7534
J. Phys. Chem. 1991, 95,1534-1538
Somerharju, P. J.; Virtanen, J. A.; Eklund, K. K.; Vainio, P.; Kinnunen, P. K. J. 1-Palmitoyl-2-pyrencdecanoyl glycerophospholipidsas membrane probes: Evidence for regular distribution in liquid-crystalline phosphatidylcholine bilayers. Biochemistry 1985, 24, 2773-278 I . Sturtevant, J. M.; Ho,C.; Reimann, A. Thermotropic behavior of some fluorodimyristoylphosphatidylcholines. Proc. Narl. Acad. Sci. U.S.A. 1979,76,2239-2243. Sugar, I. P.;Monticelli, G.Landau theory of two-component phospholipid bilayers: phosphatidylcholintphosphatidylethanolaminemixtures. Biophys. Chem. 1983, 18, 281-289.
Tamm, L. K.Lateral diffusion and fluorescence microscope studies on a monoclonal antibody specifically bound to supported phospholipid bilayers. Biochemistry 1988, 27, 1450-1457. Vanderkooi, J. M.; Callis, J. B. Pyrene. A probe of lateral diffusion in the hydrophobic region of membranes. Biochemistry 1974, 13, 4000-4006.
Vauhkonen, M.;Sassaroli, M.; Somerharju, P.; Eisinger, J. Dipyrenyl phosphatidylcholines as membrane fluidity probes. Relationship between intramolecular and intermolecular excimer formation rates. Biophys. J . 1990. 57, 291-301.
Vaz, W. L. C.; Derzko. Z. 1.; Jacobson, K. A. Photobleaching measurements of the lateral diffusion of lipids and proteins in artificial phospholipid bilayer membranes. Cell Sur!. Rev. 1982,8, 83-136. VonDreele, P. H. Estimation of lateral species separation from phase transitions in nonideal two-dimensionallipid mixtures. Biochemistry 1978,17. 3939-3943.
Wu, E. S.;Jacobson, K.; Papahadjopoulos, D. Lateral diffusion in phospholipid multibilayers measured by fluorescence recovery after photobleaching. Biochemistry 1977,16, 3936-3941. Xu, H.; Huang, C.-h. Scanning calorimetric study of fully hydrated asymmetric phosphatidylcholineswith one acyl chain twice as long as the other. Biochemistry 1987,26, 1036-1043.
Correlation between Measured and Calculated Mobilities of Ions: Sensttlvity Analysis of the Fitting Procedure Zvi Berant, Wed Shahal, and Zeev Karpas* Physics Department, Nuclear Research Center, Negev, P.O. Box 9001,Beer-Sheva, Israel 841 90 (Received: November 21, 1990;In Final Form: March 19, 1991)
The sensitivity of the procedure for calculating reduced mobilities, based on a model employing a hard-core potential, was evaluated by comparing calculations with experimentally determined reduced mobilities in three homologous series of compounds: normal primary amines, normal tertiary amines, and cyclic amines. It was found that the choice of the parameter representing the separation between the center of charge and center of mass of the ion, a*, strongly affected the magnitude of the interaction potential but had relatively little effect on the calculated reduced mobilities. The choice of the constants ro and z had a marked effect on the agreement between calculated and measured reduced mobilities, especially in light drift gases and over a large range of ion masses. However, over a limited range of values, a erroneous choice of one of these parameters could be compensated for by the other parameter. The extent of clustering, n, had a relatively small effect on the calculated parameters, especially for heavy ions, but affected the calculated reduced mobilities, through the reduced mass.
