Anal. Chem. 2008, 80, 7947–7956
Theoretical Analysis of Microscopic Ohmic Drop Effects on Steady-State and Transient Voltammetry at the Disk Microelectrode: A Quasi-Conformal Mapping Modeling and Simulation Christian Amatore,*,† Alexander Oleinick,†,‡ and Irina Svir*,†,‡ De´partement de Chimie, Ecole Normale Supe´rieure, UMR CNRS-ENS-UPMC 8640 “PASTEUR”, 24 Rue Lhomond, 75231 Paris Cedex 05, France, and Mathematical and Computer Modelling Laboratory, Kharkov National University of Radioelectronics, 14 Lenin Avenue, 61166 Kharkov, Ukraine The effect of uncompensated solution resistance on steadystate and transient voltammograms at the disk microelectrode was for the first time treated theoretically and numerically at the microscopic level using specific quasi-conformal mapping for the case of absence of electric migration. It has been shown that microscopic distributions of electric potential and current density at a disk microelectrode affect the voltammetric waves at different degrees across the electrode surface due to the variation of elementary resistances and elementary current fluxes over the electrode surface which leads to nonlinear effects that have not been discussed in existing theoretical treatments of ohmic drop at microelectrodes. The analysis of steady state voltammetry in strongly resistive media under Nernstian conditions has allowed justification by appropriate analytical derivations of the widely used potential-shift correction of steady state voltammograms by plotting i vs (E - iRe). One of the popular advantages of ultramicroelectrodes is their ability to perform measurements in resistive solutions due to their relative immunity to ohmic drop (iRe).1 Indeed, whenever a microelectrode is small enough to achieve steady state, the overall ohmic drop it experiences is independent of the electrode shape and dimension2 and is the smallest one that can be observed at the same electrode in the same electrolyte.1 Indeed, under transient voltammetric conditions, the Faradaic current i grows as v1/2 where v is the potential scan rate. Conversely, the resistance, being imposed by electrostatics, is independent of the scan rate provided that no migration is involved in the transport of molecules to and from the electrode surface, so that the ohmic drop, iRe, varies also as v1/2. However, when applying these usual considerations, one deems implicitly that the overall ohmic drop is given by the direct product of the global current, i, by the global cell resistance, Re. * To whom correspondence should be addressed. E-mail: christian.amatore@ ens.fr (C.A.);
[email protected] or
[email protected] (I.S.). † Ecole Normale Supe´rieure. ‡ Kharkov National University of Radioelectronics. (1) Amatore, C. Electrochemistry at Ultramicroelectrodes. In Physical Electrochemistry: Principles, Methods and Applications; Rubinstein, I., Ed.; M. Dekker: New York, 1995; Chapter 4, pp 131-208. (2) (a) Bruckenstein, S. Anal. Chem. 1987, 59, 2098–2101. (b) Oldham, K. B. J. Electroanal. Chem. 1987, 237, 303–307. 10.1021/ac8010268 CCC: $40.75 2008 American Chemical Society Published on Web 10/01/2008
This is certainly true under steady state because when only diffusional transport matters, the second Fick’s law (which commands ultimately the current distribution over the electrode, see below) becomes identical to the Laplace equation (which commands ultimately the distribution of resistances between each electrode point and the reference equipotentials,3 see below). Hence, elementary currents and elementary resistances for any elementary surfaces of the electrode are exactly reciprocal to each other (see below). It ensues that the effective ohmic drop is the same at each point of an electrode under diffusional steady state. Hence, the above macroscopic considerations are strictly identical to microscopic ones. However, when steady state is not achieved, the second Fick’s law and Laplace equation have not the same formulation. Hence, the ohmic drop may depend on the exact location of the active area on the electrode surface, so that the above macroscopic consideration does not respect the physics of the real system. Evidently, when perfect transient behavior is reached, viz., when edge(s) currents are totally negligible, the current over the electrode surface is imposed by planar diffusion and its value is therefore constant. Consequently, the macroscopic view seems to regain validity through reflecting the microscopic physics. However, this usual reasoning is intrinsically wrong since it neglects the fact that the elementary resistances are different at each point of the microelectrode surface. Henceforth, the ohmic drop varies over the electrode surface, from where it ensues that the effective potential difference vs the reference varies over the electrode surface; hence, the current experiences a distribution which is neglected by the classical view. Because of the large use of microelectrodes nowadays, especially under resistive conditions or outside of their steady state (3) Goude´, G., Electricite´. In Cours de Physique Ge´ne´rale de G. Bruhat; Masson & Co.: Paris, France, 1967. (4) (a) Amatore, C.; Oleinick, A.; Svir, I. J. Electroanal. Chem. 2006, 597, 69– 76. (b) Oleinick, A.; Amatore, C.; Svir, I. Electrochem. Commun. 2004, 6, 588–594. (c) Amatore, C.; Oleinick, A.; Svir, I. J. Electroanal. Chem. 2004, 564, 245–260. (d) Amatore, C.; Oleinick, A.; Svir, I. J. Electroanal. Chem. 2005, 575, 103–123. (e) Amatore, C.; Oleinick, A. I.; Svir, I. J. Electroanal. Chem. 2006, 597, 77–85. (f) Amatore, C.; Oleinick, A. I.; Svir, I. B. J. Electroanal. Chem. 2003, 553, 49–61. (5) (a) Amatore, C.; Oleinick, A.; Svir, I. Electrochem. Commun. 2004, 6, 1123– 1130. (b) Bazant, M. Z. Proc. R. Soc. London, Ser. A 2004, 460, 1433– 1452.
