How Much Supporting Electrolyte Is Required to ... - ACS Publications

J. Phys. Chem. C 2009, 113, 11157–11171

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How Much Supporting Electrolyte Is Required to Make a Cyclic Voltammetry Experiment Quantitatively “Diffusional”? A Theoretical and Experimental Investigation Edmund J. F. Dickinson, Juan G. Limon-Petersen, Neil V. Rees, and Richard G. Compton* Department of Chemistry, Physical and Theoretical Chemistry Laboratory, Oxford UniVersity, South Parks Road, Oxford OX1 3QZ, United Kingdom ReceiVed: February 22, 2009; ReVised Manuscript ReceiVed: May 07, 2009

Theory is presented for cyclic voltammetry at a hemispherical electrode under conditions where the electric field is nonzero and migration is significant to mass transport. The nonlinear set of differential equations formed by combining the Nernst-Planck equation and the Poisson equation are solved numerically, subject to a zero-field approximation at the electrode surface. The effects on the observed voltammetry of the electrode size, scan rate, diffusion coefficient of electroactive and supporting species, and quantity of supporting electrolyte are noted. Comparison is drawn with experimental voltammetry for the aqueous system [Ru(NH3)6]3+/2+ at a Pt macroelectrode with varying levels of supporting electrolyte KCl. The approximations concerned are shown to be applicable where the ratio of supporting (background) electrolyte to bulk concentration of electroactive species (support ratio) exceeds 30, and general advice is given concerning the quantity of supporting electrolyte required for quantitatively diffusion-only behavior in macroelectrode cyclic voltammetry. In particular, support ratios are generally required to be greater than 100 and certainly substantially greater than 26, as has been suggested for the steady-state case. 1. Introduction Conventionally, electroanalytical experiments such as cyclic voltammetry are performed in solutions containing a large excess of inert supporting electrolyte; KCl in water and tetrabutylammonium perchlorate in acetonitrile are typical examples. The purpose of this supporting electrolyte is to ensure that the ionic strength of the solution is high and hence that the electric field is homogeneous and near-zero and is not perturbed by the oxidation or reduction of the analyte concerned. Under suitable experimental time scales where convection can be neglected, the theory of cyclic voltammetry is therefore reduced to a diffusion problem, and the interpretation of experimental data is greatly facilitated. The use of a large amount of supporting electrolyte may introduce other problems, however; if either ion adsorbs specifically to the electrode, it will alter capacitive (non-Faradaic) currents, and the introduction of large quantities of salt is often inappropriate for analytical measurements concerning biological compounds. Further, in some otherwise appealing solvents, the dissolution of sufficient electrolyte is impossible. Consequently, it is of considerable importance to establish theoretically, and to demonstrate in practice, the minimum quantity of supporting electrolyte required for voltammetry to be indistinguishable from a diffusion-only limit and additionally to be able to model and interpret voltammetry recorded outside of the fully supported regime. The mathematical description of migration and the role of supporting electrolyte was discussed in detail by Oldham and Zoski in the context of electrochemical mass transport,1 although the suggested minimum support ratio of 26 derived at steady state in their work will be shown below to be a substantial underestimate under transient conditions. Ciszkowska et al. reviewed developments in weakly supported * To whom correspondence should be addressed. Fax: +44 (0) 1865 275410. Tel: +44 (0) 1865 275413. E-mail: richard.compton@ chem.ox.ac.uk.

microelectrode voltammetry up to 1999.2 Other past theoretical work includes extensive and insightful studies by Stojek and co-workers of the effect in such a case of diffusion coefficients for reactant and product species at microelectrodes,3,4 and Amatore et al. recently published a theoretical discussion of microscopic ohmic drop effects.5 To our knowledge, the theoretical treatment of weakly supported cyclic voltammetry at macroelectrodes has been neglected due to the implicit practical problems of high ohmic drop associated with any experimental exploitation; the only examples we note in the literature involve self-support for highly charged species at high concentrations, where convective effects cannot be ruled out.6,7 Consequently, the exact quantity of supporting electrolyte required to achieve “diffusion-only” mass transport, to within a certain tolerance, has not yet been well established for cyclic voltammetry. A variety of past theoretical developments for chronoamperometry and voltammetry in low supporting electrolyte conditions have been achieved by employing the approximation of electroneutrality throughout solution.8-12 The recent development by Streeter et al. of a complete numerical solution system for chronoamperometry at a hemispherical electrode using the combined Nernst-Planck-Poisson (NPP) equation system,13 without recourse to approximations of electroneutrality, demonstrated that for suitably large (greater than nanoscale) electrodes, the approximation of a negligibly small double layer, with zero electric field at the surface, is accurate and optimal in terms of simulation efficiency, without obliging electroneutrality in solution. This negligible double layer NPP model has already been successfully and quantitatively applied to experimental chronoamperometric data.14 This paper represents the extension of this latter NPP model from chronoamperometry to cyclic voltammetry, in which the applied potential is not stepped but rather varies linearly with time. Theoretical results are presented, demonstrating the dependence of the voltammetric waveshape on the electrolytic

10.1021/jp901628h CCC: $40.75  2009 American Chemical Society Published on Web 06/02/2009

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support ratio, electrode size, inert electrolyte diffusion coefficient, and scan rate. Comparison is then drawn with experimental cyclic voltammetry for the reduction of the hexaammineruthenium(III) ion, [Ru(NH 3) 6]3+, at a Pt macroelectrode in aqueous solution, under varying degrees of support by KCl. The range of applicability of the model is discussed, and directions for future study are noted. 2. Theoretical Model 2.1. Establishment of the Model. We consider a solution containing an electroactive species A capable of undergoing electron transfer to form species B