Introduction In previous studies'+ a fitting procedure was employed to correlate the experimentally measured reduced mobilities of ions with values calculated according to a model in which a hard-core potential was used to represent the interaction between the ions and drift gas molecule^.^ This model was found to be superior to other models, such as the rigid-sphere and Langevin models,$ employed to derive the mobility of ions drifting through a buffer gas in an electric field. However, although the model with the hard-core potential could reproduce experimental mobility values over a limited range of ion masses, it failed to do so over a broader range of ion masses. Tt was further shown that the quality of the fit was relatively insensitive to changes in the independent parameters used in the model to represent the interaction potential over a limited range,'+ which may be a reflection on the significance of the calculated physical parameters of the interaction. An example of this is the calculation of the reduced mobilities of ions drifting through helium or argon.* Varying a* (defined and explained in detail previouslyl*2*5 and briefly in the following section) from 0. to 0.4had little effect on the quality of the fit between calculated and measured reduced mobilities but strongly affected other physical parameters such as the interaction potential surface, the ion-neutral distance, and the collision cross section. ( I ) &rant, Z.; Karpas, Z . J . Am. Chem. Soc. 1989,1 1 1 , 3819. ( 2 ) Berant, Z.;Karpas, Z . 1. Phys. Chem. 1989,93, 3021. (3) Karprts, Z.; Berant, Z.; Shahal. 0.J. Am. Chem.Soc. 1989,1 1 1 , 6015. (4)Berant, Z.;Karpas, Z.; Shahal, 0. J . Phys. Chem. 1989,93, 7529. ( 5 ) (a) Revercomb. H.E.; Mason, E. A. Anal. Chem. 1975,47,970.(b) Mason, E. A. In Plasma Chromatography; Carr, T. W., Ed.;Plenum Press: New York, 1984;Chapter 2.
It was thus demonstrated that it was necessary to bear in mind the physics underlying the purely mathematical fitting procedure when attempting to derive the values of the parameters that gave the best fit. To resolve the apparent problem of lack of sensitivity to variation of the parameters of the model, an attempt was made in the present study, to evaluate the sensitivity of the calculations. These calculations were based on the results of the fitting procedure, to changes in the four independent variables (see following section) of the fit, Le., a*, ro, z, and n.'+ This was done by studying three homologous series of protonated compounds: normal primary aliphatic amines, normal aliphatic tertiary amines, and cyclic monoamines. While all members of these series readily protonate on the aminic nitrogen lone pair of electrons, they differ considerably in their structure: the primary amines have a single hydrocarbon chain 'wagging" behind the protonated aminic group, the tertiary amines consist of three hydrocarbon strands of equal length centered around the protonated nitrogen atom, and the cyclic amines have a closed saturated hydrocarbon ring from which protrudes the protonated aminic group. Thus, a methodical investigation and analysis of the sensitivity of the quality of the fit obtained on the parameters used in the fitting procedure, were carried out and are described here. Theory of Ion Mobility The mobility, K,of ions drifting through a bath gas in an electric field, E, is given byS
where q is the charge of the ion and tn its mass, N is the density
0022-3654/91 /2095-7534$02.50/0 0 199 1 American Chemical Society
The Journal of Physical Chemistry, Vol. 95, No. 19. 1991 1535
Mobilities of Ions of the drift gas molecules and M their mass, and p is the reduced mass of the ion-neutral collision pair given by p = m M / ( m M). Teffis the effective temperature, k Boltzmann constant, (Y a correction factor, generally less than 0.02 for m L M (ref, Sb, p, 50) and QD( Teff)the collision cross section, which depends on the effective temperature. Under conditions of the IMS experiment, where E / N is about 1 Td (townsend, lo-'' V an2),the more sophisticated two- and three-temperature approaches6 and corrections for the ion temperature' are not required. The effective temperature is essentially equal to the cell temperature. Following the approach of Mason and co-workers,s the collision cross section is given by
+
where a( 1 , 1 ) ( T + ) is the dimensionless collision integral that depends on the ion-neutral interaction potential and is a function of the dimensionless temperature, T* = kT/eo. Here, co is the depth of the minimum in the potential surface and r, the position of this minimum: co
= e2ap/[3rm4(1 - a*)4]
(3)
where e is the ion charge and aPthe polarizability of the drift gas molecules. u* = u/r,,,, where u represents the separation between the center of charge and the center of mass of the ion. As we are concerned mainly with polyatomic ions of complex structure, a* is not negligible. The type of interaction potential used in most cases is the 12,4 or so called hard-core potential:
V(r)
(€0/2){[(rm - a ) / ( r - a)]'' - 3[(rm - a ) / ( r (4)
where r is the distance between the ion and neutral drift gas molecule and all other parameters were defined above. In a homologous series of ions, it is reasonable to assume that the ion radius varies as the cubed root of its mass. Thus r, becomes
