Supporting electrolyte

1. 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 tetrabutylam-
monium 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
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 chronoamper-
ometry and voltammetry in low supporting electrolyte conditions
have been achieved by employing the approximation of elec-
troneutrality 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, dem-
onstrated 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 electroneu-
trality in solution. This negligible double layer NPP model has
already been successfully and quantitatively applied to experi-
mental 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
* To whom correspondence should be addressed. Fax: +44 (0) 1865
275410. Tel: +44 (0) 1865 275413. E-mail: richard.compton@
chem.ox.ac.uk.
J. Phys. Chem. C 2009, 113, 11157–11171 11157
10.1021/jp901628h CCC: $40.75  2009 American Chemical Society
Published on Web 06/02/2009

2. support ratio, electrode size, inert electrolyte diffusion coef-
ficient, and scan rate. Comparison is then drawn with experi-
mental cyclic voltammetry for the reduction of the hexaam-
mineruthenium(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
The solution is supported by a concentration Csup of a monova-
lent 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
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
hence
At any point in solution, however, the potential must additionally
satisfy the Poisson equation
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
In a hemispherically symmetric space with spatial coordinate
r, we may write, for i ) A, B, M, or X
and
The problem of simulating cyclic voltammetry under condi-
tions where migration is nonzero is therefore approached
numerically by solving simultaneously this set of coupled partial
differential equations, the Nernst-Planck-Poisson (NPP) equa-
tions, 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
where CA* is the bulk concentration of species A. Hence
Additionally, the distance coordinate is normalized to the
electrode radius, re, such that the electrode boundary occurs at
R ) 1
Hence
A a B + ne-
(1)
Ji ) -Di(∇Ci +
ziF
RT
Ci∇φ) (2)
∂Ci
∂t
) -∇·Ji (3)
∂Ci
∂t
) Di(∇2
Ci +
ziF
RT
(Ci∇2
φ + ∇Ci∇φ)) (4)
∇2
φ ) -
F
s 0
(5)
F ) F ∑
i
ziCi (6)
∂Ci
∂t
) Di(∂2
Ci
∂r2
+
2
r
∂Ci
∂r
+
ziF
RT(∂Ci
∂r
∂φ
∂r
+ Ci
∂2
φ
∂r2
+
Ci
2
r
∂φ
∂r )) (7)
∂2
φ
∂r2
+
2
r
∂φ
∂r
) -
F
s 0
∑
i
ziCi (8)
ci )
Ci
CA*
(9)
θ )
Fφ
RT
(10)
∂ci
∂t
) Di(∂2
ci
∂r2
+
2
r
∂ci
∂r
+ zi(∂ci
∂r
∂θ
∂r
+ ci
∂2
θ
∂r2
+ ci
2
r
∂θ
∂r ))(11)
∂2
θ
∂r2
+
2
r
∂θ
∂r
) -
F2
cA*
RT s 0
∑
i
zici (12)
R )
r
re
(13)
∂ci
∂t
)
Di
re
2 (∂2
ci
∂R2
+
2
R
∂ci
∂R
+ zi(∂ci
∂R
∂θ
∂R
+ ci
∂2
θ
∂R2
+ ci
2
R
∂θ
∂R))
(14)
11158 J. Phys. Chem. C, Vol. 113, No. 25, 2009 Dickinson et al.

3. Defining a dimensionless time τ as
the dimensionless Nernst-Planck-Poisson equation set may
be simply rendered
where the dimensionless variable Re 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 polariz-
ability of the solution
where I is the ionic strength and csup ) Csup/CA*. 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
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
which may be related to the current, i
2.3. Boundary Conditions. Equations 17 and 18, the di-
mensionless 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
for n ) +1 or
for n ) -1. The dimensionless heterogeneous rate constant is
defined as
and θapp is the dimensionless applied potential
Conservation of mass at the electrode surface additionally
requires
and assuming M and X to be inert and the electrode
impermeable
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
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
∂2
θ
∂R2
+
2
R
∂θ
∂R
) -re
2
F2
CA*
RT s 0
∑
i
zici (15)
τ )
DA
re
2
t (16)
∂ci
∂τ
)
Di
DA
(∂2
ci
∂R2
+
2
R
∂ci
∂R
+ zi(∂ci
∂R
∂θ
∂R
+ ci
∂2
θ
∂R2
+ ci
2
R
∂θ
∂R))(17)
∂2
θ
∂R2
+
2
R
∂θ
∂R
) -Re
2
∑
i
zici (18)
Re ) re
F2
CA*
RT s 0
(19)
rD )
RT s 0
2F2
I
I )
1
2 ∑
i
zi
2
Ci (20)
Re )
1
√2csup + |zA|(1 + |zA|)
re
rD
(21)
σ )
∂θ
∂τ
)
F
RT
re
2
D
V (22)
j ) n
∂cA
∂R |R)1 (23)
i ) 2πFCA*DAre j (24)
∂cA
∂R |R)1 ) K0
(exp((1 - R)(θapp - θ0))cA,0 -
exp(-R(θapp - θ0))cB,0) (25)
∂cA
∂R |R)1 ) K0
(exp((-R(θapp - θ0))cA,0 -
exp((1 - R)(θapp - θ0))cB,0) (26)
K0
)
k0
re
DA
(27)
θapp )
F
RT
(E - Ef
Q
) (28)
∂cA
∂R |R)1 ) -
DB
DA
∂cB
∂R |R)1 (29)
∂cM
∂R |R)1 )
∂cX
∂R |R)1 ) 0 (30)
∂θ
∂R|R)1 ) 0 (31)
Making a Cyclic Voltammetry Experiment “Diffusional” J. Phys. Chem. C, Vol. 113, No. 25, 2009 11159

