J. Phys. Chem. 1996, 100, 5913-5922
5913
Magnetic Field Effects in Electrochemistry. Voltammetric Reduction of Acetophenone at Microdisk Electrodes Steven R. Ragsdale, Jeonghee Lee, Xiaoping Gao, and Henry S. White* Department of Chemistry, UniVersity of Utah, Salt Lake City, Utah 84112 ReceiVed: October 30, 1995; In Final Form: January 22, 1996X
The influence of an external magnetic field on the electrochemical reduction of acetophenone (AP) at Pt and Au microdisk electrodes (radii ) 0.1, 6.4, 12.5, and 25 µm) is described. Voltammetric measurements in CH3CN/[n-Bu4N]PF6 solutions containing between 3 mM and 8 M AP demonstrate that the mass-transferlimited reduction of AP at a microdisk electrode may be significantly enhanced or diminished by an external magnetic field, depending on the redox concentration, the electrode radius, and the angular orientation of the microdisk relative to the field. A mechanism for the magnetic field effect is presented that considers the force arising from the divergent radial flux of electrogenerated radical anion (AP•-) through a uniform magnetic field. Viscous drag on the field-accelerated ions results in convective fluid flow that alters the rate at which electroactive AP is transported to the surface. Both lateral and cyclotron fluid motion can be established within a microscopic volume element (∼30 nL) near the electrode surface depending on the orientation of the magnetic field with respect to the microdisk. The earth’s gravitational field is demonstrated to enhance or diminish the magnetic field-induced convective flow, depending on the relative directions of the magnetic and buoyancy forces at the electrode surface.
Introduction A recent preliminary report from this laboratory demonstrated the ability to control and alter the faradaic current at an ∼10µm-radius Pt microdisk electrode using an externally applied magnetic field.1 Specifically, the mass-transport-limited rate for the reduction of a neutral and diamagnetic species, nitrobenzene, was found to be strongly affected by a magnetic field for flux densities between 0.05 and 1 T. In addition, in our preliminary account, we demonstrated that the faradaic current at the microelectrode could be either enhanced (up to ∼100%) or diminished, depending on the direction of the magnetic field with respect to the microdisk surface. Although the ability to alter currents in electrochemical cells using magnetic fields was first demonstrated over a century ago,2 and has been investigated extensively during the past several decades,3-9 all previous measurements of this phenomenon have employed conventional macroscopic electrodes. The recent development of electrochemical techniques employing electrodes with dimensions ranging from nanometers to micrometers has considerably extended the range of materials, as well as spatial and time domains, that may be explored in electrochemistry.10-12 We envision that these microelectrode techniques may open routes for using the dependence of faradaic currents on magnetic field strength and orientation in fundamental investigations of chemical kinetics and transport phenomena, as well as in the development of novel analytical methods. For instance, in the current report, cyclotron-like ion motion occurring in a microscopic solution volume near the electrode surface is shown to result from the interaction of a uniform magnetic field with the radial-divergent fluxes of electrogenerated ions at disk-shaped microelectrodes. The ability to control transport processes at the microscopic level may find application in solution-phase focusing and trapping of electrochemically-generated ions and molecules. It is generally accepted that the primary effect of a magnetic field on electrochemical reactions is to alter the rate of transport X
Abstract published in AdVance ACS Abstracts, March 15, 1996.
0022-3654/96/20100-5913$12.00/0
of ions in the cell. (Only a few reports of magnetic field effects on the interfacial rate of electron transfer have been reported, and these effects appear negligibly small.)13-18 In addition to the entropic and viscous forces which control diffusion of an electroactive species, electric (E) and magnetic (B) fields will generate a force on solution ions, imparting a net drift to these ions in a direction that depends on the field vectors, as well as the charge and velocity of the ions. Classical electromagnetics gives this force in terms of the Lorentz equation
F ) q(E + v × B)
(1)
where q is the charge on the ion and v is the velocity at which the ion is moving through solution. By definition, the quantity qv in eq 1 is finite in an electrochemical cell whenever a faradaic reaction is occurring. Thus, it may be anticipated that an externally imposed magnetic field, if sufficiently large, will directly affect the fluxes of reactants and/or products in electrochemical experiments. Although eq 1 describes the interaction of the magnetic field with the ionic constituents of the solution, it does not provide a description of how the observed electrochemical currents may be altered by a magnetic field. While most investigations of magnetic field effects have employed charged electroactive species as probe molecules (e.g., Cu2+, Fe(CN)64-), it has been demonstrated in several instances, including our preliminary report, that the rate of transport of an electrically neutral species may also be significantly enhanced by a magnetic external field.1,4c,19,20 Since the Lorentzian force acting on a neutral species is zero, the field-induced enhancement of the flux of a neutral molecule must occur by an indirect mechanism(s). The consensus on this issue among active researchers is that the observed flux and current enhancements are associated with convective solution flow induced by the magnetic external field. This approach is described by classical magnetohydrodynamic (MHD) theory. At a molecular level, the solution flow described by MHD theory results from momentum transfer from fieldaccelerated ions to neighboring ions and solvent molecules. Consequently, MHD theory describes the overall process in © 1996 American Chemical Society