r, = ro[l + b(m/M)'/3] where b is a constant representing the relative density of the ion and neutral molecule, generally taken as unity, and ro is a constant. It was previously demonstrated that the reduced mass term is insensitive to small changes in the mass of heavy ions.' Thus, even in a homologous series of ions, the calculated mobility does not quantitatively follow the experimentally determined values. The addition of a semiempirical correction term, mz, to the interaction potential and the collision cross section, through modification of the expression for rm,greatly improved agreement between the calculated and measured values over a broad ion mass range in a variety of drift gases? The physical significance of this correction term is the underlying assumption that a better representation of the position of the minimum of the interaction potential is obtained by adding a term that depends on the ion mass and accounts for the compressibility of the collision pair. Thus the dependence of r, on the ion mass is not just through the ratio of the cubed root of the masses of the ion and neutral gas molecule. Thus, a modified expression for rm was obtained:
r, = (ro + mz)[I
+ (m/M)'/'I
In addition to this, it was found that ions drifting in polarizable gases, especially at low temperatures, tend to form clusters with the drift gas molecule^.^ Thus, the mass of the ion may be incremented by clustering, so that its effective mass becomes
mer = m
+ M,
(7)
(6) (a) Viehland, L. A.; Lin, S. L. Chem. Phys. 1979, 43, 135. (b) Viehland, L. A.; Fahey, D. W. J . Chrm. Phys. 1983, 78, 435. (7) (a) Ellis, H. W.; Pai, R. Y.; McDaniel, E. W.; Mason, E. A.; Viehland, L. A. A?. Nuel. Dofa Tables 1976, 17, 177. (b) Ellis, H. W.; McDaniel, E. W.; Albritton, D. L.; Lin, S. L.; Viehland. L. A.; Mason, E. A. /bid. 1978, 22, 179. (c) Ellis, E. W.: Thackston, M. 0.;McDaniel, E. W.; Mason, E. A. Ibid. 1984, 31, 113.
where n is the average of the number of drift gas molecules clustered on the core ion as it drifts through the buffer gas.
Methodology The reduced mobilities of ions in the three homologous series mentioned above were experimentally measured by use of an ion mobility spectrometry (IMS). A fitting procedure was used to determine the parameters a*, r, z, and n of the calculation outlined above that best reproduced the experimental values. From these, the effective ion mass could be calculated by eq 7 , the position of the minimum of the interaction potential, r,, and its depth, to, could be calculated by eqs 6 and 3, respectively. The collision cross section, Q,, was calculated from eq 2. Finally, the reduced mobility itself was calculated from eq 1. However, it is common practice to rewrite eq 1, by substitution of the appropriate constants and units, and use the inverse reduced mobility:' The units of KO are cm2 V-' s-l when the masses are expressed in atomic mass units, r,,, in angstroms, and T in kelvin. It must be stressed that the four parameters u*, ro, z , and n were not truly independent. For example, the value of u* was affected by the other parameters, as they determined the value of rm (eqs 6 and 7 ) . Also, a large chosen a* value would lead to a calculated stronger interaction through smaller calculated rm values. Finally, it should be noted that there is little or no experimental data from other independent techniques on the values of the parameters used in the fitting procedure or calculated from it. It was therefore important when using reduced mobility values to derive ion-molecule interaction parameters to practice caution and judiciously choose the initial values of the independent parameters, bearing the mind the physics underlying the fitting procedure.
Experimental Section All the primary and cyclic amines studied were commercially available and used with no further purification. The tertiary amines were available either as the free base or as quaternary salts or bases that upon introduction into the heated IMS source and ionization underwent dissociation to form the protonated tertiary amine.9 The ions were not mass identified in the pesent study, but in previous studies9 it was shown that the protonated molecule was the predominant ion in all these amines. Mobility measurements were carried out with a Phemto-Chem 100 ion mobility spectrometer (IMS) made by PCP, Inc., FL. The instrument and experimental procedure were described in detail previou~ly.~*~ Spectra were acquired and averaged with Computerscope (RC Electronics) hardware and software. Reduced mobilities in air were measured relative to a standard reference compound9, protonated 2,4-lutidine, in which KOwas taken as 1.95 cm2 V-' s-', Reduced mobilities of ions in other drift gases8 were calculated from
KO = ( d / E t ) ( 2 7 3 / T ) ( P / 7 6 0 )
(9)
where t is the time the ion traverses the distance d, T i s the IMS cell temperature, and P is the ambient pressure. The error in the measured reduced mobilities is below 1% with the reference compound and below 2% for the values calculated according to eq 9. The fitting procedure was outlined above and described in detail previous1y.I
Results and Discussion The reduced mobility in air, at 250 O C , of the three homologous series of protonated compounds are summarized in Table I. The ions studied were protonated primary normal amines, C,H2,+lNHs+, where n = 1-12, protonated normal tertiary (8) Cohen, M. J.; Spangler, G. E. In Plasma Chroma; Carr, T. W., Ed.; Plenum Press: New York. 1984; Chapter 1. (9) Karpas, Z. Anal. Chem. 1989, 61, 684.
