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Anal. Chem. 1988, 60,652-656
Diffusional Distortion in the Monitoring of Dynamic Events Royce C. Engstrom,*’ R. Mark Wightman, and Eric W. Kristensen Department of Chemistry, Indiana University, Bloomington, Indiana 47405
The effect of dlffuslon between the source of a chemical event and the senslng element of a probe was evaluated theoretkally and experhentally. Convokrtlon of the waveform representlng the orlglnal chemlcal event wlth an lmpluse response function derlved from dlffuslon equations was used to predlct the observed process. Dlffuslonal distances of less than 30 pm were consldered and several geometries Important to analytlcal probes were studled. Experlmental study of dlffuslonal dlstortlon was carried out by using an ultramlcroelectrode probe to monitor concentration events produced at a generator electrode located microscopic distances from the probe. Squarcwave, trlangular-wave, and repetltlve events were produced and monltored at varylng interelectrode distances. The effect of coatlng the probe wlth a thln polymer film was evaluated experlmentally and theoretically.
When a dynamic chemical event is monitored, the observed process may be distorted if diffusion of analyte from the source of the event to the sensing element is involved. The extent of distortion depends upon the time course of the event, the transit time required of the analyte, and the diffusional geometry. As will be shown, even microscopic diffusional distances can generate significant distortion of events that occur on a subsecond time scale. Recognition of the problem arose in this laboratory as a result of in vivo monitoring of neurochemical release and uptake with ultramicroelectrodes, where the site of neurotransmitter release is at some distance away from the sensing surface of the electrode (1-4). Other analytical devices subject to diffusional distortion would include membrane-based electrodes, such as the Clark oxygen electrode (5)and various enzyme electrodes (H), in which analyte must traverse the membrane to enter the sensing region. Probes based on fiber optics terminated with a section of semipermeable hollow fiber require diffusion of analyte to the interior of the hollow fiber where eventual interaction with light occurs (9, 10). Thermistor-based sensors (11) should exhibit similar behavior, except heat transfer instead of mass transfer is involved. Furthermore, the concepts and methodology presented here are applicable to diffusion problems outside the area of analytical chemistry. Whenever a substance originates at one site and interacts at another, diffusion between the sites will influence the time course of the event at the second site. For example, surface diffusion, membrane transport, and synaptic transmission may be subject t o the considerations described herein. In this paper, we describe a theoretical and experimental investigation of diffusional distortion. The theoretical approach is based on convolution theory, in which the diffusion process is viewed as operating on a primary event of arbitrary waveform to produce the observed event some distance away. Convolution theory has become a common tool for the interpretation of detector response, and that application as well as others relevant to analytical chemistry has been reviewed (12-14). Comprehensive treatments of the theory, methoPermanent address: Department of Chemistry, University of South Dakota, Vermillion, SD 57069.
dology, and application of the convolution technique have appeared (15, 16). Recent advances in software permit the execution of convolution operations in a matter of seconds on a personal computer, making the technique applicable to the routine solution of problems involving complex input waveforms and geometries for which analytical solutions do not exist. The experimental approach described here has been used to study distortion in the case of planar diffusion and makes use of a microelectrode probe technique (17 , 18), in which an ultramicroelectrode is positioned within the diffusion layer of mother, larger electrode. The large electrode, referred to as the generator, is used to create a concentration event of controllable amplitude, duration, and form, while the microelectrode is used to monitor that event at a known and adjustable microscopic distance away.
