Uncompensated Resistance. 1. The Effect of Cell Geometry

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Anal. Chem. 2000, 72, 3972-3980

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Uncompensated Resistance. 1. The Effect of Cell Geometry Jan C. Myland and Keith B. Oldham*

Department of Chemistry, Trent University, Peterborough, Ontario, Canada K9J 7B8

The meaning and significance of uncompensated resistance are carefully explained. Many factors influence the uncompensated resistance and several of these are explored in this article, using idealized models of an electrochemical cell. Among the factors whose roles have been elucidated are the shape and size of the cell, the location of the reference electrode, the shape of the working electrode, and the size and position of the counter electrode. Worthwhile compensation is shown to be impossible with microelectrodes. The recent trend in electrochemistry toward studies in solutions of low conductivity or low ionic strength1 has renewed interest in uncompensated resistance, which is a more serious problem when cell solutions contain only low concentrations of electrolytes. Uncompensated resistance can arise from an inherent resistivity in the material of which the working electrode is constructed or in the electrical lead connecting it to the electrochemical instrumentation. In this article, however, we restrict attention to the unwelcome presence of uncompensated resistance arising from an inadequate conductance of the cell solution. In the electrochemical literature, there are very many publications dealing with the interference that uncompensated resistance causes and with techniques for circumventing the difficulty, particularly in the context of electrode kinetics, but we have found rather few articles2-5 that address the geometric factors that determine the magnitudes of resistances in electrochemical cells. The presence of uncompensated resistance U corrupts voltammetric experiments in three ways. First, it makes the determination of the electrode potential E more difficult because an ohmic polarization term UI must be subtracted from the measured potential Eapp that is applied to the cell

E ) Eapp - UI

(1)

Though this correction is easily made, precision suffers unless (1) Ciskowska, M.; Stojek, Z. J. Electroanal. Chem. 1999, 466, 129. (2) Nanis, L.; Kesselman, W. J. Electrochem. Soc. 1971, 118, 454. (3) Newman, J. Electrochemical systems; Prentice Hall: Englewood Cliffs, NJ, 1973; Section 116. (4) Winkler, J.; Hendriksen, P. V.; Bonanos, M.; Mogensen, M. J. Electrochem. Soc. 1998, 145, 1184. (5) Svensson, A. M.; Nisancioglu, K. J. Electrochem. Soc. 1999, 146, 1840.

3972 Analytical Chemistry, Vol. 72, No. 17, September 1, 2000

both the current I and the uncompensated resistance U are accurately known. The second difficulty is more insidious. Most voltammetric experiments apply a programmed potential Eapp(t) (or, less usually, a programmed current) to the cell and record the resulting current I(t) (or potential). Chemical conclusions are then drawn from an analysis of the interrelation between the three measurable variables: I, (Eapp - UI), and t. The analysis relies on a mathematical model of the experiment. However, what is a simple E(t) program in the absence of uncompensated resistance becomes distorted when U cannot be ignored. Mathematical models are extremely difficult to solve algebraically if the timedependent variables E and I are not only interrelated by the events occurring at and close to the electrode but are also constrained by being linearly linked via eq 1. Even simulative modeling encounters difficulties in these circumstances, especially if U is regarded as a parameter or if there is complication from the participation of migratory transport, which is often a further consequence of low ionic strength. The third way in which uncompensated resistance impedes voltammetry, but one that will not be pursued here, is through its interaction with that other bane of the electrochemist’s life, double-layer capacitance. When concentration polarization greatly exceeds ohmic polarization, the latter is of no consequence. Hence, limiting currents are unaffected by uncompensated resistance, enabling some studies6,7 to avoid confronting the resistance problem, even in the total absence of supporting electrolyte. Thus, in some electroanalytical applications of voltammetry, uncompensated resistance may also be legitimately ignored. On the other hand, even small values of U may interfere in measurements of fast rate constants.8,9 Just how small the uncompensated resistance must be before it is negligible depends on the purpose of the experiment. There exist many experimental methods of measuring uncompensated resistance, among which ac methods10 and currentinterruption techniques11 are paramount. Electronic compensation using positive feedback,12,13 sometimes even without explicitly (6) Cooper, J. B.; Bond, A. M.; Oldham, K. B. J. Electroanal. Chem. 1992, 331, 877. (7) Myland, J. C.; Oldham, K. B. Electrochem. Comm. 1999, 1, 467. (8) Andrieux, C. P.; Save´ant J. M. In Investigations of the rates and mechanisms of reactions, 4th ed.; Bernasconi, C. F., Ed.; Wiley: New York, 1986. (9) Petersen, R. A.; Evans, D. H. J. Electroanal. Chem. 1987, 222, 129. (10) McDonald, J. R. Impedance spectroscopy; Wiley: New York, 1987. (11) Newman, J. J. Electrochem. Soc. 1970, 117, 507. (12) He, P.; Faulkner, L. R. Anal. Chem. 1986, 59, 717. 10.1021/ac0001535 CCC: $19.00