5914 J. Phys. Chem., Vol. 100, No. 14, 1996 terms of a magnetic body force acting on a solution element, through which a distributed and continuous current, i, passes, rather than in terms of the direct interaction of the field on discrete current-carrying ions. In principle, the velocity of convective flow and, thus, the convective-diffusion transport of electroactive ions and neutral molecules may then be computed by solving the equation of motion (Navier-Stokes equation) encompassing the magnetic body force. In practice, MHD theory has primarily been employed to guide and interpret empirical correlations between current, field strength, and solution composition. An alternative approach to describing the magnetic field effect in electrochemistry, more recently developed by Olivier and colleagues, is a statistical mechanical treatment of the motion of individual ions which takes into account their diffusional properties in the presence of a magnetic field.21 The development of this theory is motivated in part by the failure of MHD to predict various magnetic field phenomena in dilute ionic solutions. However, to the best of our knowledge, this latter theoretical treatment has not yet been formulated in terms of working equations that predict a magnetic field enhancement of faradaic currents. In the present report, we present the first detailed description of magnetic field effects on the steady-state voltammetric response at microelectrodes having radii between 0.1 and 25 µm. The 1-e- reduction of acetophenone (AP) in acetonitrile (CH3CN) has been used to explore the effect of an externally applied magnetic field on the microelectrode response as a function of electrode radius, redox concentration, field strength, and field orientation. AP is a liquid at room temperature and is completely miscible with CH3CN. Thus, this particular redox system, combined with the use of microelectrode techniques, allows magnetic field effects to be investigated over an unprecedented range of redox concentrations (3 mM to 8 M AP). This experimental capability is especially significant in fundamental investigations of magnetic field effects, since it has been reported that the largest effects of a magnetic field are observed in concentrated redox solutions.1,3b,4c,9 In addition, because of the small size of microelectrodes, mass transport by convective solution flow near the electrode surface can be greatly diminished (or entirely eliminated). For instance, density-driven natural convective flow has been recently demonstrated to be negligibly small when the radius of the microdisk is reduced below ∼5 µm.22 This characteristic property of microelectrodes should also hold true for magnetic field-induced convective flow, thus offering an experimental means of distinguishing between current enhancements resulting from MHD flow or the increased flux of discrete ions. Experimental Section A one-compartment, three-electrode cell containing a Ag/ AgxO reference electrode, a Pt wire auxiliary electrode, and a Au or Pt microelectrode was used throughout the study. Au (6.4-µm nominal radius) and Pt (12.5- and 25-µm radius) microdisk electrodes were constructed by sealing Au or Pt wire in a glass tube and grinding down one end of the tube to expose the metal microdisk. Electrodes were polished with Al2O3 down to 0.01-0.02 µm and then placed in an ultrasonic H2O bath for ∼2 min to remove polishing debris. Smaller Pt electrodes (∼100 nm) were purchased from Nanomics, Inc., and used as received. The exposed tips of these electrodes are reported by the manufacturer to have a cone-shaped geometry. A GMW Associates Model 5403 electromagnet was used to apply a uniform magnetic field, B, across the electrochemical cell (see Figure 1). The magnet poles (7.6-cm diameter) were separated by ∼2 cm. The magnetic field strength, B ) |B|,
Ragsdale et al.
Figure 1. Experimental arrangement for measuring the voltammetric response of a microdisk electrode as a function of the magnetic field strength and orientation. The schematic of the electrochemical cell and electromagnet is a top view drawn approximately to scale.
Figure 2. Side view of the orientation of the microelectrode relative to the magnetic field: (a) horizontal electrode surface facing downward and defining the fixed angle θ ) 90o between the net current (i) and magnetic field (B) vectors; (b) vertical electrode surface, rotated about the variable angle θ defined by i and B. The current vector i is drawn assuming an electrochemical reduction (by convention, positive current flows into the electrode surface for an electrochemical reduction). In both a and b, the angle θ is consistent with definition of the magnetic force, Fmag ) i(l × B), where l is the unit displacement vector in the direction of i. Data presented in Figures 2 through 6 correspond to the horizontal surface orientation depicted in a; data presented in Figures 8 and 9 correspond to the vertical surface orientation depicted in b.