THEORY The original event occurs as a change in concentration at the site of origin and is designated C,(t),=,. The diffusional operator produces an observed concentration that is a function of both time and distance from the site of origin, C,(x,t) C,(t),=, diffusion C,(x,t) (11
-
-
Within a given geometry and a t any given distance, the diffusional operator is a linear, time-independent one, meaning that the form of the operator does not change with C,(t) or with time. Under those conditions, the output function is related to the input through the convolution integral (15, 16)
In eq 2, T is a variable of integration. The equivalent operation to eq 2 is multiplication of the Fourier transforms of the functions C,(t) and Z(x,t) followed by taking the inverse transform of the product. The function Z(x,t) is the “impulse response function” of the system, defined as the output, C,(x,t), when the input, C,(t),=,, is equal to the delta function, 6(t). In many cases, it is inconvenient or impossible to directly evaluate the impulse response function from its definition. Instead, it may be obtained indirectly from the “step response function”, A ( x , t ) ,which is the output when the input is equal to the Heaviside step function, through the relationship (15) Z(x,t) = (d/dt)A(x,t)
(3)
For diffusional processes, the responses to step changes in concentration for several geometries are known and tabulated (19),so it is a relatively simple matter to convert them into impulse response functions. Once Z(x,t)has been evaluated for a given geometry and distance from source to point-ofobservation, it can be used to predict the observed response for any input function. Step response functions obtained from ref 19 and the resultant impulse response functions obtained by differentiation are given in Table I for several geometries of analytical interest. Implementation of the convolution process involved creating one data array containing the impulse response function evaluated at some time interval per point and a second array of the input concentration function, C,(t),=,, evaluated by using the same time interval. In the work shown here, the impulse response functions and the input functions were defined over 1024-point arrays at 5 ms per point. The two
0003-2700/SS/0360-0652$01.6~/0 @ 1988 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 60, NO. 7, APRIL 1,
1988 653
Table I. Response Fuctions for Diffusional Geometrieso geometry
step response functionb
1. planar 2. spherical divergent 3. spherical
convergent 4. thin layer
impulse response function'
C ( x , t ) / C o= erfc [~/2(Dt)''~](eq 2-45) I ( x , t ) = x exp (-~~/4Dt)/2(*D)'/*t~/~ C ( r , t ) / C o= ( r o / r ) erfc[(r - ro)/2(Dt)1/2] (eq 6-60) I(r,t) = [(ro/r)(r- ro)/2(rD)1~2t3/2] exp[-(r - ~ , ) ~ / 4 D t C(t),,,/C" = 1 + 21n=1Y-1)n exp(-Dn2r2t/r2) (eq 6-19) I(t),=o = 2 ~n=1"(-1)"(-on2n2/r02) exp(-Dn2r2t/r,Z) C(t),=o/C" = 1 - (4/r)x,,=O" [(-1)"/(2n
exp[-D(2n
+ 1)2~2t/412] (eq 4-17)
+ l)] X
I(t),=o = (4/T)xn,o"(-l)"[o(2fl exp[-D(2n 1)2~2t/412]
+
+ 1)r2/412]X
Symbols: C, concentration of observed species; C", maximum concentration obtainable; I , impulse response function; D, diffusion coefficient; t , time. In case 1, x is the distance between source and point-of-observation. In case 2, ro is the radius of the spherical source and r is the distance from center of that source. In case 3, ro is the radius of the sphere that material diffuses in from, and the point-ofobservation is at the center of the sphere. In case 4, the thin layer has a length, 1, and the point-of-observation is at the boundary opposite that of incoming material. Obtained from ref 19, equation number from that reference shown in parentheses. 'Obtained by differentiation of step response function.
Picoommeter Positloner
B i p's ta t I
.
\I
1
Cel I A
Generator Electrode
Flgure 1. Schematic of the microelectrode probe apparatus.
arrays were convolved by using the CONV routine available in the fast Fourier transform utility package, 87FFT, marketed by Microway (Kingston, MA).