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domains, it appears then of interest to revisit at microscopic levels the effects of ohmic drop at microelectrodes. However, the microscopic analysis of ohmic drop at a microelectrode is extremely complicated by the sharpness of both the current and the resistance distributions near its edges. Indeed, edge areas account for a large fraction of the fluxes except at extremely high scan rates required for a pure planar behavior to be achieved, while these are the areas where the elementary resistances are the smallest (see below). Therefore, any error in numerical evaluation of the current and of the resistance distributions over the microelectrode surface may totally alter the outcome of the analysis. This is evidently “easiest” when the electrode shape (viz., disk or partial sphere) allows analytical formulations to be derived for the current and the resistance distributions, so accuracy is perfect. However, as explained above this is valid only for the pure steady state conditions. To solve this problem we then resorted to conformal mapping methods since these are perfectly suited to this kind of problem. Conformal or quasi-conformal mappings are coordinate transformations with unique properties. What we are going to use here is that they allow one to transform a domain allowed for diffusion4a-f (and sometimes convection and migration)5a,b of electroactive species into a simpler one where the concentration distribution is linear or almost linear. Even if such a mapping does not solve all the difficulties (or does not always produce a linear solution) we construct it in such a way that it solves the most important ones.4a-f,5a For instance, the most difficult problem when solving mass transport equations for electrochemical problems at microelectrodes is that of edge effects or, in other terms, singularities due to discontinuous current density at the boundary of the considered area where conducting electrode material meets insulator. In this case, a standard approach to the solution of diffusion equation in real coordinates using finitedifference methods exhibits extremely slow convergence. It occurs because of poor approximation of rapidly changing concentration profiles on a finite grid of nodes near the singularity. A remedy to improve the situation is to straighten the highly curved flux lines and lines of equal concentration upon projecting the whole system into an adequately transformed space and performing the numerical simulation in it. In the best case we may even define the transformed space so that concentration distributions become linear in this new space. In this article we therefore apply an ideally suited transformed space4a,b to numerically solve the problem of microscopic distribution of ohmic drop on a disk microelectrode under steady state or transient conditions. This will allow us to exactly evaluate the overall outcome of ohmic drop on voltammograms and to show that the usual macroscopic considerations mask several nonlinear microscopic phenomena of importance, particularly when neither a pure steady state nor a pure planar regime is achieved. MATHEMATICAL MODEL General Statements. The following assumptions are made for modeling the system at hand. A simple electron transfer reaction (ET) at the surface of the microdisk electrode obeying the Butler-Volmer kinetics is considered. The reference electrode is located sufficiently far away from the working one in order to consider that its equipotential is infinitely distant and symmetric vs the disk electrode axis. 7948
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To fully dissociate the problem examined here from any effect related to migration, an excess of dissociated supporting electrolyte is considered to be present in the solution. Though the resistivity of the solution may be large due to the solvent dielectric constant, the solutions with extremely low dielectric constants are not considered here since they preclude electrolyte dissociation. Similarly, we assume that even under transient conditions, the scan rates of interest are small enough for the cell time constant ReCd to be negligible. This is a consideration which is not at all required by our approach. Indeed, our model may easily accept taking into account the distribution of elementary time constants over the electrode surface. However, it will allow us to focus only on the effects which are controlled by microscopic distribution of ohmic drop only. Finally, we consider the voltammetry for a simple redox couple with formal potential E0, and that the microelectrode potential is swept from a value, Emin, corresponding to a negligible ET reaction rate to potentials where the voltammogram reaches its plateau. For simplicity we assume that only one species is present initially in the solution and that the diffusion coefficients of each member of the redox couple are equal. In this case the mathematical model of the process requires only the diffusion equation for one of the two species (here for the species generated at the electrode surface). Because of the symmetry of the system, in the real space the second Fick’s law is formulated in cylindrical coordinates. Then, a zero concentration condition applies at infinity (viz. at the reference electrode equipotential) and a zero flux condition at the insulator and the symmetry axis.4a,b The boundary condition at each point of the electrode surface is given by6