© 2000 American Chemical Society Published on Web 07/29/2000

measuring U, is a standard method of mitigating the effects of uncompensated resistance. Some modern instruments incorporate devices for measuring both U and I and applying a compensatory correction to Eapp. It is sometimes possible to carry out a series of experiments, varying the cyclovoltammetric scan rate, for example, and/or the analyte concentration, and include U among the parameters to be evaluated in the data analysis. Another approach14 is to add a well-understood “marker” analyte and use its voltammogram to measure the uncompensated resistance. In the present study, employing both mathematical analysis and finite element simulation, we explore how the geometry of the cell, and of the reference electrode itself, affects uncompensated resistance. It is demonstrated how, for certain simple cell geometries, U may be calculated from the conductivity κ of the solution and the physical dimensions of the cell and the electrodes. We do not imply that such calculations should be used as a substitute for the measurement of uncompensated resistance. They may, however, provide a useful guide for the estimation of U and to aid in the design of cells and experiments. In part 1 of the study, the role of cell geometry is examined, the reference electrode being treated as “ideal”. In part 2, we examine the effect of the finite size of the reference electrode. DEFINITION OF UNCOMPENSATED RESISTANCE There appear to be misconceptions as to the meaning of “uncompensated resistance”. One sometimes reads that “a cell’s uncompensated resistance is the resistance between the working electrode surface and the tip of the reference electrode probe”, which is inaccurate. Even some textbooks fail to define uncompensated resistance adequately. Although the concept of uncompensated resistance arises in the context of potentiostatically controlled three-electrode cells, it is appropriate first to consider two-electrode cells. Figures 1 and 2 show two idealized cell designs. In these, and all subsequent, cell diagrams, solid black represents the electrodes and hatching denotes an insulating cell wall or gas/solution boundary. The upper electrode WE in each of Figures 1 and 2 is the working electrode, while each lower electrode CE, which is to be regarded as perfectly depolarized, fills the twin roles of reference and counter electrodes. Except in the appendixes, we shall invariably use a cylindrical coordinate system with its origin at the center of the working electrode. The z-axis, r ) 0, is oriented toward the center of the counter electrode, increasing in this direction. The radial r-axis is normal to the z-axis, r taking increasingly positive values as one proceeds away from the z-axis. In all the situations considered, there is rotational symmetry about the major axis, r ) 0, relieving us of the necessity to designate a third coordinate. For the sake of uniformity, we adopt this (z,r) cylindrical system in the main text, even when another coordinate system would be mathematically advantageous. The subscripts W, C, and R refer to the working, counter, and reference electrodes. Thus, we use rW to denote the radius of the working electrode, except when WE is spherical, in which case a designates its radius of curvature. Similarly zC denotes the distance separating the centers of WE and CE, while (13) Roe, D. K. In Laboratory techniques in electroanalytical chemistry; Kissinger, P. T., Heineman, W. R., Eds.; Dekker: New York, 1996. (14) Bond, A. M.; Oldham, K. B.; Snook, G. Anal. Chem., in preparation.