A a B + ne-

(

(

∂Ci ∂2Ci ziF ∂Ci ∂φ ∂2φ 2 ∂Ci + + ) Di + + C i ∂t r ∂r RT ∂r ∂r ∂r2 ∂r2 Ci

)

z iF Ji ) -Di ∇Ci + C ∇φ RT i

∂Ci ) -∇ · Ji ∂t

∂2φ 2 ∂φ F + )r ∂r s0 ∂r2

(3)

)

z iF ∂Ci ) Di ∇2Ci + (C ∇2φ + ∇Ci∇φ) ∂t RT i

∑ ziCi i

(8)

i

ci )

Ci C*A

(9)

θ)

Fφ RT

(10)

(

(6)

(

∂ci ∂2ci ∂ci ∂θ ∂2θ 2 ∂θ 2 ∂ci ) Di 2 + + zi + ci 2 + ci ∂t r ∂r ∂r ∂r r ∂r ∂r ∂r

))

(11)

F2c*A ∂2θ 2 ∂θ + ) r ∂r RTs0 ∂r2

∑ zici

(12)

i

Additionally, the distance coordinate is normalized to the electrode radius, re, such that the electrode boundary occurs at R)1

R)

(5)

where s is the dielectric constant of the solvent medium, 0 is the permittivity of free space, and F is the local charge density, achieved by summing the charges of all species present

F)F

∑ ziCi

where C*A is the bulk concentration of species A. Hence

(4)

At any point in solution, however, the potential must additionally satisfy the Poisson equation

F s0

(7)

The problem of simulating cyclic voltammetry under conditions where migration is nonzero is therefore approached numerically by solving simultaneously this set of coupled partial differential equations, the Nernst-Planck-Poisson (NPP) equations, subject to boundary conditions which appropriately describe the experimental system. 2.2. Dimensionless Coordinates. Various dimensionless coordinates are introduced to simplify the equations above and to reduce the number of independent variables in the system. First, concentration and potential are normalized as follows

hence

∇2φ ) -

))

(2)

where for species i, Ji is the flux vector, Di is the diffusion coefficient, Ci is concentration, zi is the species charge, φ is the potential, F is the Faraday constant, R is the gas constant, and T is temperature. From Fick’s second Law, the space-time evolution of the concentration of i is given

(

2 ∂φ r ∂r

and

(1)

The solution is supported by a concentration Csup of a monovalent inert salt MX, which is completely dissociated in solution. A has an arbitrary charge zA such that zB ) zA + n, where n is the number of electrons lost by A and is +1 or -1; n ) +1 represents an oxidation, whereas n ) -1 is a reduction. Where zA * 0, the bulk concentration of M or X must be augmented to maintain charge balance; the counterion for A and the ion of the same charge in the supporting electrolyte are presumed to be equivalent. The flux of any species i at any point in solution is described by the Nernst-Planck equation. Assuming convection to be negligible, this equation has the form of the sum of two terms, one describing diffusion and the other migration

(

In a hemispherically symmetric space with spatial coordinate r, we may write, for i ) A, B, M, or X

r re

(13)

Hence

(

(

))

∂ci ∂ci ∂θ Di ∂2ci 2 ∂θ ∂2θ 2 ∂ci + ) 2 + zi + ci 2 + ci 2 ∂t R ∂R ∂R ∂R R ∂R re ∂R ∂R (14)

Making a Cyclic Voltammetry Experiment “Diffusional”

F2C*A ∂2θ 2 ∂θ 2 + ) -re R ∂R RTs0 ∂R2

∑ zici

J. Phys. Chem. C, Vol. 113, No. 25, 2009 11159

i

Defining a dimensionless time τ as

τ)

DA r2e

(16)

t

the dimensionless Nernst-Planck-Poisson equation set may be simply rendered

(

))

(

∂ci ∂ci ∂θ Di ∂2ci ∂2θ 2 ∂θ 2 ∂ci + + ci ) + z + c i i 2 ∂τ DA ∂R2 R ∂R ∂R ∂R R ∂R ∂R (17) ∂2θ 2 ∂θ + ) -R2e 2 R ∂R ∂R

∑ zici

(18)

F2C*A RTs0

Re )

RTs0

I)

2

2F I

1 2

∑ zi2Ci

(20)

i

(21)

where I is the ionic strength and csup ) Csup/C*A. Therefore, where re . rD and hence the zero-field approximation is valid at the electrode surface, Re is, in general, large. For a cyclic voltammetry experiment, the applied potential E is varied linearly in t at some scan rate V; the associated linear variation of applied θ in τ is a dimensionless scan rate σ, defined as 2

σ)

∂θ F re ) V ∂τ RT D

(22)

and therefore, for a scan between a start and an end potential θi and θf, respectively, θapp is swept upward linearly in τ at rate σ across this range and then linearly back to the start potential. Surface flux is recorded simply as

j)n

∂cA ∂R

|

R)1

which may be related to the current, i

|

R)1

) K0(exp((1 - R)(θapp - θ0))cA,0 exp(-R(θapp - θ0))cB,0)

(25)

for n ) +1 or

∂cA ∂R

|

R)1

) K0(exp((-R(θapp - θ0))cA,0 exp((1 - R)(θapp - θ0))cB,0) (26)