7536 The Journal of Physical Chemistry,Vol. 95, No.19, 1991
Berant et al.
TABLE I: Experimclltrlly Measured Reduced Mobilities in Air, at 250 OC, of Protonated Normd Primary Amines, Tertiary Amines, and Cyclic Amines
amine
1 2 3 4 5 6 7 8 9 10 11 12 15 21 24 36
2.65 2.41 2.20 2.02 1.85 1.72 1.61 1.50 1.42 1.35 1.28 1.23
2.38 2.14 1.94 I .84 1.72 1.64
1.95 1.62 1.35 1.19 0.95 0.85 0.64
1.36
6
8
12
10
16
14
18
r (AOI Figure 1. Calculated potential energy surface for protonated octylamine ions (130 amu) drifting through air at 250 O C . Curve a was calculated with u* = 0.1, r,,, = 7.988 A, and Q = 3.1 1 meV, curve b was calculated with u* = 0.2, r , = 7.143 A, and to = 7.79 meV, and curve c with a* = 0.4, r,,, = 7.143 A, and to = 7.79 meV.
-
0.7-
TABLE II: Semitirity of tk Rtt& Procedure to tbe Vdue of a *, An JuQsd bv tk Relative xs Value in Each Series
n-primary amines n-tertiary amines
cyclic amines
0 1.26 2.74 0.153
0.1 0.91 2.90 0.147
0.2 0.34 5.9 0.497
0.3 38 57 50
0.4 8.8 123 4.2
amines, (C,HwJ3NH+, where n = 1-5,7,8, 12, and protonated cyclic amines, c-C.HhNH3+, where n = 4-8,12. Some reduced mobility values obtained in other drift gases, such as helium' and C 0 2 published previously? were also used. The effects of the choice of each of the independent parameters of the fitting procedure a*, ro, z,and n, on the calculated values of r, eo, V(r),and QD, and finally on the reduced mobility will be discussed. Choke of a*. Values of a* covering the range Cb1 in increments of 0.1 were taken from the tables generated by Mason and coworkers.I0 Values of a* > 0 indicate separation between the center of charge of the ion and its center of mass, while a* = 0 means that they coincide. The fitting procedure outlined above was carried out in each series of compounds. The value of a* was fixed, while the values of rm z, and n, were not restricted, until the best fit, determined by the best xz, was obtained. The results are summarized in Table 11. From these results it appears that the best fit in the series of primary amines was obtained for a* = 0.2. In the tertiary amine series, taking a* = 0 was only slightly better than a* = 0.1, while in the cyclic amine series the situation is reversed. This is consistent with what would be intuitively expected if the charge is localized mainly on the nitrogen, which is the preferred site of protonation. Thus, in tertiary amines, the center of charge and center of mass are on the nitrogen atom and almost coincide, in cyclic amines they are slightly separated, while in the primary amines they are further apart than in the other series. The choice of a* strongly influences the calculated potential energy surface, V(r),through its effect on r, and eo ( q s 6 and 3, respectively). For example, shown in Figure 1 is the potential energy surface (eq 4) calculated for the interaction of protonated octylamine (1 30 amu) with the drift gas molecules in air at 250 OC. Curve a was calculated by using a* = 0.1, r, = 7.988A, and co = 0.31 1 meV, while in curve b a* = 0.2, r, = 7.142 A, and e0 = 0.779 meV. These values of r, and eo were obtained from the calculation. The stronger interaction calculated with a* 0.2 is reflected in the larger depth and closer position of the minimum in the potential surface. It is interesting to note that f
(IO) Mason, E. A,; Ohara, H.; Smith, F.J. J . f h y s . B, Ar. Mol. fhys.