EXPERIMENTAL S E C T I O N Apparatus. The instrumentation for the microelectrode probe technique was a modification of that described earlier (17,18) and is shown schematically in Figure 1. Ultramicroelectrodes prepared from 10 pm diameter Thornell P-55 carbon fibers (Union Carbide, New York) were beveled (20) on a commercial beveling apparatus (WP Instruments, Inc., New Haven, CT) a t an angle of approximately 45O to yield a smooth, flat, and reproducible tip geometry with an overall diameter of approximately 25 pm. Experiments involving Nafion films made use of microelectrodes dipcoated in solutions of Nafion (Du Pont, Mendenhall, MS) with a previously described procedure (21). The generator electrode was a 0.5 mm diameter platinum disk sealed in glass and polished to a final finish with 0.05-pm alumina. The generator electrode was placed in a cell so that its surface faced upward and the microelectrode was mounted over the generator a t an angle of 45" so that the two electrode surfaces were parallel to one another. Positioning of the microelectrode was accomplished with a Klinger Model CC1.2 controller (Richmond Hill, NY) equipped with stepper-motor positioners capable of motion in 1-pm increments. A laboratory-built bipotentiostat (22) was used to control the potentials of the microelectrode and the generator electrode, with the potential waveforms of the generator provided from an IBM PC (Boca Raton, FL) equipped with a Labmaster Interface (Scientific Solutions, Solon, OH). The microelectrode potential was set a t a constant value appropriate for the detection of the electrolysis product of the platinum electrode. A reed relay (Gordos 47059, Bloomfield, NJ) was placed between the microelectrode and its current amplifier, a Kiethley Model 427 (Cleveland, OH), so that the microelectrode could be held a t open-circuit between measurements. This was found necessary to minimize "recycling" of electroactive material between the two electrodes at close distances of separation. The switching was controlled by computer-generated logic pulses, and unless stated otherwise, the microelectrode was switched "on" for 5 ms at intervals of 50 ms to make the current measurements. The current was read a t the end of the 5-ms sampling period. Reagents. Solutions of 1mM potassium ferrocyanide in 1M potassium chloride were prepared fresh daily from reagent-grade
0
L 0
0.2
0.I
Time, s Flgure 2. Impulse response functions for (A) various diffusional geometries all evaluated at a distance from source to pointdfobservation of 15 pm and a diffusion coefficient of 5 X lo-' cm2/s and (B) planar diffusion at distances shown on figure and with a diffusion coefficient of 7.6 X lo-' cm2/s.
Table 11. Response Times of Microelectrodes (in milliseconds) interelectrode distance, pm 0
5 10 15 20 25
bare electrode
Nafion-coated electrode
difference
59 (18)" 76 (24) 136 (28) 213 (18) 316 (18) 422 (24)
84 (22) 172 (52) 264 (77) 366 (82) 426 (28) 540 (38)
25 96 128 153 110 118 1216
Values shown are mean of three electrodes. Standard deviation is given in parentheses. bAverage. The value taken at 0 pm was excluded from the average based on the Q test. chemicals and doubly distilled water. Solutions of 3,4-dihydroxybenzylamine (Sigma Chemical Co., St. Louis, MO) were prepared in pH 7.0 phosphate-citrate buffer.
RESULTS Impulse Response Functions. Before the response to a particular waveform is examined, it is instructive to look at the impulse response functions themselves. Figure 2A shows the impulse response functions from Table I1 all evaluated
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 7, APRIL 1, 1988 ~~
I
iI
Convolution (0-25~.1m) A
Experimental
c/co 0 0
2
4 0
2
4
Time, s
Figure 4. Response to a triangle-wave concentration event in planar diffusion. I n each panel, highest curve represents the original event and each successively lower curve is at a 5-pm interval.
Time, s Figure 3. Response to a square-wave concentration pulse in planar diffusion. I n each panel, highest curve represents response at 5 pm and each successively lower curve is at a 5-pm interval.