D
∂c ) -kf(c0 - c) + kbc ∂z
(1)
where c is the concentration of the species generated at the electrode surface due to the ET reaction, c0 is the bulk concentration of the electroactive species initially present in the solution, D is the (common) diffusion coefficient, z is the coordinate normal to the electrode surface, and kf and kb are the forward and backward rate constants of the ET reaction. For the case of oxidation, the latter is represented as6
[
kf ) k0 exp (1 - R)
[
kb ) k0 exp -R
nFE* RgT
nFE* RgT
]
]
(2a)
(2b)
where k0 is the standard rate constant; R is the transfer coefficient; n is the number of transferred electrons; F is the Faraday constant; Rg is the gas constant; T is the temperature. E* is the effective overpotential at each electrode surface point, i.e., which governs the local values of kf and kb. In the following, though we develop a general formulation, for the sake of simplicity and space restriction, we will present and thoroughly discuss only the results that correspond to rather fast electron transfer rate constants (viz., around 1 cm/s). However, this does not mean that we enforce a thermodynamic equilibrium for the two redox species at each (6) Bard, A.; Faulkner, L. Electrochemical Methods: Fundamentals and Applications, 2nd ed.; Wiley: New York, 2001.
Figure 1. Sketch of the system indicating parameters used in the resistance definition (eq 4) in (a) real and (b) transformed spaces. Asterisks in part b mark transformed equivalents of corresponding quantities shown in part a.
microelectrode location since we always maintain the general kinetic conditions given in eqs 2a and 2b. Since we do not consider any contribution related to the local capacitance charging, E* may differ from (E - E0), where E ) Emin + vt is the electrochemical potential applied to the electrode metallic phase vs the reference electrode and t is the time elapsed since the beginning of the voltammetric scan, only because of the local ohmic drop. This leads to recasting the habitual form of the overpotential (E - E0) into the following one: E* ) E - E0 - di(r) dRe(r)
(3)
where r is the radial distance from the electrode axis, di(r) is the current through the infinitesimally thin ring of radius r and width dr, and dRe(r) the resistance of the current tube supported by this ring and connecting it to the reference equipotential. Equation 3 gives the actual potential distribution (affected by the solution resistance) in response to the applied potential. It is clear from the formulation (eq 3) that, under present conditions, the actual potential distribution is not uniform along the electrode surface but is a function of the position at the disk surface since both current density and resistance vary with the radial coordinate. In order to implement the boundary condition (eqs 1-3) in computations it is then helpful to derive an expression for the last term in eq 3, which is the subject of the next section. Potential Distribution, Elementary Current, and Resistance. The elementary resistance distribution for a disk electrode is known.7 However, since in this work we need to recast it in the conformal space that will be used, we wish to reintroduce the problem from the physical definition of the resistance Re of any electrical conductor3 limited by two electrical equipotentials (here, one is the metallic electrode surface and the other is the reference electrode equipotential). Thus 1 ) Re
∫∫
s1
ds1 dl cl ds γ ds1
∫
(4)
this current line cl; ds is the elementary area of the cross-section (i.e., made along electrical isopotentials) of the current tube issued from the elementary surface ds1 at any value of l (l ) 0 on the electrode surface and l f ∞ when the reference equipotential is reached). See Figure 1 for the illustration of geometrical quantities involved in eq 4. In order to make eq 4 explicit, it is necessary to consider the electrical potential distribution in the system since this defines the isopotential and current lines involved in eq 4. Before proceeding further, we recall again that the electron transfer reaction is considered to occur under an excess of supporting electrolyte so that any electric migration effects are suppressed. Hence, the conductivity of the solution may be assumed constant everywhere. On the other hand, the electrical potential is established much faster (viz., at the velocity of light in the medium) than any diffusion associated one. This ensures that the isopotential and current lines are at steady state at any applied potential and remain invariable during any electrochemical experiment.8 Hence, the potential (V) distribution is independent of concentration distributions and is described by the Laplace’s equation: ∆V ) 0
(5)
with the following boundary conditions: r e rd, r > rd, r ) 0,
z ) 0,
V ) E;
z ) 0,
0 < z < ∞, r2 + z2 f ∞,
(disk electrode)
∂V ) 0; ∂z ∂V ) 0; ∂r V ) 0;
(insulator) (symmetry axis) (infinity)
(6a) (6b) (6c) (6d)
where rd is the disk radius and E is the applied potential. The solution of the problem (eqs 5-6) can be obtained using the quasi-conformal mapping defined by4a,b π ξ ( 2 ) R) π cos( η) 2 sin
where γ is the conductivity of the medium; cl is a current line issued from an elementary surface element of area ds1 located onto the microelectrode surface; dl is the differential length along (7) Oldham, K. Electrochem. Commun. 2004, 6, 210–214.