1972, 5. 169.
I
I
20
60
I
100
I
1
1LO
180
ION MASS (amu 1
Figure 2. Experimentally determined inverse reduced mobility of protonated normal primary amines as a function of ion mass. Curves a-c represent the calculated fit obtained with u* = 0, 0.2, and 0.4, respectively.
using given values of r, and c,, and changing a* only slightly affects the shape of the potential surface, as indicated by curve c in Figure 1, which was calculated with a* = 0.4 but with the same r, and eo values as curve b. However, as seen from Figure 2, a* has only a relatively small effect on the calculated KO-'values. Changing a* from 0 (curve a) to 0.2 (curve b) barely affects the fit with the experimental values, but increasing a* to 0.4 (curve c) noticeably diminishes the quality of the fit. As pointed out above, the choice of a* strongly affects the values calculated for r, eo, and QD. As the optimal a* value changes from 0 in the tertiary amines to 0.2 in the primary amines, comparison between these series can be made only if the physical parameters are calculated with the same a* value. The value of choice is a* = 0.1, which is close to the best value for all three series. In each series r, increases with ion mass, as expected from q 6. However, for a given ion mass rm is largest for the primary amines and smallest for the cyclic amines. This, again, corresponds to our intuitive understanding of the shape of these ions. Namely, the primary amines can be vistkalized as a long chain wagging behind the protonated amino group, while the cyclic and tertiary amines have more compact structures. The depth of the minimum in the interaction potential curve, eo, decreases with ion mass in the three series. For a given ion mass, eo is smallest in the primary amines and slightly larger in the cyclic amines than in the primary amines. This is a manifestation of the trend in the value of r, as expressed by eq 3. The trend in the value of the collision cross sections, QD, also reflects the effect of r,, expressed in eq 2. Namely, the cross section increases with ion mass in all series, and for a given ion mass is largest in the primary amines and smallest in the cyclic amines. As mentioned above, increasing a* leads to a marked decrease in r, and therefore in QD, while eo increases. This is demonstrated in Figure 3 for primary amines, where changing a* from 0.1 ( c u m a) to 0.2 (curve c) strongly affects r, by decreasing its value by
The Journal of Physical Chemistry, Vol. 95, No. 19, 1991 1531
Mobilities of Ions
200
100
I
20
300
LOO
500
ION MASS (amu) I
60
I
100
I
1 LO
I
180
ION MASS (amu) Figure 3. Calculated r,,, values of protonated primary and cyclic amines as a function of ion mass, Curves a and c were calculated with a* = 0.1 and 0.2, respectively, for the same experimental data set of primary amines, and curve b was calculated, with u* = 0.1 for cyclic amines. 0.7 I
60
I
I
I
I
I
,
80 100 120 1LO 160 180 ION MASS (amu)
Figure 5. Dependence of the quality of the fit between experimental and calculated reduced mobilities on the value of n. Curve a was calculated with n = 0, curve b with n = 0.44 and curve c with n = 2. Figure 5a shows ions with masses up to 522 amu, and curve 5b expands up to 200 amu.
I
I
I
I
I
60
100
1 LO
180
ION MASS ( a m ) Figure 4. Effect of choice of ro on the quality of the fit for primary amines in air at 250 OC. All curves were Calculated with u* = 0.2. Curve a with ro = 2.28 A and z = 0.003 16 A/amu (fully optimized), curve b with ro = 2.48 A and z = 0.003 16 A/amu, curve c with ro = 2.08 A and z = 0.003 16 A/amu, curve d with ro = 2.48 A and z = 0.001 86 A/amu, and curve e with ro = 2.08 A and z = 0.0045 A/amu.