(19)

re + |zA |(1 + |zA |) rD 1

√2csup

∂cA ∂R

for n ) -1. The dimensionless heterogeneous rate constant is defined as

Re represents the relative scale of the electrode compared to the Debye length, rD,15 and as such is, in effect, an inverse dimensionless measure of the electric susceptibility or polarizability of the solution

rD )

(24)

2.3. Boundary Conditions. Equations 17 and 18, the dimensionless NPP equations, must be solved subject to suitable boundary conditions. There are 10 boundary conditions in total, one for each species, plus potential, at each of the two boundaries R ) 1 and R f ∞. At R ) 1, Butler-Volmer kinetics are applied, relating the flux of A normal to the electrode to the surface concentrations of A and B in the form

i

where the dimensionless variable Re is defined as

Re ) re

i ) 2πFC*ADAre j

(15)

(23)

K0 )

k0re DA

(27)

and θapp is the dimensionless applied potential

F (E - EQf ) RT

θapp )

(28)

Conservation of mass at the electrode surface additionally requires

∂cA ∂R

|

R)1

)-

DB ∂cB DA ∂R

|

R)1

(29)

and assuming M and X to be inert and the electrode impermeable

∂cM ∂R

|

R)1

)

∂cX ∂R

|

R)1

)0

(30)

The boundary condition for the potential at the electrode surface assumes that the double layer is negligible in extent compared to the diffusion layer, and therefore, the electric field (potential gradient) at the electrode surface is vanishing

∂θ ∂R

|

R)1

)0

(31)

This simplifying assumption has been demonstrated by past work13 to be valid where the diffusion layer extends beyond a few nanometres and therefore is suitable in general, except for the case of nanoelectrodes, where this approximation becomes inappropriate. The R f ∞ boundary is represented by limiting the simulation space to the maximum extent of the diffusion layer for a

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diffusion-only system, being Rmax ) 6(τmax)1/2 or Rmax ) 6[(DB/ DA)τmax]1/2, whichever is greater;16 the requirement of electroneutrality in bulk solution ensures that even in a weakly supported system, this is still an appropriate limiting point for the simulation space. The concentrations of A, B, M, and X are set to their bulk values at this boundary:

cA ) 1

(32)

cB ) 0

(33)

cM ) c*M

(34)

cX ) c*X

(35)

The bulk concentrations of M and X must be augmented appropriately from the supporting electrolyte ratio in order to conserve charge in bulk solution

csup )

zA g 0

Csup C*A

(36)

c*M ) csup

c*X ) csup + zA

(37) zA < 0

c*M ) csup - zA

c*X ) csup

(38)

Finally, the potential boundary condition in bulk solution is set such that electroneutrality is maintained outside of the diffusion layer; again, past work17 presents this as a valid approximation compared to a fuller treatment in which a coordinate transform allows φ ) 0 to be set at infinite r. The Poisson equation for a uniformly electroneutral medium is

|

R)Rmax

)0

(40)

which is used as the R f ∞ boundary condition. 2.4. Numerical Methods. The NPP equations are solved using a finite difference method across an irregular grid of points, expanding in R away from the electrode. The expanding grid corresponds to that used in previous work,17 with a region of extremely dense regular grid spacing close to the electrode that expands proportionally to R beyond some characteristic switching point Rs. Mathematically, the R position of a point Rj is defined as

R < Rs

(42)

where R0 ) 1 is the point precisely at the electrode surface. A regular time grid was employed, with a parameter τPT given the number of time steps per unit θ. Optimal parameters were established using detailed convergence studies at varying σ and csup; for the majority of systems, these were γR ) 5 × 10-3, Rs ) 2 × 10-3, and τPT ) 100. The NPP equations are discretized in this simulation space according to the Crank-Nicolson method,18 which is characterized as stable and accurate for 1D simulations.19 The resulting set of coupled nonlinear simultaneous equations is solved using the iterative Newton-Raphson method, detailed in Appendix A. All simulations were programmed in C++ and run on various desktop computers (processors ranging from Intel Pentium 4, 3 GHz, to Intel Core2 Quad, 2.4 GHz, and RAM ranging from 1 to 2 GB), with running times of 2-20 min per voltammogram being typical. 3. Theoretical Results 3.1. Fully Supported Voltammetric Analysis. The fully supported limit is defined as a situation where the electric field in solution is negligible, such that the migration term in the Nernst-Planck equation is 0 everywhere and mass transport is exclusively diffusive. Under these conditions and assuming infinitely fast (Nernstian) electrode kinetics, the typical experimentally observable parameters, peak current, ipf, and peak-topeak separation, ∆Epp, of a cyclic voltammogram are governed by well-established expressions. In a macroelectrode limit (high σ) where diffusion is predominantly planar, ipf is described by the Randles-Sˇevcˇ´ık equation20 and is proportional to V1/2; in the same limit, ∆Epp is constant with scan rate and ∼57 mV (at 298K). In practice, the finite potential switching window tends to fractionally increase ∆Epp from this ideal limit to ∼59-62 mV. In a microelectrode limit (low σ), the waveshape is sigmoidal; therefore, ∆Epp has no real meaning, and a limiting current iss is noted, which is independent of scan rate. For a hemispherical geometry, this has the analytically determined value21

(43)

(39)

Solving subject to the condition that θ f 0 as R f ∞ yields the expression

∂θ ∂R

Rj ) Rj-1 + γR(Rj-1 - 1)

iss ) 2πFDAC*Are

∂2θ )0 ∂R2

θ(R ) Rmax) + Rmax

R g Rs

Rj ) Rj-1 + γR(Rs - 1)