Figure 1. Two-electrode cell with planar symmetry, with simple associated circuitry. The locations of four equipotential surfaces are shown by dashed lines. The resistances between adjacent equipotential surfaces all have a value equal to 1/5 of the total cell resistance.

Figure 2. As Figure 1, but for a cell with hemispherical symmetry.

the coordinates (zR, rR) locate the tip of the reference electrode, RE. In Figure 1, a cell of volume πr W2zC holds a solution of uniform composition and conductivity κ. The resistance of an element of the solution, which has a cross-sectional area πr W2 and thickness dz, is dR ) dz/πr W2κ and the total resistance is therefore

R)

∫

z)zC

z)0

dR )

1 πrW2κ

∫

zC

0

dz )

zC πrW2κ

(2)

In Figure 2, the cell has hemispherical symmetry and a volume of 2π(r C3 - a3)/3. The resistance of each hemispherical shell of radius dr is dR ) dr/(2πr 2κ), the total resistance therefore being

R)

∫

r)rC

r)a

dR )

1 2πκ

∫

rC -2

a

r

dr )

rC - a 2πrCaκ

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If the radius a of the hemispherical working electrode is sufficiently small in comparison with the dimensions of the cell, as is always true for hemispherical microelectrodes, R is well approximated by 1/(2πaκ). The flow of current through the resistive solution causes the development of an electrostatic potential φ at all points in the solution. Its magnitude at any point in the solution can be found by an appropriate integration of Ohm’s law. In the case of the cell in Figure 1, one finds the potential at a distance z from the working electrode surface to be given by

φ(z) )

∫

z)z

z)0

dφ ) -Ι

∫

z)z

z)0

dR )

-I πrW2κ

∫

z

0

dz )

-Iz (4) πrW2κ

while for the hemispherical geometry of Figure 2

φ(F) )

∫

F)F

F)a

dφ ) -Ι

∫

F)F

F)a

dR )

-I 2πκ

∫F

F -2

a

dF ) -(F - a)I (5) 2πaFκ

where F ) (r2 + z2)1/2. In performing these integrations, the solution immediately adjacent to each working electrode has been assigned an electrostatic potential of zero. Elsewhere in solution the potential is then negative if I is positive, i.e., if the working electrode is anodic. In each of Figures 1 and 2, examples of four equipotential surfaces are illustrated by dashed lines. Notice that, in accord with eqs 4 and 5, the equipotentials are regularly spaced in Figure 1, whereas they crowd toward the working electrode in Figure 2. The cell shapes in Figures 1 and 2 are unusually simple in that the surfaces of constant potential are parallel planes or concentric hemispheres. More generally, these surfaces obey the following three rules: (i) immediately adjacent to each electrode, they lie parallel to the electrode surface; (ii) they obey the Laplace equation

∇2φ )

∂2φ ∂2φ 1 ∂φ + 2+ )0 r ∂r ∂z2 ∂r

(6)

(iii) they intersect insulating surfaces at right angles. Because both involve solving Laplace’s equation, the problem of determining the steady-state voltammetric current in a two-electrode electrochemical cell when the counter electrode is much larger than, and remote from, the working electrode, is mathematically equivalent to finding the cell resistance. With use of standard notation, the equivalence is

Isteady state nFDcb

)

(a characteristic quantity with dimensions of length) )

1 Rκ (7)

An example is provided by Saito’s expression15 4nFDcbrW for the (15) Saito, Y. Rev. Polarogr. 1968, 15, 177.