a fixed decrement of about 0.9 A and giving lower values than those calculated for the cyclic amines with an a* value of 0.1 (curve b). This effect is relatively larger for the smaller ions. It has been shown previously that the value of a* depends not only on the nature of the ion but also on the properties of the drift gas? The observed increase of a* with the polarizability of the drift gas was attributed to the dipole induced in the drift gas molecules by the ion. Choice of ro and z. It has been previously shown that the addition of a correction factor is especially important when the ion mass range is large and when the drift gas has a low molecular weight.2 This is due to the lack of sensitivity of the reduced mass term to changes in the ion mass when m >> M . The correction factor, mz, compensates for this by adding an additional massdependent term to the expression for r,. Choice of z will thus affect the calculated value of r, and therefore of 6,V(r),and fl,. The choice of ro strongly affects the calculated reduced mobilities if the correction factor, mz, is set to 0, as was demonstrated previously for ions drifting through air and heliume4 It was also found that for a homologous series of ions, spanning a broad range of ion masses, the values of both parameters, ro and z, must be optimized in order to obtain a good fit between calculated and measured mobilities. However, it is of interest to test the sensitivity of the calculations, when looking at homologous ions covering a limited mass range, such as the primary amines (32-186 amu). As shown in curve a of Figure 4,changing the optimized value of ro (ro = 2.285 A, z = 0.003 16 A/amu) without imposing restrictions on the other parameters, a*, z, and n, barely affects the quality of the fit with experimental values (curves b and c). On the other hand, changing ro by less than IO%, while using the same optimized value of z = 0.003 16 A/amu, leads to a significant deterioration of the quality of the fit as seen in curves d and e. A similar phenomenon is observed when z is interchanged by ro. Thus, over a limited
mass range an erroneous choice of ro (within reason) may be compensated for by z, without noticeably affecting the quality of the fit. However, over a broad range of ion masses (up to 500 amu), the quality of the fit is more sensitive to the choice of both parameters. Choice of n. The experimental data shown above were measured at a relatively high temperature (250"C) and in drift gases with low polarizability. However, when carrying out mobility measurements a t temperature below 100 O C or in gases with a tendency to form clusters, such as COz and SF6, the assumption that the effective mass of the ion is that of the core ion is no longer validO4Thus, while neglecting n is not always justified, it is so in the calculations made above. A different case is shown in Figure 5, where the mobilities were measured in COz at 154 OC, and where curves a-c were obtained with n = 0,0.44,and 2, respectively. In Figure Sa the full mass range is shown, and the fit with measured mobilities appears to be fair in all cases. However, a closer look a t the lower end of the mass scale (up to 200 amu), Figure Sb, shows that underestimating the extent of clustering (curve a, n = 0) or overestimating it (curve c, n = 2), leads to a marked decrease in the quality of the fit. Thus, as was also previously demonstrated,' the quality of the fit depends also on the choice of n.
Summary In the fitting procedure, which is purely mathematical, four parameters are used to fit a set of experimental data points. However, the physical significance of the choice of these parameters is manifested in the values the parameters obtain. The most striking example is the value derived for a* from these calculations. For example, the fact that a* increases with the polarizability of the drift gas and with the asymmetry of the ion indicates that it can be used to derive structural and physical information. Furthermore, as the choice of a* strongly affects the calculated value of rm, and through it of V(r),eo and QD, this must be carefully done, bearing in mind the physics underlying the experimental data. For example, due to the small effect of the choice of a* on the agreement between experimental and calculated reduced mobility values, especially in helium,' choosing a* > 0.1 would lead to an unreasonably large attractive interaction between the ion and helium atom. The quality of the agreement between the experimental and calculated reduced mobilities shows a much more pronounced
1538
J. Phys. Chem. 1991, 95, 7538-1542
dependence on the choice of r, and z . Improper choice of these parameters leads to a good quantitative fit only over a limited range of ion masses. Changing ro and z directly affects the calculated value of rmrand through it of to, V(r),and Q,. The sensitivity of the fitting procedure to the choice of n is not as
dramatic as with the other parameters, as its effect is manifested only in the reduced mass and reduced mobility. The effect becomes important only under conditions of low temperatures with highly polarizable drift gases, and affects light ions more strongly than heavy ions.
Chronoamperometry at Channel Electrodes: A General Computational Approach Adrian C. Fisher and Richard C. Compton* Physical Chemistry Laboratory. South Parks Road, Oxford OX1 3QZ, United Kingdom (Received: January 29, 1991)
A conceptually simple and computationally efficient implicit approach to the simulation of current/time transients at the channel electrode is presented. The generality of the method is illustrated with reference to the simple case of a step between potentials mmponding to zero and transport-limited currents for a simple redox reaction (where good agreement with existing analytical theory is found), to ECE and to DISPl processes, and finally to the case of double-potential steps. The latter are confirmed to be a possible means of distinguishing between ECE and DISPl reactions.