4. diffusion-only system, being Rmax ) 6(τmax)1/2
or Rmax ) 6[(DB/
DA)τmax]1/2
, whichever is greater;16
the requirement of electro-
neutrality 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:
The bulk concentrations of M and X must be augmented
appropriately from the supporting electrolyte ratio in order to
conserve charge in bulk solution
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
Solving subject to the condition that θ f 0 as R f ∞ yields
the expression
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
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 character-
ized 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 experi-
mentally observable parameters, peak current, ipf, and peak-to-
peak 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-Sˇevcˇı´k 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
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 differ-
ence between the working electrode and solution as
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 >
cA ) 1 (32)
cB ) 0 (33)
cM ) cM* (34)
cX ) cX* (35)
csup )
Csup
CA*
(36)
zA g 0 cM* ) csup cX* ) csup + zA
(37)
zA < 0 cM* ) csup - zA cX* ) csup (38)
∂2
θ
∂R2
) 0 (39)
θ(R ) Rmax) + Rmax
∂θ
∂R|R)Rmax
) 0 (40)
R < Rs Rj ) Rj-1 + γR(Rs - 1) (41)
R g Rs Rj ) Rj-1 + γR(Rj-1 - 1) (42)
iss ) 2πFDACA*re (43)
E ) EWE - Eref + iR (44)
11160 J. Phys. Chem. C, Vol. 113, No. 25, 2009 Dickinson et al.

5. 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; conse-
quently, 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 voltam-
mograms, 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 peak-
to-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 <1 mV.1
For the case csup ) 10, a plot of the simulated variation of
current, ohmic drop, and solution resistance during the scan is
shown in Figure 3. Additionally, Figure 4 shows the variation
of EWE, the potential difference between the working electrode
and the solution at the working electrode surface, with time
plotted against the same variation in applied potential E, noting
deviations from the perfect triangular waveshape resulting from
a nonzero ohmic drop; a similarity is noted here with the similar
plots produced by dynamic iR compensation produced by
Amatore et al.23
Here, the approximation of time-independent
uncompensated solution resistance, Ru, throughout the scan is
accurate; for the high σ regime, there is therefore no migrational
effect to suggest inadequacy in compensating ohmic drop by
simple arithmetic correction with a constant Ru, as reviewed by
Britz.24
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 X-
required close to the electrode surface to maintain electroneutrality.
Making a Cyclic Voltammetry Experiment “Diffusional” J. Phys. Chem. C, Vol. 113, No. 25, 2009 11161

6. 3.4. Dependence of jpf and ∆θpp on csup. The numerically
predicted quantitative variation of the dimensionless observables
jpf and ∆θpp with csup was studied by producing surfaces in which
the change in the observable is plotted in the z-axis against
variation in csup and σ, the dimensionless scan rate. The macro-
and microelectrode limits are hence both considered. Figures 5
and 6 show the variation with csup for the one-electron oxidation
of a neutral species (e.g., ferrocene to ferrocenium).
In Figure 5, the recognizable transition with increasing σ from
steady-state behavior to Randles-Sˇevcˇı´k behavior is observed
at high csup. As csup decreases, the gradient in the
Randles-Sˇevcˇı´k region is reduced from 0.5, and convex
deviations from a straight line are noted; this results as the rate
of migration becomes rate-limiting at high σ, and hence, some
limiting current is approached as σ f ∞. By comparison, the
peak current is not significantly affected in the low σ regime.
We note that the variation of σ does not strictly correspond to
a variation in re as Re has been held constant at 105
(macro-
electrode); it will be demonstrated below, however, that varying
Re has no observable effect on the voltammetry even over the
wide range of re implied by this range of σ, given that σ ∝ re
2
.
Therefore, the low σ regime may conveniently be treated as
equivalent to a microelectrode regime without error, and the
conclusion that voltammetry is not strongly affected by csup at
low σ is therefore as expected. We emphasize that this
conclusion cannot hold for nanoelectrodes where the Debye
length becomes significant on the scale of the electrode.
Figure 6 demonstrates the increased ∆θpp resulting from
increased ohmic drop at large electrodes (high σ) in conditions
of decreased csup. Under typical macroelectrode voltammetry
conditions of σ > 103
, a significant positive deviation from ∆Epp
Figure 3. Simulated current, ohmic drop, and resistance plotted against time for a cyclic voltammetry experiment performed for the reversible
system A a B + e-
, with D ) 10-5
cm2
s-1
, re ) 1 mm, CA* ) 1 mM, V ) 10RT/F V s-1
≈ 400 mV s-1
, s ≈ 40.
Figure 4. Applied potential, E, and actual potential difference at the
working electrode, EWE, against time for the same system.
Figure 5. log jpf versus log σ and log csup for A0
a B+
+ e-
at a
hemispherical electrode with Re ) 105
.
Figure 6. ∆θpp versus log σ and log csup for A0
a B+
+ e-
at a
hemispherical electrode with Re ) 105
and switching potential θf )
40.
11162 J. Phys. Chem. C, Vol. 113, No. 25, 2009 Dickinson et al.