(41)

3.2. Qualitative Effects of Incomplete Support. The electric field resulting from incomplete electrolytic support will tend to alter the observable parameters introduced above, as migration effects alter the rate of mass transport to the electrode surface, and hence the perceived current at a given potential and scan rate. In particular, ipf becomes a function of csup and Re. Further, ∆Epp will be affected by the nonzero resistance of the solution. The applied (and hence recorded) potential, E, at the working electrode surface is altered from the observed potential difference between the working electrode and solution as

E ) EWE - Eref + iR

(44)

where Eref is the constant reference electrode potential and EWE is the potential difference between the working electrode and the solution at its surface. Where i, the current passed at the working electrode, and R, the solution resistance, are significant, the potential applied and the potential difference between the working electrode and the solution at its surface will differ by a value iR, denoted as the ohmic drop. In such a case, ∆Epp >



Figure 1. Exemplar voltammetry for the reversible one-electron oxidation of neutral A to B+ at a macroscale hemispherical electrode (σ ) 104, Re ) 105, K0 ) 105) at support ratios varying from “full” support (csup ) 1000) to weak support (csup ) 1).

Figure 2. Concentration profiles at θ ) 10 (forward sweep) for the weakly supported reversible one-electron oxidation of neutral A to B+ at a macroscale hemispherical electrode (σ ) 104, Re ) 105, K0 ) 105, csup ) 1), showing the depletion of M+ and increased concentration of Xrequired close to the electrode surface to maintain electroneutrality.

60 mV even in the limit of infinitely fast kinetics. In general, microelectrode voltammetry is expected to be substantially less sensitive to changes in csup as the passed current is very small, and therefore, the ohmic drop becomes insignificant; consequently, microelectrode voltammetry in weakly supported media is already a well-established technique.22 3.3. Exemplar Voltammetry and Variation of the Ohmic Drop over the Scan. Three exemplar macroelectrode voltammograms, simulated under conditions ranging from full to weak electrolytic support, are shown in Figure 1. The limiting effect on current and the marked effect of the ohmic drop on the peakto-peak separation are clear; notable is the “ramping” effect on the observed voltammetry under weak support, where current increase with potential becomes approximately linear rather than near-exponential due to the contribution of uncompensated solution resistance. In this case, the forward peak is not encountered within the scan range; a concentration profile (Figure 2) taken in the prepeak region shows clearly that migration of M+ and X- limits the flux of A toward the electrode. Additionally, the perceptible predicted broadening of

the peak-to-peak separation to over 70 mV even in the csup ) 100 case is notable, in contrast to the past conclusion that at steady-state, and by extension transient conditions, csup ) 26 is sufficient to limit ohmic drop to 103, a significant positive deviation from ∆Epp



Figure 7. log(-jpf) versus log σ and log csup for A+ + e- a B0 at a hemispherical electrode with Re ) 105.

Figure 9. log(-jpf) versus log σ and log csup for A3+ + e- a B2+ at a hemispherical electrode with Re ) 105.

Figure 8. ∆θpp versus log σ and log csup for A+ + e- a B0 at a hemispherical electrode with Re ) 105 and switching potential θf ) -40.

Figure 10. ∆θpp versus log σ and log csup for A3+ + e- a B2+ at a hemispherical electrode with Re ) 105 and switching potential θf ) -40.

≈ 60 mV (∆θpp ≈ 2.3) is noted below relatively high support ratios, with distinguishable ohmic drop beginning below 10 < csup < 100. The extent of the ohmic drop at the lowest support ratios is such that no diffusion-limited peak occurs within the range of the scan for which the simulated switching potential was ∼EfQ + 1.5 V; consequently, some jpf data values in Figure 5 are absent. It is in particular notable that at all scan rates, the limit of infinite resistance for a solution with csup f 0, and hence no mobile charges, is achieved; infinite peak-to-peak separation would be observed. 3.5. Effect of zA and the Concept of Self-Support. Where zA * 0, the electroactive species and its counterion contribute to compensating solution resistance and act themselves as a supporting electrolyte. Additionally, if its charge is appropriate, the electroactive species can compensate for the change in solution charge for its own oxidation or reduction by migrating toward the electrode surface. Where zA * 0 and csup f 0, it is likely that migration of the electroactive species and its counterion dominate mass transport are crucial in allowing electron transfer by migration in order to maintain electroneutrality. Consequently, a minimum “intrinsic” support level exists for the case of zA * 0, being the degree of support provided by the electroactive species itself. Figures 7 and 8 show the variation with csup for the one-electron reduction of a singly charged species (e.g., cobaltocenium to cobaltocene). Figures 9 and 10 show the variation with csup for the one-electron

reduction of an ion with zA ) +3 (e.g., hexaammineruthenium(III) to hexaammineruthenium(II)). In place of the infinite limit of Figure 6, a self-supported limiting maximum ∆θpp is approached as csup f 0. Rooney et al. reported “approximately reversible” selfsupported macroelectrode voltammetry for the [Fe(CN)6]3-/4couple at glassy carbon, Au, and Pt electrodes.6 Examination of the published voltammograms suggests ∆Epp . 60 mV is in fact observed, and as noted, at the high concentrations of the electroactive species and low scan rates employed, convection is likely to significantly influence mass transport. Our prediction is that in a self-supported regime, a finite proportion of mobile charges is present at any given concentration of A, and hence, the ohmic drop observed in cyclic voltammetry is not a function of the electroactive species concentration, provided there is sufficient ionic strength in solution to render the Debye length much smaller than the diffusion layer. This is confirmed below in the determination that for macroscale voltammetry, the parameter containing C*A, Re, does not affect peak-to-peak separation. 3.6. Effect of Re. The dimensionless parameter Re, which arises in the removal of dimensionality from the Poisson equation, was introduced above as representative of the scale of the electrode compared to the Debye length. Detailed surfaces (see Supporting Information) were produced, showing the effect