by using a distance from source to point-of-observation of 15 pm and a diffusion coefficient of 5 x lo4 cm2/s. For the divergent case, the radius of the source was set a t 5 pm, although that choice influences only the amplitude and not the time course of the function. The diffusional geometry has an important effect on the time course of the impulse response function. The response functions for those cases where material diffuses into a finite volume (convergent and planar thin layer) are of shorter duration than when diffusion takes place into an infinite or semiinfinite volume, implying that a lower degree of diffusional distortion should result for the finite volume cases. Since impulse response functions result from the input of a delta function, and since mass must be conserved, the area of the impulse response function must be equal to unity when evaluated over infinite time. However, when evaluated over a finite time, the area under the function may be significantly less than 1 and the output function amplitude will be attenuated relative to the input function. For example, the amplitude of an output function that has been subjected to planar semiinfinite diffusion will be less than that subjected to convergent diffusion over the same distance as long as the duration of the event is finite. Note that the planar and divergent cases are identical in time course and vary in amplitude. The amplitude of the divergent case depends upon the size of the spherical source, ro, and tends toward the planar case as ro becomes very large. Impulse response functions for the case of semiinfinite planar diffusion evaluated over a distance range of 5-15 pm with a diffusion coefficient of 7.6 X lo4 cmz/s (the diffusion coefficient of ferricyanide (23))are shown in Figure 2B. The functions clearly show that the response to a given primary event will be slowed as the distance between the source of that event and the probe increases. Over the time interval shown, the areas under the functions decrease with increasing distance so the amplitude of the output function should be expected to be lower at increasing distances. Response to a Square Concentration Pulse. The first input function, C,(t),=,, to be studied was a square-wave concentration pulse since the results of that situation are well-known from equations describing double-potential step chronoamperometry (21)and the waveform is easily generated
at an electrode. Thus, the square wave provided a test system to validate both the convolution and experimental approaches. The top panel of Figure 3 shows the convolution of a square-wave concentration pulse of amplitude 1 and duration of 1s using the semiinfinite planar impulse response functions of Figure 2B (except evaluated out to 5 s) for distances of 5-25 pm. The distortion predicted from the convolution appears as a decrase in amplitude, a time delay at the foot of the wave indicative of the time needed for diffusion from source to point-of-observations, a time delay at the end of the concentration pulse before the response begins its downward trend, and decreased slopes on both rising and falling portions of the signal. These effects are also predicted by direct calculation (22) as shown in the center panel of Figure 3. The families of curves obtained by convolution and direct calculation are in fact superimposable, establishing the validity of the convolution technique. Experimental results, shown in the lower panel of Figure 3, were obtained by placing a solution of 1 mM potassium ferrocyanide and 1 M potassium chloride in the cell and stepping the potential of the platinum generator electrode from 0.0 to 0.8 V for 1 s and then back to 0.0 V. The carbon fiber microelectrode was switched on at a potential of 0.0 V at the interval specified in the experimental section, and the interelectrode distance was adjusted to the values used in the theoretical cases. All of the features predicted from convolution and direct calculation were observed experimentally, although the time course of the leading edge was noticeably slower at close distances than that predicted, perhaps due to uncertainty in the positioning of the microelectrode or to “shielding” of the generator by the microelectrode. The relative amplitudes of the responses at the end of the 1-s pulse agree quite well with those amplitudes predicted theoretically. The results show that the diffusional process has some functional analogy to a low-pass electrical filter, in that the high-frequency components of the input signal are attenuated. (The time delays evident in the diffusional process are not present in the low-pass filter.) The effective “time constant” of the diffusional filter increases as the distance between source and point-of-observation increases. Response to a Triangular Waveform. With the convolution technique validated, it was applied to other waveforms for which direct calculation of diffusional distortion is difficult or impossible. One such waveform of interest in our laboratory is that associated with the release and uptake of neurotransmitter in the mammalian brain (1-4). As a result of stimulated release and enzyme-mediated uptake, the primary concentration event has been predicted under a variety of experimental conditions associated with in vivo monitoring ( 4 ) . Under most conditions, the release-uptake event resembles a triangle with the rising portion equal in length to the duration of stimulation. Figure 4 shows the diffusional distortion of a triangle waveform having a total duration of 2 s as predicted by convolution and as observed experimentally.
ANALYTICAL CHEMISTRY, VOL. 60, NO. 7, APRIL 1, 1988
Convolution
Experimental
lime, s Flgure 5. Response to a repetitive waveform in planar diffusion. Distance away from source is shown in lower right corner of convolution panels.