Introduction Channel electrodes are now recognized as well-characterized and advantageous flow-through hydrodynamic electrodes both for the study of electrode processes and for electroanalytical purp0ses.l Their merits include the huge range over which mass transport can be varied, operation under chemostatic conditions, and the mechanistically discrimating power conferred by the nonuniformity of the diffusion layer over the electrode surface.2 Hitherto the great majority of channel electrode studies have confined themselves to steady-state conditions, although of course the extra information available from time-dependent work is established.) Accordingly, timedependent theory for channel electrode problems has begun to emerge, albeit rather slowly. Thus we have derived and experimentally verified a theory for the ac voltammetry of an electrochemically reversible redox couple a t the channel and this work has been extended by Currane6 Additionally, Aoki has treated theoretically the problem of chronopotentiometry a t the channel electrode’ while Johnson* has devised a pulsed anodic technique for electroanalytical purposes: a full, review of the area appears in ref 1. The purpose of the paper is to provide a general implicit computational strategy for the calculation of the chronoamperometric responses arising from potential-step experiments at channel electrodes. The method is applicable to a wide range of electrode reaction mechanisms and, by way of example, theoretical results are given for single- and double-potential-step experiments for ECE and DlSPl reactions as well as for a simple electron transfer. The capability for general extension to other electrode processes should be evident. The ability to treat mechanistically complex processes arises from the computational efficiency of the implicit calculations used here, as opposed to the explicit finite-difference treatment developed by Marcoux for channel flow systems? We (1) Unwin, P. R.; Compton, R. G. Compr. Chem. Klner. 1989, 29, 173. (2) Compton, R. G.; Fisher, A. C.; Tyley, 0. P. J . Appl. Electrochem. 1991, 21, 295. ( 3 ) Sluyters-Rehbach, M.; Sluyters, J. H. Compr. Chem. Kiner. 1986, 26,
203. (4) ComPton, R. G.; Sealy, G. R. J . Elecrroanal. Chem. 1983, 145, 35. ( 5 ) Compton, R. G.; Laing, M. E.; Unwin, P. R. J . Elecrroanal. Chem. 1986, 207, 309. (6) Kingsley, E. D.; Curran, D. J . Electroanalysis 1990, 2, 273.
(7) Aoki, K.; Matsuda, H. J . Electroanal. Chem. 1978, 90,333. (8) Polta, J . A.; Johnson, D. C. Anal. Chew. 1985, 57, 1373.
0022-3654/91/2095-7538$02.50/0
note that for the very simple case of a single potential step in the absence any homogeneous kinetics Aoki et a1.I0 have derived an approximate analytical theory valid under conditions where the LEvQue approximation (i.e., treating the convective flow velocity profile as linear near the electrode surface) is applicable: this will be shown to be in excellent agreement, under the appropriate conditions, with our computations.
Theory We consider first the computation of the current/time transient resulting from a step between a potential at which no current flows and one corresponding to the transport-limited reduction or oxidation of a single solution-phase species, A: A&e+B The convective-diffusion equation describing the distribution of A in time ( 1 ) and space is
where D is the diffusion coefficient of A and the Cartesian coordinates x and y can be understood with reference to Figure 1 which is a schematic diagram of a channel electrode. u, is the solution velocity in the x direction, the components in t h e y and z directions being zero. Provided a sufficiently long lead-in length’ exists upstream of the electrode so as to allow the full development of Poiseuille flow then v, is parabolic 0,
=
.( ( 1-
;)2)
vv = 0,0, = 0
(2)
-
where h is the half-height of the cell (see Figure l ) , y ’ * h y , and vo is the velocity of flow at the center of the channel. The boundary conditions defining the problem specified above are t < 0, y = 0,O < x < x,: [A] = [A]hlk (3)
t l O , y = O , O < x < x , : [A]=O all t , y = 2h, all X: a [ ~ ] / a=y o (9) Flanagan, J.
E.; Marcoux, L. J . Phys. Chem. 1974, 78, 718.
(4) (5)
(IO) Aoki, K.; Tokuda, K.; Matsuda, H. J . Elecrrwnal. Chem. 1986.209,
241.
0 1991 American Chemical Society