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Figure 3. Three-electrode potentiostatically controlled cell equipped with a reference electrode and Luggin capillary.

steady-state current at an inlaid disk electrode of radius rW and Newman’s expression16 1/(4rWκ) for the resistance of such a cell, each corresponding to a characteristic length of 4rW. Equivalence 7 can be helpful in transferring information from one area of electrochemical endeavor to another, and we will use it later in this paper. As pointed out elsewhere17,18 in the context of microelectrodes, it is obvious from eq 7 that the ohmic polarization of a steady-state cell is

RIsteady state ) nFDcb/κ

(8)

and is independent of the shape of the working electrode or its size, provided it is much smaller than the counter electrode. With that background on two-electrode cells, let us now progress to consider the introduction of a third electrode. When introduced into an operational two-electrode cell, an ideal reference electrode RE would have the property of sampling the potential at a point in the vicinity of the working electrode, without perturbing the preexisting potential and current density distributions in any way. A real electrode may fail to live up to that ideal in at least three ways: first, by acting faradaicly, providing either a source or sink of charge; second, by occupying space that was formerly occupied by solution, thereby changing the cell geometry; third, by providing a pathway by which current may travel en route from WE to CE. If the reference electrode is connected, as shown in Figure 3, only to a well-designed potentiostat, then there should be little danger of a faradaic effect, except perhaps transitorily. The space-occupying problem is difficult to surmount completely, but it is minimized by the design shown in Figure 3, in which a thin capillary, with insulating walls and an orifice at its tip, is placed facing the working electrode. Such a device is called (16) Newman, J. J. Electrochem. Soc. 1966, 113, 501. (17) Bruckenstein, S. Anal. Chem. 1987, 59, 2098. (18) Oldham, K. B. J. Electroanal. Chem. 1987, 237, 303.

a “Luggin probe” after the glassblower who first constructed one. Whether it perturbs the preexisting equipotential surfaces or not, the tip of the reference electrode probe will intersect one of the equipotential surfaces established by the current flow between the working and counter electrodes. Let φR be the potential in question. At last we are in a position to define uncompensated resistance. The equation

U ) -φR/I

(9)

provides the definition. In words, we can say that the uncompensated resistance is the resistance between the working electrode and the entire equipotential surface that traverses the tip of the reference electrode.19 In the circumstance diagrammed in Figure 3, ∼80% of the total resistance would be compensated, U/R ) 1/5. In principle, the uncompensated resistance of a cell is a function of time, unless WE is “uniformly accessible”. This temporal variability, which is responsive to the reversibility of the electrode reaction and to the nature of the voltammetric program, arises because the distribution of current density across the face of WE generally changes as the experiment progresses, causing a redistribution of the electric field in the vicinity of the working electrode and a realignment of the equipotential surfaces. Changes with time will not be addressed in part 1 of this study; we shall assume that the current density distribution is that appropriate to a steady state. Severe temporal variability of the uncompensated resistance may arise from quite a different causesa progressively increasing nonuniformity of the conductivity κsif supporting electrolyte is not present in excess, but this circumstance is not addressed here, either. DEPENDENCE ON THE LOCATION OF THE REFERENCE PROBE If the Luggin probe in Figure 3 were of infinitesimal thickness then, by analogy with eq 2, the uncompensated resistance would be

U ) zR/πrW2κ

(10)

In principle, the uncompensated resistance could be reduced indefinitely by bringing such an ideal reference probe close enough to WE. In practice, of course, the difficulty of maintaining a very small, fixed interelectrode separation precludes this. Moreover, the shielding imposed by a reference electrode that is too close brings other problems in its wake. Part 2 of this study will address this space-occupying aspect. For the geometry of Figure 2, with RE’s tip positioned at (zR, rR) in our cylindrical coordinates, the corresponding equation is

U)

(

1 1 2πκ a

1

xzR

2

)

(11)

+ rR2

For the geometries of Figures 1 and 2, and based on eqs 2, 3, 10, and 11, Figure 4 shows the fraction of the total cell resistance (19) Oldham, K. B.; Myland, J. C. Fundamentals of electrochemical science; Academic Press: San Diego, 1993; p 211.