was varied between 0 and 0.8 T by adjusting the current through the electromagnet. Field strength and uniformity were measured using a gauss meter (F. W. Bell, Model 4048). The electrochemical cell was aligned within the electromagnet such that the Au or Pt microdisk electrode was positioned directly at the center of the poles. The magnetic field varied by less than 0.01 T over a radial distance of 1 cm from the center of the poles; thus, a small error in positioning the cell has a negligible effect on the magnetic field applied across the surface of the microelectrode. Because the electrode surface areas are very small ( 0.05 M. Specifically, we find that the experimentally measured currents are significantly smaller than the values predicted by eq 3. In addition, ilim obtains a maximum value when the concentration of the electroactive species is ∼2 M. At higher concentrations (>∼2 M), the steady-state current decreases as the concentration of AP increases, Figure 4. This unusual dependence of ilim on redox concentration has also been observed for the reduction of nitrobenzene26 and benzophenone27 in CH3CN solutions. The maximum in the ilim-[AP] plot is due to an increase in the bulk solution viscosity at higher redox concentrations, resulting in a decrease in the rate of diffusive transport of the electroactive molecule. Stated differently, the diffusion coefficient employed in eq 3 must be treated as a concentration-dependent variable when [AP] > 0.05 M. A detailed analysis of this behavior has been presented in a separate report.27 Application of the magnetic field at θ ) 90° (horizontal surface orientation, see Figure 2a) results in an increase in the limiting current for AP reduction. For instance, Figure 3 shows that an ∼100% increase in ilim occurs at a 25-µm radius Pt disk in a 2 M AP solution upon increasing B from 0 to 0.6 T. Although a similar field-induced enhancement of ilim is obtained using 6.4-µm Au and 12.5-µm Pt microdisks, the magnitude of the effect is a strong function of both the electrode radius and the redox concentration (vide infra). Figure 4 shows the dependence of ilim on [AP] for measurements with B ) 0.7 T. Similar to the results obtained in the absence of a field, ilim reaches a maximum value when [AP] ∼ 2 M. In addition, Figure 4 demonstrates that the magnetic field effect (i.e., the enhancement in the current) is very small at both low and high redox concentrations. The external magnetic field in our experiments is many orders of magnitude larger than the internal magnetic field induced by the faradaic current. By applying Ampere’s law to this
problem (∫B‚dl ) µ0ilim), and using the largest measured value of ilim (∼30 µA), we compute a maximum internal field strength of ∼3 × 10-7 T. This field is a factor of ∼105 smaller than the smallest (non-zero) external field used in our investigations (0.025 T). Thus, the internal field generated by faradaic currents will be ignored throughout our discussion. A key component in the analysis of the observed enhancement of the current, Figures 3 and 4, is the fact that the steady-state fluxes of supporting electrolyte ions are zero, regardless of the relative concentrations of the redox species or supporting electrolyte.23 On the other hand, electroneutrality requires that the charge created within the depletion layer by electrogeneration of AP•- be balanced by an increase and/or decrease in the concentrations of the supporting electrolyte cation and anion, respectively. Either of these processes may be accomplished by the rapid transient flux of electrolyte ions on the slow timescale of the voltammetric experiment. In dilute AP solutions, the quantity of AP•- generated within the depletion layer is small in comparison to the concentration of supporting electrolyte, and thus, the supporting electrolyte cation and anion concentrations in the depletion layer remain relatively unchanged from their bulk solution values. However, as the concentration of AP is increased, the generation of AP•- results in a significant increase in the concentration of n-Bu4N+ near the surface. In the limiting case in which the bulk concentration of supporting electrolyte is much less than that of the redox species, it can be readily shown that electroneutrality within the depletion layer is maintained by the inward transient migration of the supporting electrolyte cation. Thus, the following equality holds within the depletion layer when [AP] >> [(n-Bu4N)PF6].28
[AP•-] ≈ [n-Bu4N+] We focus our discussion on this limiting case, since the largest magnetic field effects are observed in relatively concentrated AP solutions, Figure 4. Based on the above description, the depletion layer at the microdisk can be characterized as containing AP, AP•-, n-Bu4N+, and solvent. Of these, only AP and AP•- have nonzero steady-state fluxes, and only AP•- is electrically charged. Thus, it follows from the Lorentz equation that the only species whose flux may be directly altered by the magnetic field is the electrochemical product, AP•-. However, since the observed voltammetric current is limited by molecular transport of reactant, AP, any mechanism of the observed magnetic field enhancement of the current requires a description of how the interaction of the product ion AP•- with the field affects the transport of the neutral reactant AP. We will assume that the fluxes of the electrochemical reactant (AP) and product (AP•-) are coupled by convective flow resulting from viscous and electrostatic drag by the solvent and electrolyte cation n-Bu4N+, respectively, on AP•- ions accelerated by the magnetic field. Although the effect of convective flow on voltammetric currents is significantly reduced when employing electrodes of micrometer dimensions, we have recently shown that density-driven natural convection22 and forced convection29 result in measurable increases in the limiting current at microdisks. Both natural and forced convection increase the rate at which electrochemical reactants (neutral or ionic) are transported to the surface. In general, the convective flow pattern at the electrode and, thus, the limiting current will depend on the electrode geometry and the direction of the force responsible for the convection (e.g., buoyancy force). As shown later in subsection III, the ability to control the orientation of the magnetic field relative to the microdisk surface provides a directional control over both the flow pattern and voltammetric current.
Magnetic Field Effects in Electrochemistry
J. Phys. Chem., Vol. 100, No. 14, 1996 5917
Figure 6. Dependence of iB - i0 on the magnetic field strength (θ ) 90°, horizontal surface orientation) as a function of electrode radius (6.4, 12.5, and 25 µm). Data correspond to the voltammetric reduction of AP in a CH3CN/0.2 M [n-Bu4N]PF6 solution containing 2 M AP.
Figure 5. Dependence of the current enhancement, iB - i0, on the magnetic field strength, B, at θ ) 90° (horizontal surface orientation). iB and i0 are the voltammetric limiting currents for AP reduction measured in the presence and absence of an externally applied magnetic field. (a) iB - i0 vs B for 0.1 M < [AP] e 1 M; (b) iB - i0 vs B for 2.5 M e [AP] e 8 M. All solutions contained 0.2 M [n-Bu4N]PF6.