(7a)
(8) Newman, J. J. Electrochem. Soc. 1966, 113, 501–502. (9) Amatore, C.; Pflu ¨ ger, F. Organometallics 1990, 9, 2276–2282.
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7949
( π2 ξ) tan( π2 η)
Z ) cos
(7b)
where R ) r/rd and Z ) z/rd are the dimensionless cylindrical coordinates; ξ and η are the coordinates in the transformed space (Figure 1). Upon application of the transform (eqs 7a and 7b), the solution of eqs 5-6d can be written as4a,b V ) E[1 - η(R, Z)] 2 2E arcsin ) π 2 2 √Z + (1 + R) + √Z2 + (1 - R)2
[
]
∫
) 2πrd
∫R
) π rd
1
0
∫
lξ(ξ, 0)dξ dL Cl dS γ dS1
∫
cos
1
1
0
∫
( π2 ξ)dξ
( ) ∫ dL dS γ
π sin ξ 2
d
(9)
Cl
From the latter formula, one obtains the elementary resistance of the infinitesimal ring of average radius R and width dR (see Appendix II for the details):
dRe(R) )
√1 - R2 4rdγR dR
1
)
( )
π ξ dξ 2
2πrdγ sin
) dRe(ξ)
1
i ) -2πnFDc0rd
0
0 d
0
R dR
Z)0
( π2 ξ)dξ
sin
η)0
( ∂C ∂Z )
di(R) ) -2πnFDc0rd
( ∂R∂ξ ) + ( ∂Z∂ξ ) π π π ) 1 - sin ( ξ) cos ( η) π 2 2 2 cos( η)
R dR
( )
∂C π sin ξ dξ ∂η 2
2
(10)
Note that lξ(ξ, 0) ) (π/2) cos((π/2)ξ). The integral in the denominator of the integrand in eq 9 is equal to (see Appendix I for the details of the derivations)
7950
Combining eqs 13 and 15, one obtains the formula for the product of the differential current and differential resistance, i.e., of the ohmic drop of the current tube: di(R) dRe(R) ) di(ξ) dRe(ξ)
2
Cl
(15)
2
2
∫
(14)
Z)0
) di(ξ) where R, dR, dL, dS, dS1, and Cl are the normalized values of the corresponding “lower case” quantities; lξ is the metric coefficient given by 2
(13)
When expressed in dimensionless cylindrical coordinates, this expression of the elementary resistance is the same as that given in ref 7, although it was derived here in the conformal space of Figure 1b. The ionic current which flows into the system is ultimately commanded by that which evolves at the electrode surface because of the Faradaic reaction. The expression for the current is then the classical one:4a,b,6,7
) -2πnFDc0rd
dS1
lξ(ξ, η) )
(12)
where C ) c/c0 is the dimensionless concentration of the species generated at the electrode. This is true for any elementary current tube issued from the electrode surface, and by definition of a current tube, one has (∂ C/∂ η) ) (∂ C/∂ η)η ) 0 at any η value owing to the orthogonality of current lines and equipotentials in the transformed space in Figure 1b. Hence, the elementary current passing through the infinitesimally thin ring located on the disk surface is
dR dL Cl dS γ dS1
0
1
0
1
∫ R(ξ, 0)
) 2πrd
2
∫
0
∫ sin( π2 ξ)dξ ) 4r γ
∫ ( ∂C ∂Z ) ∂C ) -2πnFDc r ∫ ( ) ∂η
dr r dl cl ds γ ds1
rd
1 ) 2πrdγ Re
(8)
The elegant view of the solution is ensured by the intrinsic properties of the quasi-conformal mapping, namely, it transforms the half-infinite space onto the unit square in such a way that isopotential lines correspond to straight horizontal lines and current lines are mapped onto vertical straight lines in the transformed space.4a,b Hence any current tube is represented by a rectangle of the unit height in the quasi-conformal space (see Figure 1), the fact which greatly facilitates the evaluation of the integral (eq 4). Indeed eq 4 can now be written in the (ξ,η)-space (after the normalization of all distance quantities by the disk radius and area quantities by the squared radius) as 1 ) 2π Re