Figure 4. Fraction of the total resistance that is compensated by an ideal reference electrode positioned at various distances from the working electrode. The two curves relate to the geometries shown in Figures 1 and 2, the latter having a WE/CE gap 9 times the radius of WE.

that remains uncompensated as the reference electrode position is varied. Notice how much more crucial is the positioning of RE in the hemispherical case. For example, to compensate half the total resistance, the distance separating WE and RE must be reduced to 1/11 of the WE/CE gap with the hemispherical geometry, but to only 1/2 of WE/CE gap for planar geometry. With small electrodes benefitting from convergent transport, worthwhile compensation is impossible unless a midget reference electrode is positioned extremely close to the working electrode. Because of the symmetry of their geometries, the cells of Figures 1 and 2 have uncompensated resistances that depend solely on the distance separating the working and reference electrodes. Such “uniformly accessible” geometries are rare. With other geometries there may also be a dependence of the electrostatic potential φR on the angular location of the reference electrode with respect to the major axis of the cell. If the diffusioncontrolled steady-state voltammetry of the working electrode has been a subject of prior study, information on the equipotential surfaces may be available by analogy with the steady-state equiconcentration surfaces. Just as with eq 7, so the useful equivalence

csteady state cb

)

(a dimensionless position-dependent quantity) )

-φ (12) IR

holds, because the Laplace equation is obeyed in each circumstance. Here csteady state is the concentration of the electroreactant at some point in the solution during the flow of a steady-state voltammetric limiting current, cb being the bulk concentration during that experiment. φ is the electrostatic potential at the same point during a different experiment in which a current I, which is not necessarily steady-state or diffusion controlled, flows through the cell resistance R. Because the voltammetry of this electrode has been the subject of much theoretical study, we have been able to exploit eq 12 in Analytical Chemistry, Vol. 72, No. 17, September 1, 2000

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Figure 5. Inlaid disk working electrode and three possible positions for the reference electrode.

Figure 7. (a-e) Five examples of sphere-cap working electrodes. The extreme members correspond to a pendant sphere and an inlaid disk; (c) is a hemisphere. (f) Definitions of the dimensions employed in the text.

Figure 6. Fraction of the total resistance that is compensated for each of the RE positions shown in Figure 5.

the case of the inlaid disk working electrode diagrammed in Figure 5. If the cell is “large” and the counter electrode is “large” and “remote” (we shall discuss the significance of these terms in the next section), the total resistance of the cell is known16 to be R ) 1/(4rWκ). Figure 5 shows three possible locations for the tip of the reference electrode: on the major axis of the cell, below the edge of the disk, and in the plane of the disk. We have calculated20 the uncompensated resistance for each of these locations for a reference probe of infinitesimal thickness. These data are presented in Figure 6 for a range of interelectrode spacings. Notice that, for a given distance of approach, the uncompensated resistance is least for the reference electrode positioned directly in front of the working electrode. The reason for this relates to the disparities in current densities in the zone between WE and RE. The current density is least in the center of the disk and greatest along the insulator surface. From that viewpoint, the third position is the worst. However, from other considerations, positioning the Luggin probe through the insulating support has several advantages: there is no space-occupation problem, reproducibility from one experiment to the next is ensured, and fabrication could often be simpler, requiring only the drilling of a fine hole. The placement of REs in this location is common in solid-electrolyte cells.4,5 The data in Figure 6, and the equations20 on which they are based, derive from solving Laplace’s equation for the geometries (20) Oldham, K. B.; Myland, J. C. Supporting Information, part A.

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in question and take no cognizance of the temporal variability of U discussed earlier. When applied to voltammetry, they are, in fact, exact only for an experiment at constant Eapp as t f ∞ with a reversible electrode reaction. Nevertheless, we believe them to be fully adequate for any voltammetric experiment in the normal repertoire, with any experimentally realistic spacing between RE and WE. A similar caveat applies to most of the data presented later in this article. Mercury is a popular electrode material, and because of the very large surface Gibbs energy of its interfaces with other liquids, it usually forms electrodes with shapes that are sphere caps. A variety of sphere-cap electrodes is depicted in Figure 7; the first four of those illustrated have the same radius a of curvature. Note that the pendant sphere (a), the hemisphere (c), and the inlaid disk (e) are all special cases of the sphere cap. The pendant sphere and the hemisphere correspond respectively to the special values h ) 2a and h ) a of the height h of the cap. The disk is a limiting case in which h approaches zero and a approaches infinity but in such a way that (2ha - h2)1/2, the radius rW of the basal disk, remains constant. When the cell is large and the counter electrode is remote, the total cell resistance for any sphere cap WE, found by solving21 Laplace’s equation, is