II. Correlation between the Magnetic Field Enhancement and the Voltammetric Current Measured in the Absence of a Magnetic Field. The dependency of ilim on B for a 25-µmradius disk (horizontal orientation, θ ) 90°, Figure 2a) is shown in Figure 5 as a plot of iB - i0 vs B, where iB ) ilim measured in the presence of the magnetic field and i0 ) ilim measured with the field off. The quantity iB - i0 represents the enhancement in the voltammetric current due to the external magnetic field. The curves plotted in Figure 5 correspond to six different CH3CN solutions containing AP at concentrations between 0.1 and 8 M. We observe that iB - i0 always increases as the field strength is increased when the electrode surface is positioned horizontally, regardless of the redox concentration or electrode radius. Close inspection of the data plotted in Figure 5 shows that iB - i0 has a slight nonlinear dependence on B. At small B, iB - i0 increases very slowly, and in some instances, a threshold value of B is necessary for the current enhancement. A rapid increase in iB - i0 is observed at intermediate values of B. Finally, at the largest fields available for the investigations, we observe that iB - i0 increases approximately linearly with B. Figure 6 shows the results from similar experiments in which the concentration of AP was held constant (1 M) and the electrode radius was varied between 6.4 and 25 µm. Nonlinear iB - i0 vs B plots were obtained for each electrode, similar in shape to that described above. In addition, these data demonstrate that the enhancement in current drops precipitously as the electrode radius is decreased. For example, iB - i0 measured at B ) 0.8 T is ∼25 times smaller for the 6.4-µm-radius electrode than for the 25-µm-radius electrode. In similar experiments, we were unable to detect a significant magnetic field enhancement of the limiting current using a 0.1-µm-radius Pt microelectrode. We note that the dependence of the enhancement of the limiting current on the microdisk radius closely parallels that observed for current enhancements resulting from natural convection. In the latter case, we have observed that density-driven convective flow has a negligibly small effect on the limiting current for microdisks having radii less that ∼6 µm.22 The similar critical values of the microdisk
Figure 7. Plot of current enhancement, iB - i0, as a function of the voltammetric limiting current in the absence of an applied magnetic field, i0. Symbol keys: r0 ) 6.4 µm (]); r0 ) 12.5 µm (4); r0 ) 25 µm (O).
radius below which an enhancement is not observed is consistent with our contention that the magnetic field enhancement results from convective flow. The dependencies of iB - i0 on redox concentration (Figures 4 and 5) and electrode radius (Figure 6) are complex and do not readily lend themselves to the interpretation of the magnetic field effect. However, a more interesting and apparently general relationship exists between iB - i0 and the value of the limiting current in the absence of the magnetic field, i0. For instance, Figure 7 shows a plot of iB - i0 vs i0 constucted using data recorded at constant flux density (B ) 0.7 T) using 6.4-, 12.5-, and 25-µm-radius electrodes in solutions containing 0.01 < [AP] < 8 M. By inspection of this figure, it is clear that there is a strong correltation between the current enhancement and io measured in the absence of the field. Thus, the critical parameter operative in determining the magnitude of the enhancement in current is i0, rather than the electrode radius or redox concentration. To our knowledge, no similar relationship has been previously established using electrodes of macroscopic dimensions. The solid line fitted to the data in Figure 7 is described by the parabolic expression, iB - i0 ) 0.0675i02. However, the accuracy and precision of our voltammetric measurements limit the useful working range of this empirical correlation to io values greater than ∼1 µA. Physically, the observed parabolic dependence of iB - i0 on i0 suggests that the magnetic-fieldinduced enhancement in the voltammetric limiting current is not proportional to the magnetic force, Fmag, exerted on the charge carrying ions, i.e., AP•-. This conclusion follows from the Lorentz equation (eq 1), which indicates that Fmag ∝ i0. However, because the observed parabolic dependence of the
5918 J. Phys. Chem., Vol. 100, No. 14, 1996 CHART 1: Direction of Magnetic Force (Indicated by Arrows) on Electrogenerated Ions at a Microdisk Electrodea
a Top: magnetic field parallel to the electrode surface. Bottom: magnetic field perpendicular to surface. Lines originating on the electrode surface represent the diffusion-migration paths of electrochemically generated anions. By convention, positiVe current flows into the surface for an electrochemical reduction.