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Figure 11. ∆θpp versus DM/DA and DX/DA for A0 a B+ + e- at a hemispherical macroelectrode with csup ) 10 (weak support), σ ) 104, Re ) 105, and switching potential θf ) 20.

of Re and σ on jpf and ∆θpp for a one-electron oxidation of a neutral species with csup ) 5 and 0.2. In both cases, Re is shown to have no significant effect on the voltammetry; csup is sufficient to make the Debye length negligible on the electrode scale, even at low Re and low σ, such that electroneutrality is maintained until a distance very close to the electrode surface, and therefore,

Dickinson et al. perceptible migration effects are negligible, with ohmic drop dominating the effect of csup on the voltammetry. The gradient in the Randles-Sˇevcˇ´ık “straight-line region” is ∼0.435 here, with marked convexity at the highest values of σ. A comparison with a similar plot for the one-electron reduction of a species A+ (Supporting Information) showed no different results; it is therefore evident that provided Re is large, its magnitude does not further quantitatively affect the voltammetry. 3.7. Effect of DB. The effect of the ratio of diffusion coefficients, DB/DA, was studied by examination of variations in θpf and ∆θpp. These parameters are affected by DB only where csup is very low, and hence, the migration of the product species following its creation becomes significant to maintaining charge balance (see Supporting Information). For the one-electron oxidation of neutral A, increased DB shifts the peak positions, considered from θpf, to more negative potentials, as is well established in the diffusion-only case.21 This effect is enhanced under weakly supported conditions where the diffusion-migration of B becomes more significant to charge balance and hence affects the rate of electrolysis to a greater extent. 3.8. Effect of DM and DX. In cases where one or both of the supporting electrolyte species has a high diffusion coefficient compared to the electroactive species and therefore can migrate quickly relative to the rate of electrochemically driven diffusion to the surface, solution resistance is compensated more rapidly, and hence, the ohmic drop is correspondingly less. A plot of ∆θpp versus DM and DX is shown in Figure 11. The symmetry of this plot indicates that the compensation of solution charge

Figure 12. Comparison of experiment and simulation of the theoretical model for the process [Ru(NH3)6]3+ + e- a [Ru(NH3)6]2+ at a 1 mm radius Pt disk, V ) 50 mV s-1, C *A ) 5 mM. (A) Csup ) 2 M. (B) Csup ) 500 mM. (C) Csup ) 150 mM. (D) Csup ) 50 mM.



Figure 13. Comparison of experiment and simulation of the theoretical model for the process [Ru(NH3)6]3+ + e- a [Ru(NH3)6]2+ at a 1 mm radius Pt disk, V ) 100 mV s -1, C *A ) 5 mM. (A) Csup ) 2 M. (B) Csup ) 500 mM. (C) Csup ) 150 mM. (D) Csup ) 50 mM.

is equally well achieved at the electrode surface by the migration of either like charges away from the surface or opposite charges toward it. The importance of a relatively high supporting electrolyte diffusion coefficient to effective compensation of solution resistance will be emphasized in our conclusions below. 4. Experimental Section All solutions were prepared with ultrapure water with resistivity > 18.2 MΩ cm-1 (at 298 K) and degassed for 30 min with N2 (BOC, high-purity oxygen-free) before starting each experiment. The temperature was maintained constant at 298 K using a water bath (W14, Grant). Hexaammineruthenium(III) chloride (HexRu(III), Aldrich, 98%) and potassium chloride (Aldrich, >99%) were used without further purification. A platinum disk with radius of 1 mm was used as a working electrode, a platinum foil was used as a counter electrode, and a saturated calomel electrode (SCE) was used as a reference electrode. Diamond spray (0.3 and 0.1 µm, Kemet International, U.K.) on soft lapping pads (Buehler, U.S.A.) was used to polish the working electrode surface. Solutions containing 5 mM hexaammineruthenium(III) chloride with a range of concentrations of supporting electrolyte (2000, 500, 150, and 50 mM KCl) were prepared, and cyclic voltammetry was performed across a range of scan rates (50, 100, 200, 500, and 1000 mV s-1) using an analog potentiostat (built in house) which was connected to an oscilloscope (TDS 3034B, Tektronix, U.S.A.) to record the data. The analog

potentiostat rather than a digital equivalent was employed to prevent artifacts known to be introduced by the staircase voltage waveform, especially the broadening of peak-to-peak separation, which is an essential observable for the study of migration. These effects in staircase cyclic voltammetry were first noted by Bilewicz et al.25 in purely diffusive systems and were investigated theoretically for disk electrode systems by Barnes et al.26 The capacitive current was subtracted from the cyclic voltammograms using blank voltammetry recorded in solutions without the electroactive species but containing the same degree of electrolytic support. The excess of chloride ions introduced by the hexaammineruthenium chloride into the experimental solutions was compensated for by adding 15 mM potassium chloride to the corresponding concentration of supporting electrolyte in the blank solutions. The diffusion coefficients for the hexaammineruthenium(III) and hexaammineruthenium(II) cations in the aqueous KCl system were investigated via double potential step chronoamperometry using a 25 µm radius platinum disk, according to the procedure established by Klymenko et al.;27 fitted data are available in the Supporting Information. The values determined were DHexRu(III) ) 9.0 × 10-6 cm2 s-1 and DHexRu(II) ) 10.0 × 10-6 cm2 s-1. These compare well to values for HexRu(III) already in the literature ranging from 8.5 × 10-6 to 9.1 × 10-6 cm2 s-1,28-30 all measured in 0.1 M KCl(aq) at 298 K.