∫ (1 -

R ) 1/[4πx2ha - h2κ

∞

0

tanh{πµ} tanh{2µ arccos xh/2a}) dµ] (13) As expected, this reduces to 1/(2πaκ) when h ) a and to 1/(4rWκ) when h ) 0, corresponding to the hemisphere and disk, respectively. The integral is infinite for the pendant sphere, the h ) 2a case, but its product with (2ha - h2)1/2 remains finite and it leads21 to a total resistance R of 1/(4π ln{2}aκ), in agreement with the (21) Oldham, K. B.; Myland, J. C. Supporting Information, part B.

Figure 8. Fraction of the total resistance that is compensated for four of the sphere cap WEs shown in Figure 7. Circles represent data derived from an analytic expression for the pendant sphere, while triangles are based on a published23,24 approximation.

findings of Bobbert et al.22 and differing by only 4% from the approximate formula ln{2}/2πaκ derived by Taylor and Barradas.23 Based on eq B5,21 the curves in Figure 8 show the ways in which the uncompensated resistance varies for three of the sphere caps portrayed in Figure 7. For the case of the pendant sphere, we have deduced a simple analytic expression for the potential on the major cell axis and thereby for the uncompensated resistance, which is reported elsewhere21 and as the circular points in Figure 8. In that figure, we also compare our exact results for the pendant sphere with a corrected23 formula based on the approximate method of Britz and Bauer.24 Notice that their method underestimates the uncompensated resistance somewhat. Figure 8 relates to a reference electrode located, as shown in Figure 7f, on the major axis of the cell, facing WE. Though we do not report data to validate the observation, an interesting inversion occurs as one proceeds through the family of spherecap electrodes. For early family members, and especially the pendant sphere, the “best” position for the reference electrode, i.e., the one that compensates the largest fraction of the total resistance for a given distance of RE’s tip from the surface of WE, is the one we have studied, on the r ) 0 axis. For the hemisphere, the location is immaterial. For family members with h < a, and especially for the disk electrode which we treated earlier, the best position for RE is a “sideways” approach. We conclude that, especially for electrodes that enjoy convergent diffusion, the uncompensated resistance declines dramatically as the Luggin probe approaches WE; however, the angle of approach may be important. As throughout part 1 of this study, this conclusion holds for an ideal reference electrode, one that does not perturb the distribution of electrostatic potential or current. Reference electrodes of finite size cause “shielding”, to be discussed in part 2, and this may impose a limit to the benefit of close RE/WE spacing. DEPENDENCE ON THE COUNTER ELECTRODE One purpose of the third electrode is to make the characteristics of the counter electrode irrelevant to the performance of (22) Bobbert, P. A.; Wind, M. M.; Vlieger, J. Physica 1987, 141A, 58. (23) Taylor, D. F.; Barradas, R. G. J. Electroanal. Chem. 1969, 23, 166. (24) Britz, D.; Bauer, H. H. J. Electroanal. Chem. 1968, 18, 1.

Figure 9. Geometry employed in investigating the effect of the position of a large counter electrode on the uncompensated resistance. Three alternative positions of the reference electrode are shown.