current enhancement on i0 can be rewritten as (iB - i0)/i0 ) 0.0675i0, it follows that both Fmag and (iB - i0)/i0 are proportional to i0. Thus, (iB - i0)/i0 must also be proportional to Fmag. This latter relationship is used in the next section to determine the dependence of Fmag on the angular orientation. A caveat in the above reasoning is that the transport of AP•away from the microdisk electrode is radially divergent. Thus, both i and Fmag are position-dependent quantities that vary throughout the depletion layer. Clearly, the direction and magnitude of the local magnetic force acting on the ions contained within a small solution volume element will depend on the flux of ions within that volume element. For instance, as shown below, because of the symmetrical radial flux of ions away from a microdisk, it is possible for the net magnetic force summed over all current carrying ions to be zero, although the force on individual ions remains finite. This situation is not encountered using macroscopic planar electrodes, where the current and ion fluxes may be assumed to be essentially uniform and independent of spatial position in a steady-state voltammetric measurement. However, several interesting phenomena arise due to the spatial dependency of the ion flux near a microdisk electrode. As detailed in the following section, these effects are readily explored by examining the dependence of iB - i0 on the orientation of the magnetic field relative to the electrode surface. III. Angular Dependence of the Magnetic Field Effect. The effect of varying the orientation of the external magnetic field, relative to the electrode, on the magnetic force exerted on ions within the depletion layer is qualitatively depicted in Chart 1. The top drawing in Chart 1 shows a situation in which the magnetic field is oriented parallel to the disk surface, corresponding to θ ) 90°. To determine the direction of the magnetic force at any point in solution, we recall that electrogenerated AP•- is transported away from the microdisk surface in a quasi-radial pattern and that the direction of positive current opposes that of the anion flux (i.e., positive current is directed inward toward the surface for this reaction). The direction of the magnetic force within any volume element in the depletion layer is then determined from eq 1. For B parallel to the electrode surface, Fmag points from left to right for essentially all regions of the solution layer adjacent to the electrode. Thus, the magnetic force tends to accelerate essentially all product ions from left to right across the electrode surface. (This
Ragsdale et al. description is not exactly correct; upward- and downwarddirected forces occur near the left- and right-hand edges of the electrode, respectively, as shown in the schematic drawing.) In addition, because of the radial divergence of the flux of AP•-, the current density at any point in solution is approximately inversely proportional to the distance from the electrode at that point, eq 4. Thus, Fmag for any solution element is also expected to decay rapidly as one moves from the electrode surface into the bulk solution. Consequently, the magnetic force acting on the charge-carrying ions is localized to the depletion layer region adjacent to the microdisk surface. Arbitrarily defining the depletion layer thickness, δ, as being equal to the distance away from the surface where the current density falls to 10% of its value at the surface, we calculate δ ≈ 250 µm for a 25-µmradius microdisk. Thus, assuming a hemispherical depletion layer geometry, the magnetic field effect is localized to a solution element of ∼30 nL. In analogous fashion, when B is oriented perpendicular to the electrode surface (bottom of Chart 1, corresponding to θ ) 180°), AP•- will again be accelerated in a direction parallel to the surface. However, the magnitude and direction of the force for this orientation vary as a function of the distance from the electrode and the position across the electrode surface. For instance, from electrostatics, the current flowing into an equipotential surface (e.g., the microdisk surface) must be directed orthogonal to the surface. Thus, B will be parallel to the flux of AP•- in the solution element immediately adjacent to the surface. Consequently, the magnetic force will vanish in this region. For the same reason, Fmag is zero for solution elements situated directly on the flux line defined by the center axis of the electrode. In addition, and as previously discussed, the current density and magnetic force rapidly decay as one moves toward the bulk of the solution. On the other hand, Fmag will have a finite value for positions off of the center axis (i.e., toward the edge of the electrode) and at intermediate distances from the surface. However, because of the symmetrical radial diffusion pattern established at a microdisk, the direction of the force rotates as one moves in a circular direction at a fixed distance from the centerline axis. Consequently, the magnetic field will induce a net rotational motion (about the center of the electrode) of electrogenerated AP•- as this species diffuses radially away from the electrode surface. Because Fmag is zero at the center of the electrode, as well as directly on the electrode surface and at distances far from the electrode, only the ions within a “doughnut” shaped solution volume directly above the electrode will be accelerated by the magnetic field. Within this doughnut-shaped volume, the net effect of the field will induce a rotational or cyclotron-like convective flow. It is important to note here that this motion is superimposed on the normal diffusion/migration of product ions away from the electrode. Thus, although the magnetic field will induce a net rotational motion of the fluid above the electrode surface, individual ions will not remain indefinitely in a circular path. Instead, radial diffusion and migration, coupled with the rotational convective flow, must result in electrogenerated AP•- being transported (on average) along a helical path that spirals outward away from the electrode surface. The above concepts were explored by measuring the voltammetric limiting current as a function of the angular orientation using the electrode geometry depicted in Figure 2b. Rotation of the “bent” microdisk electrode allows the angle θ to be varied between 0 and 360° (see Experimental Section). Parts a and b of Figure 8 show the dependence of voltammetric currents on θ for different field strengths and electrode radii, respectively. The data were plotted as (iB - i0)/i0 vs θ, where (iB - i0)/i0
Magnetic Field Effects in Electrochemistry
Figure 8. Angular dependence of the magnetic field effect (i.e., (iB i0)/i0 vs θ) as a function of (a) field strength and (b) electrode radius. Data in part a correspond to CAP ) 2 M and ro ) 12.5 µm. Data in part b correspond to [AP] ) 1 M and B ) 0.7 T.