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5. Results and Discussion Experimental voltammetry is presented in Figures 12-16 across a range of scan rates from 50 mV s-1 to 1 V s-1. In each case, voltammogram A is recorded at Csup ) 2M, B at Csup ) 500 mM, C at Csup ) 150 mM, and D at Csup ) 50 mM. These correspond to support ratios of csup ) 400, 100, 30, and 10, respectively. Simulated theoretical fits of the voltammetry are shown by the closed circles. The following parameters were used in the simulation: re ) 0.71 mm to correspond area to area with a disk electrode with re ) 1 mm; C*A ) 5 mM; DHexRu(III) ) 9 × 10-6 cm2 s-1 as characterized from our microelectrode double potential step chronoamperometry as described above; DHexRu(II) ) 1 × 10-5 cm2 s-1 as also characterized from microelectrode double potential step chronoamperometry; and DK+ ) 1.8 × 10-5 cm2 s-1 and DCl- ) 1.95 × 10-5 cm2 s-1, which are suitable literature values in the concentration range employed.15,31 Parameters for electrode kinetics of k0 ) 1 cm s-1 and R ) 0.5 were used, which render the system effectively reversible under all pertinent conditions.32 Additionally, the following parameters were used: s ) 78.54 (H2O)33 and T ) 298 K. The formal potential used, EQf , was different for each support ratio; in each case, reversible voltammetry was available, which permitted establishment of the value. The formal potential is noted to vary consistently with ionic strength as follows: -213 mV versus SCE for Csup ) 2 M; -190 mV versus SCE for Csup ) 500 mM; -175 mV versus SCE for Csup ) 150 mM; and -170 mV versus SCE for Csup ) 50 mM. Since the diffusion coefficients of the two species are characterized to within a close

tolerance, the shift in both forward and back peaks, which is consistent in all scans, must be attributed to variation in EfQ. At this stage, we note this ionic strength dependence of EfQ as an experimental observation, but in general, we may attribute this variation to dependence of the ratio of the activity coefficients (γHexRu(II)/γHexRu(III)) on ionic strength as the species have different charges; in a high ionic strength system, these activity coefficients will respond differently to variations in ionic strength, as described by the Robinson-Stokes equation. In general, we note from Figures 12-16 that the closeness of fit between the theoretical and experimental data is accurate for cases A-C but that peak-to-peak separation is underestimated by the theoretical model in case D, beyond the tolerances of experimental error. In case D, changes in the voltammetry are still predicted qualitatively but not quantitatively; this is made clear by a plot of ∆Epp against log V for the theoretical and experimental data, in Figure 17. A variety of the approximations in this model are likely insufficient under weakly supported conditions, but under typical experimental conditions of csup g 30, the theoretical model accurately predicts the extent of broadening of peak-to-peak separation caused by uncompensated solution resistance. At this stage, we are therefore able to answer the question we posed in the Introduction, at least in part; quantitatively, how much supporting electrolyte is required to achieve “diffusional” cyclic voltammetry? In this limit, peak current and peakto-peak separation are as predicted by a diffusion-only model, and therefore, conclusions drawn from the observables ipf or ∆Epp are not prejudiced by the presence of an electric field in




solution. Defining experimental distinguishability as we desire in terms of changes in ipf or ∆Epp, we can define, for any given combination of re, V, DA, DB, DM, and DX, the minimum value of csup required to render the voltammetry indistinguishable from the diffusion-only case. According to our comparisons with experimental voltammetry, the theory is accurate to within an experimental error for the analog potentiostat system of (5 mV for ∆Epp, in the region of csup g 30, and in the region where a disk-shaped macroelectrode may be approximated by a hemisphere of the same area, roughly σ > 103. Our conclusions will therefore be quantitatively accurate in this range. For three model systems, where DA ) DB, DM ) DX, s ) 78.54 (as for water), and DM/DA, respectively, ) 1, 2, and 3, voltammetry was simulated for the hemispherical electrode on a logarithmic scale varying in (r2e /DA)V and in csup. Each voltammogram was classified in terms of the recorded ∆Epp compared to the diffusion-only case, the latter being calculated from the same program with the artificially high support ratio csup ) 106. Contours are plotted (Figures 18-20) to show the regions where the deviation (∆Epp - ∆Epp (diffusion-only)) is less than 1 mV (indistinguishable), 1-3 mV (well supported, i.e., indistinguishable within typical experimental error, for a digital potentiostat), 3-5 mV (slightly distinguishable within experimental error), 5-10 mV (insufficiently supported), and greater than 10 mV (acutely unsupported). The data are simulated here for a hemispherical electrode for which it is assumed that the voltammetry is largely indistinguishable from a disk macroelectrode. It is essential to note the