the electrochemical cell. Nevertheless, both the location and the size of the counter electrode do affect the uncompensated resistance to some extent. Naturally, CE’s size and location will have a large effect on the total cell resistance R, but the important question here is the extent to which the uncompensated resistance U is affected. We characterize CE as “remote” when moving it still further away would cause no significant change in the uncompensated resistance. To investigate how severely the proximity of the counter electrode influences the uncompensated resistance, it is convenient to study the cell configuration illustrated in Figure 9. Here the counter electrode is an infinite sheet, positioned a distance zC below the center of the spherical working electrode. The support of WE is an insulated lead, assumed fine enough not to perturb the potential distribution around the electrode. We have addressed25 the difficult mathematics of finding R when zC is finite; this reduces to R ) 1/(4πaκ) if the counter electrode is remote. We have explored the uncompensated resistance for three orientations, illustrated in Figure 9, of a reference electrode of infinitesimal thickness, for various relative positions of the three electrodes. There are two interelectrode gaps to consider: that separating CE from WE and that separating RE from WE. These two gaps correspond respectively to the abscissas and ordinates of the three contour plots in Figure 10. The contours themselves have values equal to U/UremoteCE, where the normalizing denominator is given by eq 14 below. For example, all points on the contours labeled 1.10 or 0.90 correspond to a 10% departure of the uncompensated resistance due to the counter electrode not being remote. As expected, a finite value of zC inflates the uncompensated resistance for position RE1 but leads to a lower U in the other two positions. To have the RE/WE gap greater than the CE/WE gap is physically impossible for the RE1 position, and because they would be unlikely in practice, such configurations are represented by dotted lines for RE2 or RE3 in Figure (25) Oldham, K. B.; Myland, J. C. Supporting Information, part C.

Analytical Chemistry, Vol. 72, No. 17, September 1, 2000

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Figure 11. Comparable sizing of working and counter electrodes. A convenient construction is to insert a plane midway between them.

10. Each of these graphs emphasizes that the RE/WE gap has only a minor influence on the way that the counter electrode’s position affects U. Note the advantage gained by positioning the reference electrode to the side of, or behind, the working electrode when the counter electrode is not remote. Not only is the uncompensated resistance itself less than when RE is in front of WE, but its dependence on CE’s position is also less. Our calculations show that, as zC approaches infinity, the location of the reference electrode becomes immaterial. This implies that the potential distribution around the working electrode has spherical symmetry, even though the counter electrode is an infinite plane. In fact, the potential distribution around a spherical WE is spherically symmetrical whatever the geometry of the counter electrode, provided it is remote. The uncompensated resistance is then just half of the corresponding hemisphere, as given in eq 11. In our cylindrical (z, r) coordinates this means

UremoteCE ) Figure 10. Contour plots relating to the three cells diagrammed in Figure 9. The number associated with each curve is the uncompensated resistance U divided by the value that U would adopt were the counter electrode remote. The full lines correspond to the reference electrode being closer to the working electrode than is the counter electrode. 3978

Analytical Chemistry, Vol. 72, No. 17, September 1, 2000

(

1 1 4πκ a

1

x

2

zR + rR2

)

(14)

The voltammetric advantage of having CE remote, or effectively remote, is that the current density on the spherical WE is then uniform.