represents the enhancement in current normalized to the current in the absence of the field. As shown in the previous section, the quantity (iB - i0)/i0 is linearly related to the magnetic force acting on the depletion layer. This representation of the data also allows the dependence of the magnetic field effect on θ to be more easily visualized on the same plot for different field strengths and electrode radii. All data were recorded for solutions containing either 2 M (Figure 8a) or 1 M (Figure 8b) AP. The results shown in Figure 8 demonstrate that the magneticfield-induced enhancement of the current is largest at either θ ) 90° or 270°. Both of these angles correspond to B parallel to the electrode surface (see top drawing in Chart 1). The occurrence of maxima in (iB - i0)/i0 at θ ) 90 and 270° is clearly independent of B or the electrode radius. At angles intermediate of θ ) 90 and 270°, the enhancement in the current rapidly decays. In some data sets, negative values of (iB i0)/i0 are observed at angles near θ ) 0 and 180°, corresponding to B directed normal to the electrode surface (bottom drawing of Chart 1). A negative value of (iB - i0)/i0 indicates that application of the magnetic field results in a decrease in the voltammetric limiting current. A surprisingly large asymmetry in the angular dependence is apparent in the results shown in Figure 8. Specifically, values of (iB - i0)/i0 decrease very rapidly at angles slightly off of 90°, producing a sharper maxima in (iB - i0)/i0 at 90° than at 270°. This result is unexpected since the magnitude of the magnetic force acting on the electrogenerated ions within the depletion layer is expected to be identical for any two values of θ that differ by exactly 180°. Of course, the direction of Fmag will depend on θ, i.e., Fmag is directed upward for 0 < θ < 180° and downward for 180 < θ < 360°. However, the voltammetric current should be the same for these two orientations since the magnetic force magnitude and resulting convective flow pattern should be identical in both cases. Clearly, the apparent asymmetry in Figure 8 must arise from forces other than Fmag. Figure 9 demonstrates that the asymmetry in the angular dependence is also a strong function of the concentration of the redox species. For AP concentrations greater than ∼5 M, (iB - i0)/i0 is positive for 0 < θ < 180° and negative for 180
J. Phys. Chem., Vol. 100, No. 14, 1996 5919
Figure 9. Dependence of (iB - i0)/i0 on θ as a function of AP concentration. Data correspond to steady-state voltammetric limiting currents measured at a 25-µm-radius Pt microdisk electrode in CH3CN/0.2 M [n-Bu4N]PF6 solutions, B ) 0.7 T. The solid horizontal line is where (iB - i0)/i0 ) 0.
Figure 10. Schematic drawing depicting the rotation of a microdisk electrode in the earth’s gravitational field, g. The electrode is inserted horizontally into the cell and rotated about the angle φ defined by the electrode surface normal, nˆ , and g; φ ) 0° corresponds to the electrode facing downward.
< θ < 360°. At [AP] ) 5 M, positive values of (iB - i0)/i0 are observed at values of θ near 270°; however, the maxima in (iB - i0)/i0 are more sharply peak shaped at 270° than at 90°. This particular feature is reversed at lower concentrations (e.g., [AP] ) 1 M), where the enhancement in current is more broadly distributed around θ ) 270° than at θ ) 90°. At intermediate redox concentrations ([AP] ) 3.5 M), the plot of (iB - i0)/i0 vs θ has a more symmetrical shape. The concentration-dependent asymmetry in the angular dependence of (iB - i0)/i0 can be qualitatively understood by considering the combined effects of the applied magnetic field and the earth’s gravitational field.30 In a recent report, we demonstrated that natural convection occurs at microdisk electrodes (6.4-25-µm radius) when the electrochemical reaction yields a product species that changes the solution density near the electrode surface. For instance, natural convection accounts for ∼15% of the total steady-state limiting current at a 25-µm-radius Pt disk for the oxidation of 0.5 M Fe(CN)64in aqueous solutions.22 The influence of natural convection, in the absence of a magnetic field, on the reduction of AP was demonstrated in the current investigation using a microelectrode bent at 90° and inserted horizontally into the electrochemical cell, as depicted in Figure 10. Rotation of this electrode allows the angle φ between the gravitational field, g, and the electrode surface normal, nˆ to be varied over 360°. The convention used in our
5920 J. Phys. Chem., Vol. 100, No. 14, 1996
Ragsdale et al. CHART 2: Qualitative Representation of Bouyancy Forces Acting on the Depletion Layer at a Microdisk Electrode during the Steady-State Reduction of APa
Figure 11. Dependence of normalized current enhancement, (ig - i0′)/ i0′, as a function of the angle φ. Data correspond to the steady-state voltammetric reduction of AP at a 25-µm-radius Pt disk in CH3CN solutions containing 0.2 M [n-Bu4N]PF6. i0′ corresponds to the voltammetric current measured at φ ) 0° (i.e., the surface normal is pointing in the same direction as g). The concentration of AP corresponding to each data set is indicated on the figure.
investigations is that φ ) 0° when the disk is facing downward (i.e., nˆ and g in the same direction). Voltammetric currents at a 25-µm-radius Pt disk were recorded for 0 < φ < 360° in CH3CN/0.2 M [n-Bu4N]PF6 solutions containing between 2 and 8 M AP. The results are shown in Figure 11 as plots of (ig - i0′)/i0′, where ig represents the limiting current at an arbitrary angle φ and io' represents the current at φ ) 0°. A positive value of (ig - i0′)/i0′ indicates that the current measured at the angle φ is greater than at φ ) 0°; conversely, a negative value indicates that the current is smaller than at φ ) 0°. By definition, and as indicated in Figure 11, (ig - i0′)/i0′ is equal to zero at φ ) 0°. A non-zero value of (ig - i0′)/i0′ indicates that natural convection makes a significant contribution to the flux of AP. Inspection of Figure 11 reveals that natural convection accounts for up to ∼15% of the total flux of the reactant for the reduction of AP at a 25µm-radius disk. The data plotted in Figure 11 suggest that natural convective transport has a complex dependence on the redox concentration as well as φ. For [AP] ) 5 and 8 M, maxima in the limiting current occur when the electrode is oriented in the vertical position (i.e., φ ) 90 and 270°). The current is also larger at φ ) 180° (electrode directed upward) than at φ ) 0°. The observed behavior is consistent with the hypothesis that the reduction of AP results in a solution near the electrode surface that is less dense than the bulk solution. When the electrode is oriented at 180°, the less dense depletion layer will tend to rise, resulting in the convective flow of fresh solution inward along the surface of the glass sheath surrounding the microdisk, Chart 2. Convective flow will increase mass transport of AP to the electrode, resulting in an increase in the limiting current. Conversely, at φ ) 0°, the convective upward flow of the less dense depletion layer will be inhibited by the solid electrode surface, resulting in little, if any, increase in the mass-transport rate. Completely analogous behavior has been observed for the oxidation of Fe(CN)64-.22 At AP concentrations below ∼3.5 M, a different dependence of limiting current on φ is observed. For instance, at [AP] ) 2 M, the current at φ ) 0° is greater than that at φ ) 180°, indicating that the density of the depletion layer is greater than that of the bulk solution. Thus, when the electrode is oriented downward (φ ) 0°), the depletion layer sinks under the influence of the gravitational force, resulting in an increase in limiting
a The lengths of the arrows indicate the relative magnitudes of the force at different AP concentrations.