2 2 ) 2re,hemi , and hence, correspondence, for equal area, that re,disk the horizontal scale of the plots varies by a factor of 2 depending on the geometry in which re is taken. The approximation of a disk to a hemisphere is justified provided that the reference electrode is situated sufficiently far from the working electrode, as then the electric field will be approximately hemispherical in solution, tending away from the working electrode, such that a hemispherical treatment is appropriate. The approximation is then limited to a consideration of diffusional flux at the predominantly planar electrode surface being appropriately modeled by a hemisphere, which will hold good above some value of re; the accuracy of comparison between peak-to-peak separations from theory and experiment in the range of interest for DM ≈ 2DA suggests that this approximation is valid. Similarly, the choice of switching window will affect values for ∆Epp. Here, the simulation uses a ∼1 V window, from EfQ - (20 RT/F) to EQf + (20 RT/F); it is expected that in a narrower switching window, the same support regions will be appropriate. Lastly, we note that the data are simulated exclusively for the one-electron oxidation of a neutral species, but at the support ratios considered (g30), the contribution to electrolytic support from the counterion of a charged electroactive species is negligible.

6. Conclusion The theory of cyclic voltammetry in weakly supported media has been approached using a numerical model invoking a zero-

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Figure 16. Comparison of experiment and simulation of the theoretical model for the process [Ru(NH3)6]3+ + e- a [Ru(NH3)6]2+ at a 1 mm radius Pt disk, V ) 1 V s-1, C *A ) 5 mM. (A) Csup ) 2 M. (B) Csup ) 500 mM. (C) Csup ) 150 mM. (D) Csup ) 50 mM.

Figure 17. Comparison of experimental and theoretically predicted ∆Epp for cyclic voltammetry of the process [Ru(NH3)6]3+ + e- a [Ru(NH3)6]2+ at a 1 mm radius Pt disk, C *A ) 5 mM, and support ratios A-D as above.

field approximation at the electrode surface. Expected changes in the observed voltammetric waveshape with variation of the significant intrinsic parameters of the system were plotted on working surfaces and interpreted appropriately. Comparison was then drawn with experimental cyclic voltammetry for the aqueous [Ru(NH3)6]3+/2+ system at a Pt macroelectrode; by plotting theoretical and experimental data together, the theoreti-

cal model was shown to be quantitatively accurate, within experimental error for our system, for at least the region of csup g 30, across a range of typical experimental scan rates. The contour plots presented at Figures 18-20 are intended to be instructive to the experimentalist; for any macroelectrode system, the required csup to limit deviations from the diffusiononly case to within a certain tolerance may be inferred. The



Figure 18. Contour plot showing the deviation ∆∆Epp of theoretically observed ∆Epp from the diffusion-only value at a hemispherical electrode for varying csup ) Csup/C *A and (r2e /DA)V, where DM/DA ) 1, A0 a B+ + e-, K0 ) 105, and Re ) 105.

Figure 19. Contour plot showing the deviation ∆∆Epp of theoretically observed ∆Epp from the diffusion-only value at a hemispherical electrode for varying csup ) Csup/C *A and (r2e /DA)V, where DM/DA ) 2, A0 a B+ + e-, K0 ) 105, and Re ) 105.

experimentalist is free to choose the desired tolerance, but we would expect most systems to have an experimental error of approximately (3 mV, and therefore, real (diffusion-migration) and ideal (diffusion-only) voltammetry will be indistinguishable for tolerances of ∆∆Epp e 3 mV. It must be noted that the range of applicability of these plots is confined, however, to macroelectrode voltammetry and, at this stage, to electrochemically reversible systems. Of additional interest is the observation of the thermodynamic effect of varying ionic strength in such a system, such that EfQ is a function of csup. These contour plots show clearly that the support ratio of 26 once proposed for the steady-state system1 is not equally useful for transient cyclic voltammetry but, in fact, is broadly inappropriate; the majority of macroelectrode systems require support ratios greater than 100 to avoid detectable peak broadening from ohmic drop. The current state and pace of development in analytical electrochemistry are such, however,

that it is perhaps no longer adequate to simply apply excess electrolyte and ignore the issue of migration. To the modern electrochemist, an understanding of the microscopic and macroscopic effects associated with extended electric fields in solution is increasingly important. We consequently hope to develop the theory presented above to approach the case of csup < 30, and indeed csup f 0, by examining and revising those approximations made. Acknowledgment. E.J.F.D. thanks St John’s College, Oxford, for support via a graduate studentship. J.G.L.-P. thanks CONACYT, Me´xico, for financial support via a scholarship (Grant 208508). N.V.R. thanks EPSRC for funding. Appendix A. The Iterative Newton-Raphson Method. The iterative Newton-Raphson method is a standard numerical procedure

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Figure 20. Contour plot showing the deviation ∆∆Epp of theoretically observed ∆Epp from the diffusion-only value for cyclic voltammetry at a hemispherical electrode for varying csup ) Csup/C *A and (r2e /DA)V, where DM/DA ) 3, A0 a B+ + e-, K0 ) 105, and Re ) 105.

for solving a system of coupled nonlinear simultaneous equations.34 The n equations are all rewritten in a form

fn(x0, x1, ..., xn) ) 0

(45)

n variables, this may be extended, such that for each variable m

∑ un n

∂fm(x0) ) -fm(x0) ∂xn

(51)

Let x be the vector containing the unknowns x0-xn, let f(x) be the vector containing the functions f0-fn, and let u be the vector containing the differences between x at successive iterations, such that

which may be expressed for the vectors u and f in the matrix form

u ) xz+1 - xz

J(x0)u ) -f(x0)