On the basis of Figure 10, and by quantifying the word “effectively” to mean “within 5%”, we can conclude that, for the purposes of characterizing the uncompensated resistance, a large CE is effectively remote when its distance from the working electrode is at least 5 times the radius of WE, for any likely configuration of the three electrodes. When the counter electrode does not meet this criterion, its position must be taken into account in discussions of uncompensated resistance. But what about the size of the counter electrode? Experiments do not always employ large electrodes; frequently they are no larger than the working electrode. Therefore, direct attention to the geometry of Figure 11, in which WE and CE are spheres of equal size, their centers separated by zC. By a dotted line, the figure shows a plane positioned midway between the two electrodes. Symmetry dictates that this plane be an equipotential surface at a potential of -RI/2. Compare the upper half of this cell with the cell depicted in Figure 9. It is evident that, if the zC length in Figure 11 is twice that in Figure 9, then the electrical conditions in the two configurations will exactly match. Uncompensated resistances will duplicate those in Figure 10, though the total cell resistance R in Figure 11 will be twice that in Figure 9. We conclude that when the CE is of a size comparable to that of WE, it is effectively “remote” when the interelectrode distance is at least 10 times the radius of WE. DEPENDENCE ON CELL SIZE It was taken for granted in discussing the cells illustrated in Figures 5, 7, 9, and 11 that the cell walls were so far from the r ) 0 symmetry axis that they could be regarded as being at infinity. But how far, in practice, must they be to be “effectively at infinity”? Of course, this has a bearing on the minimum cell volume needed to match a model based on an infinite cell volume. To investigate this problem, we need not explicitly address the uncompensated resistance. It will suffice to determine how wide the vessel needs to be before the total cell resistance is effectively that of an infinite vessel. If R is independent of cell size, so will U be. For our purpose we will consider the cell shown in Figure 12, in which a cylindrical vessel of radius rV houses disk-shaped working and counter electrodes, both of a much smaller radius rW. As usual, zC denotes the distance separating the two electrodes, and in accordance with the conclusions of the last section, we set zC ) 10rW. Figure 13 is the result of applying the procedure discussed elsewhere;26 it shows that the resistance is within 5% of the value for an infinitely large vessel, for radii that exceed the electrode radius 5-fold. Figure 12 was drawn with rV ) 5rW and zC ) 10rW and thus represents the minimum dimensions of an acceptable vessel. A related question is the following: if a spherical working electrode is immersed in solution, how deeply submerged must it be before it behaves as if it is effectively in an infinite reservoir of solution? Fortunately, there is a simple empirical equation (valid to better than 0.15%) that answers this question, though it was published27 in a different context. If zV is the finite distance of the center of a spherical working electrode of radius a below the gas/ solution interface, then the cell resistance is greater by the factor (26) Oldham, K. B.; Myland, J. C. Supporting Information, part D. (27) Alfred, L. C. R.; Oldham, K. B. J. Electroanal. Chem. 1995, 396, 257.

Figure 12. Cylindrical vessel with disk shaped working and counter electrodes in opposite faces. The effect of varying the rV dimension has been studied.

Figure 13. Normalized resistance of the cell in Figure 12 plotted as a function of the radius ratio rV/rW. The dotted line is a construction showing that the resistance is 5% greater when rV ∼ 5rW than when rV ) ∞. Notice that R ) zC/πrW2κ when rV ) rW.

2 4zV -2.9zV + exp a + 2zV a

{

}

(15)

than for a very deeply submerged electrode. This formula applies even if zV ) a, corresponding to a barely covered sphere. It shows that when zV ) 10a, the factor is 1.05. We conclude that an electrode must be covered by at least 10 times its own radius of solution, before it is immersed in an effectively infinite volume. SUMMARY We have investigated the way that the uncompensated cell resistance depends on a variety of geometric factors. In all cases, the reference electrode has been “ideal”, i.e., occupying negligible Analytical Chemistry, Vol. 72, No. 17, September 1, 2000

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space. The effects of nonideality will be addressed in part 2 of this series. The conclusions of the present article and its Supporting Information may be summarized as follows: (1) For working electrodes benefitting from convergent transport, worthwhile compensation is impossible unless a midget RE is positioned extremely close to WE. (2) Uncompensated resistance declines dramatically as RE approaches WE; however, the angle of approach may be important. (3) A large CE is effectively remote when its distance from the working electrode is at least 5 times the radius of WE. (4) When the CE is of a size comparable to that of WE, it is effectively remote when the interelectrode distance is at least 10 times the radius of WE. (5) The resistance is effectively that of a cell in an infinitely large vessel, if the vessel’s radius exceeds that of the electrodes 5-fold. (6) An electrode must be covered by at least 10 times its own radius of solution, before it is immersed in an effectively infinite volume.

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Throughout, we have employed “effectively” to mean “without causing a change that exceeds 5%”. Because a variety of idealized cell geometries have been used, none of which correspond precisely with typical voltammetric cells, our conclusions must be regarded as semiquantitative only. ACKNOWLEDGMENT Financial support from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. SUPPORTING INFORMATION AVAILABLE (A) electrostatic potentials in the vicinity of a disk electrode; (B) electrostatic potentials in the vicinity of a sphere-cap electrode; (C) electrostatic potentials in the region between a sphere and a plane; (D) a solution for the geometry of Figure 12. This material is available free of charge via the Internet at http://pubs.acs.org. Received for review February 7, 2000. Accepted June 5, 2000. AC0001535