current. Conversely, at φ ) 180°, the electrode surface inhibits fluid convection. The above results suggest that there is an inversion of the relative densities of the depletion layer and bulk solution as the redox concentration increases. Specifically, the data are consistent with Fbulk > Fsurf at high redox concentrations and Fbulk < Fsurf at low redox concentrations (where Fbulk and Fsurf represent the bulk and near-surface fluid densities). The results indicate the existence of an intermediate AP concentration where Fbulk ) Fsurf. Indeed, Figure 11 shows that the limiting current is essentially independent of φ in solutions containing 3.5 M AP. Thus, at this particular concentration, the gravitational force acting on the depletion layer is essentially zero, and the natural convection component of the current is correspondingly very small. Chart 2 summarizes the relative magnitude and direction of the buoyancy forces acting on the depletion layer at low, intermediate, and high AP concentrations. The asymmetry observed in the dependence of limiting currents on B, Figure 9, can be understood by assuming that the enhancement in current is proportional to the absolute value of the total force acting on the depletion layer
(iB - i0)/i0 ∝ |FT|
(6)
where FT comprises both the magnetic and gravitational forces.
FT ) Fmag + Fg
(7)
The absolute value of FT is used in eq 6 since the enhancement in current that results from convection should be independent of the flow direction (e.g., (iB - i0)/i0 must be identical regardless of whether FT points upwardly or downward). In measuring (iB - i0)/i0 as a function of θ, Figure 9, the gravitational force, Fg, remains constant, since φ ) 90° for all values of θ. As a zero-order approximation, we assume that Fg is proportional to the measured enhancement in current measured at φ ) 90° in the absence of a magnetic field, i.e.,
Fg ∝ (ig - i0′)/i0′
(8)
The magnetic force component of the total force can be written as
Fmag ) Fmag(θ ) 90°)sin θ
(9)
Magnetic Field Effects in Electrochemistry
J. Phys. Chem., Vol. 100, No. 14, 1996 5921
CHART 3: Qualitative Description of the Total Force, |FT|, Responsible for the Observed Enhancement or Diminishment of the Limiting Current as a Function of Angle θa
Figure 12. Dependence of |FT| on θ as a function of AP concentration computed from Fmag (eq 8) and Fg (eq 11) (see text). The dashed horizontal line corresponds to |FT| ) |Fg|, measured relative to the experimental reference frame. a Part a corresponds to a high [AP], where the net bouyancy force, Fg, is directed upward and is larger than the magnetic force, Fmag. Part b corresponds to the intermediate [AP], where Fg is directed upward and smaller than Fmag. Part c corresponds to low [AP], where Fg is directed downward and is smaller than Fmag.
where Fmag(θ ) 90°) is the magnetic force at θ ) 90°. The quantity Fmag(θ ) 90°) cannot be directly measured because of the interfering effects of the gravitational field, which is always present. However, an estimate of Fmag(θ ) 90°) can be obtained by subtracting the component of the enhancement in current due to the gravitational force, (ig - i0′)/i0′, from the total enhancement measured at θ ) 90°, i.e., [(iB - i0)/i0]90.
Fmag(θ ) 90°) ∝ [(iB - i0)/i0]90 - (ig - i0′)/i0′
(10)
Thus, Fmag (eq 9) at any arbitrary angle θ is obtained by combining eqs 9 and 10.