7. ≈ 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 ∼Ef
Q
+ 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 electroneu-
trality. 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., hexaammineruthe-
nium(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” self-
supported macroelectrode voltammetry for the [Fe(CN)6]3-/4-
couple 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 CA*, 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
Figure 7. log(-jpf) versus log σ and log csup for A+
+ e-
a B0
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 9. log(-jpf) versus log σ and log csup for A3+
+ e-
a B2+
at
a hemispherical electrode with Re ) 105
.
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.
Making a Cyclic Voltammetry Experiment “Diffusional” J. Phys. Chem. C, Vol. 113, No. 25, 2009 11163

8. 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,
perceptible migration effects are negligible, with ohmic drop
dominating the effect of csup on the voltammetry. The gradient
in the Randles-Sˇevcˇı´k “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 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.
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
, CA* ) 5 mM. (A) Csup ) 2 M. (B) Csup ) 500 mM. (C) Csup ) 150 mM. (D) Csup ) 50 mM.
11164 J. Phys. Chem. C, Vol. 113, No. 25, 2009 Dickinson et al.

9. 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) chlo-
ride 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 chrono-
amperometry 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.
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
, CA* ) 5 mM. (A) Csup ) 2 M. (B) Csup ) 500 mM. (C) Csup ) 150 mM. (D) Csup ) 50 mM.
Making a Cyclic Voltammetry Experiment “Diffusional” J. Phys. Chem. C, Vol. 113, No. 25, 2009 11165

10. 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; CA* ) 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 chro-
noamperometry; 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, Ef
Q
, 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 Ef
Q
. At
this stage, we note this ionic strength dependence of Ef
Q
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 coef-
ficients 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 underesti-
mated 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 ap-
proximations 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 uncom-
pensated 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 “diffu-
sional” cyclic voltammetry? In this limit, peak current and peak-
to-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
Figure 14. 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 ) 200 mV s-1
, CA* ) 5 mM. (A) Csup ) 2 M. (B) Csup ) 500 mM. (C) Csup ) 150 mM. (D) Csup ) 50 mM.
11166 J. Phys. Chem. C, Vol. 113, No. 25, 2009 Dickinson et al.

11. 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 hemi-
sphere 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 (re
2
/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 indistin-
guishable from a disk macroelectrode. It is essential to note the
correspondence, for equal area, that re,disk
2
) 2re,hemi
2
, and hence,
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 Ef
Q
- (20 RT/F) to Ef
Q
+ (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-
Figure 15. 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 ) 500 mV s-1
, CA* ) 5 mM. (A) Csup ) 2 M. (B) Csup ) 500 mM. (C) Csup ) 150 mM. (D) Csup ) 50 mM.
Making a Cyclic Voltammetry Experiment “Diffusional” J. Phys. Chem. C, Vol. 113, No. 25, 2009 11167

12. 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 diffusion-
only case to within a certain tolerance may be inferred. The
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
, CA* ) 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, CA* ) 5 mM, and support ratios A-D as above.
11168 J. Phys. Chem. C, Vol. 113, No. 25, 2009 Dickinson et al.

13. 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 macro-
electrode 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 Ef
Q
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 inap-
propriate; 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 mac-
roscopic 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 CONA-
CYT, 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
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/CA* and (re
2
/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/CA* and (re
2
/DA)V, where DM/DA ) 2, A0
a B+
+ e-
, K0
) 105
, and Re ) 105
.
Making a Cyclic Voltammetry Experiment “Diffusional” J. Phys. Chem. C, Vol. 113, No. 25, 2009 11169

14. for solving a system of coupled nonlinear simultaneous equa-
tions.34
The n equations are all rewritten in a form
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
In the simple Newton-Raphson method, the Taylor series of a
function about a trial solution x0 is considered
and hence the solution x may be derived from successive
iterations of the form
For a function of multiple variables, a trial vector x0 may be
altered similarly
and hence, defining un ) xn - xn,0
Where there are n simultaneous equations associated with the
n variables, this may be extended, such that for each variable
m
which may be expressed for the vectors u and f in the matrix
form
where J is the Jacobian matrix, being the n × n square matrix
for which the element Jmn is described
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.
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