Experimental generation of the triangular concentration waveform was accomplished by incrementing the platinum electrode potential according to the Nernst equation
E = E O’
- 0.059 log (1- Co)/Co
(4)
in which Co,the fractional concentration of ferricyanide, was increased linearly with time from 0.01 to 0.99 and then decreased back to 0.01 at the same rate. The formal potential, E O’, was found to be 0.245 V. The observation that the experimental signal a t a distance of zero is very nearly triangular indicates the assumption of equilibrium at the electrode surface was valid. Predicted and observed diffusional distortion of the triangular waveform appears as decreased amplitude, decreased slope on both the rising and falling parts of the waveform, and general rounding at each change in slope of the original signal. Since kinetic information about release and uptake processes is obtainable from the slopes of the event, it is clear that diffusional distortion would lead to errors in the estimates of kinetic parameters. For example, from the convolution results in Figure 4,the maximum slope of the falling response a t 25 pm is only 37% that of the original signal. Resolution of Repetitive Events. Diffusional distortion may limit the ability to resolve repetitive events if the time between those events becomes short compared to the time needed for diffusion from the source to the probe. Figure 5 shows convolution and experimental results when the concentration at the source is a series of pulses of 200 ms duration spaced 500 ms apart. As the probe is moved farther from the source, the signal no longer returns to the base line between pulses and the peak amplitude decreases, effectively filtering the periodic process. With a large enough diffusional time, the observed signal would approach a constant value, removing all information regarding periodicity. The same effect was observed when the frequency of the pulses was increased but a t a constant interelectrode distance. The effect can be characterized in terms of the dimensionless parameter, D T / x 2 , where D is the diffusion coefficient of the species involved, T is the interval between pulses, and x is the interelectrode
655
distance. The experimental results of Figure 5 indicate that the signal returns to within 10% of the peak amplitude from base line when D T / x 2 is approximately 2. Diffusion t h r o u g h a Membrane. Occassionally a probe is coated or covered with a membrane to lend selectivity to the device or to protect the sensing surface from its environment. This membrane may introduce an additional diffusional barrier that causes further distortion of the primary event. Nafion-coated microelectrodes used for in vivo voltammetry have been evaluated with respect to response time to dopamine in a previous study with a flow-injection system (21) and appeared to cause minimal additional distortion over that of a bare electrode. The experimental apparatus described in this paper provided a means of characterizing the diffusional distortion attributable to Nafion films with a time resolution greater than that of the flow-injection system. A l-s concentration pulse of the oxidation product of 3,4dihydroxybenzylamine (DHBA) was created at the generator electrode by stepping its potential from 0.0 to 0.8 V in the presence of 1 mM DHBA. The response time of the probe to DHBA was quantified by noting the time required for the response to reach 63% of its maximum value during the concentration pulse. This “time constant” has no physical basis since the diffusional process is not governed by the same response function as an electrical filter and since the responses were not necessarily at a steady value at the end of the l-s pulse. However, the parameter does provide a convenient means of comparing response time of electrodes with and without Nafion films. The response times of three electrodes before and after Ndion coating are given in Table I1 at various interelectrode distances. As expected, the electrodes exhibited longer response times when coated. With the exception of the distance of 0 Fm, the increase in response time caused by the film was a constant within experimental error, adding 121 ms to the response time. The significance of the added response depends upon the solution diffusion component that is also operative in the system and, of course, on the time course of the signal being monitored. Theoretical modeling of the diffusional distortion a t the Nafion-coated electrodes was accomplished with a double convolution
where the primary event was first convolved with the impulse response function for semiinfinite planar diffusion as before, using an interelectrode distance of 15 wm. The resultant ] ~ . film was function was then convolved with Z ( X , ~ ) ~ ~The modeled as a planar thin film (Table I, case 4) in which the film thickness was taken to be 200 nm and the diffusion coefficient in the film as 1 X cm2/s based on earlier independent measurements (21). Figure 6 shows the results of the convolution of the solution diffusion alone and with the Nafion diffusion added for a square pulse (A) and a triangular pulse (B). Without the Nafion, the response to the square pulse requires 233 ms to reach 63% of its maximum response, in good agreement with the experimentally observed average value of 213 ms shown in Table 11. With the Nafion present, the convolution predicts that an additional 221 ms is needed to reach the 63% level compared to the value of 121 ms found experimentally. The discrepancy could easily be due to uncertainties in the film thickness and or film diffusion coefficient. The important conclusion from Figure 6 and Table I1 is that the Nafion causes additional significant distortion of events occurring in the subsecond time domain.
CONCLUSIONS Diffusion over microscopic distances can cause significant distortion of a dynamic process, making obscure the original event, introducing error in the extraction of kinetic infor-
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I
I 1
I
Bare
A
I
merically differentiating that response, then applying it in a convolution with the desired input waveform. The success of that approach would require the removal of noise prior to differentiation.