Fmag ∝ {[(iB - i0)/i0]90 - (ig - i0′)/i0′}sin θ
(11)
Assuming that the proportionality constants in eqs 8 and 11 are nearly equal, the total force, FT, can be estimated from experimental data (Figures 9 and 11). In Chart 3, we have used the above analysis to qualitatively draw the directions and magnitudes of the vectors Fmag and Fg as a function of θ for solutions containing high (8.0 M), intermediate (5 M), and low (0.3 M) AP concentrations. Addition of Fmag and Fg (eq 7) yields the total force acting on the depletion layer. The absolute value of the total force, |FT| ) |Fmag + Fg|, which is the operative parameter in determining the observed enhancement of the current in the presence of both gravitational and magnetic fields, eq 6, is shown in Chart 3. The solid horizontal line in the schematic represents zero absolute force; because gravity cannot be “turned off” during the measurement of the reference current, i0, the gravitational force, |Fg|, must be added to zero total force in order to compare the relative angular dependencies of the |FT| and experimental values of (iB - i0)/i0. Thus, the dashed horizontal lines in Chart 3 show the baselines for |FT| after correcting for |Fg|. For [AP] ) 8.0 M (part a of Chart 3), the absolute value of FT (measured against the experimental reference frame (dashed line)) has a sinusoidal dependence on θ, with a maximum at 90° and minimum at 270°. Based on the assumption that (iB -
i0)/i0 is proportional to |FT|, maximum and minimum values of (iB - i0)/i0 are expected at θ ) 90 and 270°, respectively, in solutions containing 8 M AP. This behavior is clearly observed in the experimental results shown in Figure 9. For [AP] ) 5 M (part b of Chart 3), the absolute value of total force displays maxima at 90 and 270° with the maximum at 90° being slightly more pronounced. For [AP] ) 0.3 M (part c of Chart 3), the maxima are shown at 90 and 270°, but the maximum at 270° is more pronounced. These patterns also closely approximate the experimental results for [AP] ) 5 and 0.3 M shown in Figure 9. Therefore, the asymmetry between the rise around 90 and 270° is well explained by the relative direction and magnitude of Fmag and Fg for all [AP]. The above analysis of FT was performed for solutions containing between 0.3 and 8 M AP, with the results shown in Figure 12. These computed curves should be directly compared with the experimental results shown in Figure 9. Although this analysis is highly approximate, it clearly captures essentially all of the qualitative concentration-dependent asymmetries observed in the dependence of (iB - i0)/i0 on θ. These results also provide a physical explanation of the field-induced diminishment of the limiting current (i.e., negative values of (iB - i0)/i0). When Fmag and Fg are aligned parallel to each other, but in opposite directions, the absolute value of the total force |FT| will be smaller than the magnitude of Fg in the absence of the magnetic field (for example, at θ ) 270° in the 8.0 M AP solution). Thus, the magnetic force reduces convective mass transport of the electrochemical reactant to the electrode surface. Conclusion We have demonstrated that an externally-applied magnetic field can significantly enhance or diminish the voltammetric limiting current at microdisk electrodes having radii g 6 µm. The magnetic field effect results from convective flow, engendered by viscous drag on the electrochemical product ions which are accelerated by the magnetic force. An empirical correlation, iB - i0 ∝ i02, has been established between the magnitude of the current at a microdisk in the absence of the field and the enhancement of the current resulting from the externally-applied magnetic field. This correlation provides a basis for observa-
5922 J. Phys. Chem., Vol. 100, No. 14, 1996 tions of negligible magnetic field effects at very low and very high redox concentrations (i.e., conditions where i0 is small). The magnetic-field-induced diminishment of limiting currents at specific angular orientations and redox concentrations has been shown to result from partial cancellation of the gravitational force by the magnetic force. The ability to enhance or diminish convective mass transport in microscopic domains using an external field may find application in analytical measurements and in electrochemical synthesis of metal and semiconducting deposits under highly controlled conditions. Acknowledgment. This work was supported by the Office of Naval Research (ASSERT). References and Notes (1) Lee, J.; Gao, X.; Hardy, L. D. A.; White, H. S. J. Electrochem. Soc. 1995, 142, L90. (2) (a) Bagard, H. C. R. Acad. Sci. 1896, 122, 77. (b) Bagard, H. J. Phys. 1896, 5, 3, 499. (3) (a) A review of the effects on imposed magnetic fields in electrochemical processes is found in: Fahidy, T. Z. J. Applied Electrochem. 1983, 13, 553. (b) Mohanta, S.; Fahidy, T. Z. Can. J. of Chem. Eng. 1972, 50, 248. (c) Mohanta, S.; Fahidy, T. Z. Electrochim. Acta 1976, 21, 25. (d) Quraishi, M. S.; Fahidy, T. Z. Electrochim. Acta 1980, 25, 591. (e) Gu, Z. H.; Fahidy, T. Z. J. Electrochem. Soc. 1987, 134, 2241. (f) Fahidy, T. Z. Electrochim. Acta 1990, 35, 929. (4) (a) Olivier, A. C. R. Acad. Sci. 1970, 271, 529. (b) Peyroz-Tronel, E.; Olivier, A. J. Chim. Phys. 1980, 77, 427. (c) Olivier, A.; Chopart, J. P.; Douglade, J.; Gabrielli, C. J. Electroanal. Chem. 1987, 217, 443. (d) Olivier, A.; Chopart, J. P.; Douglade, J.; Gabrielli, C.; Tribollet, B. J. Electroanal. Chem. 1987, 227, 275. (e) Aaboubi, O.; Chopart, J. P.; Douglade, J.; Olivier, A.; Gabrielli, C.; Tribollet, B. J. Electrochem. Soc. 1990, 137, 1796. (5) (a) Iwakura, C.; Edamoto, T.; Tamura, H. Denki Kagaku 1984, 52, 596. (b) Iwakura, C.; Edamoto, T.; Tamura, H. Denki Kagaku 1984, 52, 654. (c) Iwakura, C.; Kitayama, M.; Edamoto, T.; Tamura, H. Electrochim. Acta 1985, 30, 747. (6) (a) Gak, E. Z.; Krylov, V. S. Elektrokhim. 1986, 22, 829. (b) Gak, E. Z.; Rokminson, E. K.; Bondarenko, N. F. Elektrokhim. 1975, 11, 529.
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