(46)

In the simple Newton-Raphson method, the Taylor series of a function about a trial solution x0 is considered

f(x) ≈ f(x0) + (x - x0)f' (x0) ) 0

xz+1 ) xz -

f(xz) f' (xz)

(48)

For a function of multiple variables, a trial vector x0 may be altered similarly

f(x) ≈ f(x0) +

∑ (xn - xn,0) n

∂f(x0) ∂xn

(49)

and hence, defining un ) xn - xn,0

∑ n

∂f(x0) un ) -f(x0) ∂xn

where J is the Jacobian matrix, being the n × n square matrix for which the element Jmn is described

(47)

and hence the solution x may be derived from successive iterations of the form

(52)

Jmn )

∂fm ∂xn

(53)

Equation 52 is solved iteratively and x is augmented by u, until all values u0-un are less than a characteristic parameter , at which point x is taken as the trial vector for the next time step. Equation 52 can be solved by an adapted Thomas algorithm method (LU decomposition followed by back-substitution) as the Jacobian is a diagonal matrix, with the minimum of 15 diagonals when the unknowns xn are ordered (cA,0, cB,0, φ0, cM,0, cX,0, cA,1, cB,1, φ1, cM,1, cX,1,...). Supporting Information Available: Detailed surfaces and additional plots. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes

(50)

Where there are n simultaneous equations associated with the

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Making a Cyclic Voltammetry Experiment “Diffusional” (5) Amatore, C.; Oleinick, A.; Svir, I. Anal. Chem. 2008, 80, 7947– 7956. (6) Rooney, M. B.; Coomber, D. C.; Bond, A. M. Anal. Chem. 2000, 72, 3486–3491. (7) Bond, A. M.; Coomber, D. C.; Feldberg, S. W.; Oldham, K. B.; Vu, T. Anal. Chem. 2001, 73, 352–359. (8) Amatore, C.; Deakin, M. R.; Wightman, R. M. J. Electroanal. Chem. 1987, 225, 49–63. (9) Ciszkowska, M.; Jaworski, A.; Osteryoung, J. G. J. Electroanal. Chem. 1997, 423, 95–101. (10) Amatore, C.; Knobloch, K.; Thouin, L. Electrochem. Commun. 2004, 6, 887–891. (11) Stevens, N. P. C.; Bond, A. M. J. Electroanal. Chem. 2002, 538539, 25–33. (12) Klymenko, O. V.; Amatore, C.; Svir, I. Anal. Chem. 2007, 79, 6341– 6347. (13) Streeter, I.; Compton, R. G. J. Phys. Chem. C 2008, 112, 13716– 13728. (14) Limon-Petersen, J. G.; Streeter, I.; Rees, N. V.; Compton, R. G. J. Phys. Chem. C 2008, 112, 17175–17182. (15) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworths Publications Ltd.: London, 1955. (16) Svir, I. B.; Klymenko, O. V.; Compton, R. G. Radiotekhnika 2001, 118, 92. (17) Limon-Petersen, J. G.; Streeter, I.; Rees, N. V.; Compton, R. G. J. Phys. Chem. C 2009, 113, 333–337. (18) Crank, J.; Nicolson, P. Math. Proc. Cambridge Philos. Soc. 1947, 43, 50–67. (19) Stoerzbach, M.; Heinze, J. J. Electroanal. Chem. 1993, 346, 1–27.

J. Phys. Chem. C, Vol. 113, No. 25, 2009 11171 (20) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications, 2nd ed.; John Wiley & Sons: New York, 2001. (21) Compton, R. G.; Banks, C. E. Understanding Voltammetry; World Scientific Publishing: Singapore, 2007. (22) Bond, A. M. Analyst 1994, 119, R1–R21. (23) Amatore, C.; Lefrou, C.; Pflu¨ger, F. J. Electroanal. Chem. 1989, 270, 43–59. (24) Britz, D. J. Electroanal. Chem. 1978, 88, 309–352. (25) Bilewicz, R.; Osteryoung, R. A.; Osteryoung, J. Anal. Chem. 1986, 58, 2761–2765. (26) Barnes, A. S.; Streeter, I.; Compton, R. G. J. Electroanal. Chem. 2008, 623, 129–133. (27) Klymenko, O. V.; Evans, R. G.; Hardacre, C.; Svir, I. B.; Compton, R. G. J. Electroanal. Chem. 2004, 571, 211–221. (28) Banks, C. E.; Compton, R. G.; Fisher, A. C.; Henley, I. E. Phys. Chem. Chem. Phys. 2004, 6, 3147–3152. (29) Banks, C. E.; Rees, N. V.; Compton, R. G. J. Electroanal. Chem. 2002, 535, 41–47. (30) Marken, F.; Eklund, J. C.; Compton, R. G. J. Electroanal. Chem. 1995, 395, 335–338. (31) Lobo, V; M, M.; Ribeiro, A. C. F.; Verissimo, L. M. P. J. Mol. Liq. 1998, 78, 139–149, and references therein. (32) Beriet, C.; Pletcher, D. J. Electroanal. Chem. 1994, 375, 213–218. (33) Lide, D. R., Ed.; CRC Handbook of Chemistry and Physics, 89 ed.; CRC Press: London, 2008. (34) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes. The Art of Scientific Computing, 3rd ed.; Cambridge University Press: Cambridge, U.K., 2007.

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