ACKNOWLEDGMENT Discussions with Leslie May concerning in vivo voltammetry and Paul Hale concerning convolution techniques were greatly appreciated. Registry No. Potassium ferrocyanide, 13943-58-3; 3,4-dihydroxybenzylamine, 37491-68-2; Nafion, 39464-59-0.
c IC"
LITERATURE CITED
Time, s Figure 6. Double convolution of (A) square wave and (8)triangle wave showing predicted response at a bare probe situated 15 pm from the source and at a Nafion-coated probe at the same position. The original events are shown with dotted lines.
mation from that event, and ultimately placing a lower limit on the time course of the event that can be monitored. Minimization of diffusional distortion requires a decrease in the distance between the source of the measured event and the sensing surface of the probe. Alternatively, retrieval of the original event might be accomplished through mathematical deconvolution of the observed process provided the appropriate impulse response function and distance of diffusion is known. However, as the degree of distortion increases, the error associated with deconvolution would also increase and the point would eventually be reached where not enough of the original signal character remained to obtain a unique solution in the deconvolution process. In cases where the diffusional geometry is too complicated to model, it would be possible to obtain an impulse response function experimentally by measuring the response to a step function, nu-
(1) Amatore, C.; Kelly, R. S.; Kristensen, E. W.; Kuhr, W. G.; Wightman, R. M. J. Nectroanal. Chem. 1986, 273.31. (2) Kuhr, W.G.;Wlghtman, R. M. Brain Res. 1987, 387, 168. (3) Kuhr, W. G.; Wightman, R. M.; Rebec, G. V. Brain Res., In press. (4) Wightman, R. M.; Amatore, C.; Engstrom, R. C.; Hale, P. D.; Kristensen, E. W.; Kuhr, W. G.; May, L. J. J . Neurochem., In press. ( 5 ) Clark, L. C., Jr. Trans.-Am. SOC.Artif. Intern. Organs 1956, 2 , 41. ( 6 ) Bradley, C. R.; Rechnitz, G. A. Anal. Chem 1985, 5 7 , 1401. (7) Arnold, M. A. Anal. Chim. Acta 1983, 754, 33. (8) Mell, L. D.; Maloy, J. T. Anal. Chem. 1975, 4 7 , 299. (9) Seltz, W. R. Anal. Chem. 1984, 5 6 , 16A. (10) Peterson, J. I.; Vurek, G. G. Science 1984, 224,123. (11) Mosbach, K.; Danlelsson, B. Anal. Chem. 1981, 53,83A. (12) Hleftje, G. M.; Horlick, G. Am. Lab. (Fairfield, Conn.) 1981, 73, 76. (13) Ng, R. C. L.; Horlick, G. Spectrochim. Acta, Part B 1981, 366,529. (14) Horlick, G.; Hlftje, G. M. Contempory Topics in Analytical and Clinical Chemistry: Plenum: New York, 1978; Vol. 3. (15) Bracewell, R. N. The Fourier Transform and Its Applications; 2nd ed.; McGraw-Hill: New York, 1978. (16) Brlgham, E. O., Jr. The Fast Fourier Transform; Prentis-Hall: New York, 1974. (17) Engstrom, R. C.; Weber, M.; Wunder. D. J.: Burgess, R.; Whquist, S. Anal. Chem. 1986, 58, 844-848. (18) Engstrom, R. C.; Meaney, T.; Tople, R.; Wightman, R. M. Anal. Chem. 1987, 59,2005-2010. (19) Crank, J. The Mathematics of Diffusion, 2nd ed.; Clarendon: Oxford, 1975; pp 21-102. (20) Kelly, R.; Wightman, R. M. Anal. Chim. Acta 1986, 187, 79. (21) Kristensen, E. W.: Kuhr, W. G.; Wightman, R. M. Anal. Chem 1987, 59, 1752. (22) Bard, A. J.; Faulkner, L. €lectfochemical Methods: Fundamentals and Applimtlons; Wlley: New York, 1980; p 566. (23) Adams, R. N. Nectrochemistry at Solid .Electrodes; Marcel Dekker: New York, 1969; p 219.
RECEIVED for review July 28, 1987. Accepted December 7, 1987. This research was supported in part by the National Science Foundation, Grant No. CHE-8411000 and BNS8606354.
