bruylants, ref 2, by Mangelsdorf, ref 3, by Swinbourne, ref 4, and by Glick, Brubacher, and Leggett, ref 8.
In this paper the phrase “kinetic analysis” refers to the procedure whereby measurements are made of the decay variable z vs. time t , and these data are used to estimate the magnitudes and the statistical uncertainties of unknown parameters. The simplest exponential decay scheme is char-
acterized by the well-known nonlinear expression z = y exp(-lzt) (1) where the unknown parameters are y, the preexponential factor, and k , the rate constant, or l/k, the first-order lifetime. We will assume here that the analyst is quite certain of the functional form of the time dependence of z. He knows that eq 1 or one of the generalizations to be discussed is the proper model. While the problem of deducing the kinetic order or the functional form of the model is sometimes of legitimate concern in kinetic investigations, this problem will not be addressed here. Linearization of eq 1 is not a prerequisite for calculating the parameters since several alternative procedures are available, but this is no doubt the most popular technique. Logarithmic transformation yields In z = In y - kt (2) which is a straight line on In z vs. t coordinates with the rate constant becoming the negative slope and the preexponential factor being calculated by exponentiating the intercept on the In z axis. In many instances the primary decay variable z is not measured directly but rather some other physical property
0003-2700/81/0353-0206$01.00/00 1981 American Chemlcal Society