Substituting eq 11 into eq 9, we obtain the overall resistance of the disk electrode-electrolyte system:8
dL π 1π cos ξ ) γ2 2 dS γ dS1
( )
π nFDc0 ∂C √1 - R2 2 γ ∂Z nFDc0 ∂C )γ ∂η )-
Thus the overpotential in eq 3 can then be explicitly rewritten in transformed coordinates as
(11)
Analytical Chemistry, Vol. 80, No. 21, November 1, 2008
(16)
E* ) E - E0 +
nFDc0 ∂C γ ∂η
(17)
The initial and boundary conditions are (τ ) 0):
or in the dimensionless form
θ* ) θ + ν
∂C ∂η
0 e ξ e 1,
(18)
C)0
0 e η e 1,
(22a)
(τ > 0):
where θ ) [nF(E - E )]/(RgT) and ν ) [(nF)/(RgT)][(nFDc0)/ γ]. Equation 16 shows that the resistive effect depends on solution properties (parameter ν) and concentration gradient in the transformed space. Since ∂C/∂η is expected to vary with ξ, the expressions in eqs 17 and 18 show that the potential distribution varies with the radial coordinate except when ∂C/∂η is constant with ξ. Interestingly, this peculiar situation occurs when the microelectrode performs under the pure steady state regime provided that the rate of electron transfer is fast enough for a Nernstian condition to apply at each electrode point. Indeed, under steady state conditions, viz., when ∂c/∂t ) 0, we will establish below that irrespective of the degree of ohmic drop the solution of the diffusional problem is C(ξ,η) ) C(ξ,0)(1-η) with C(ξ,0) ) Γ being constant at a given potential. Hence, ∂C/∂η ) -Γ so that from eq 14, ∂C/∂η ) -i/(4nFD c0rd) at each point of the solution. From eq 17 it ensues then that at steady state: 0
E* ) E - E0 - i ⁄ (4rdγ) ) E - E0 - iRe
(19)
is independent of ξ which is tantamount to say that the ohmic drop across the electrode is the same and equal to the product of the overall current by the overall resistance. This peculiar behavior justifies the usual practice for correcting a steady state voltammogram affected by ohmic drop by plotting the experimental i/(E) current-potential curve as i/(E - iRe) (see, e.g., a full description of the method in ref 9). However, this is a very particular situation in which the microscopic view and the macroscopic ones afford the same result. When steady state is not fully achieved, ∂C/∂η is not expected to be constant across the electrode surface and needs to be evaluated by fully solving the coupling of the diffusional problem. Coupling with Diffusion: A Mathematical Model in Transformed Coordinates. The resulting mathematical model of the process at hand has the following expression in the transformed space:4a,b
[
]
∂C ∂2C π ∂2C ∂C ) ∆ξ,η + cos2 η + Fξ,η 2 ∂τ 2 ∂η2 ∂ξ ∂ξ
( )
(20)
where τ ) Dt/rd2 is the dimensionless time; ∆ξ,η and Fξ,η are metric coefficients that are given by
[ ] [ ] [ ]
π cos2 η 4 2 ∆ξ,η ) 2 π 1 - sin2 π ξ cos2 π η 2 2
2 Fξ,η ) π
[ π2 ξ]
(21a)
cot
[ ]
[ ]
π π sec2 η - sin2 ξ 2 2
(21b)
0 e ξ e 1,
η ) 0,
(hη ⁄ Khet)
∂C ) exp[-Rθ*]C ∂η
exp[(1 - R)θ*](1 - C); ξ ) 0,
0 < η < 1,
ξ ) 1,
∂C ) 0; ∂ξ
0 < η < 1,
0 e ξ e 1,
η ) 1,
(disk electrode) (22b) (symmetry axis)
∂C ) 0; ∂ξ
(22c)
(insulator)
(22d)
(infinity)
(22e)
C ) 0;
where hη(ξ, 0) ) (2/π) sec(πξ/2) is a metric coefficient; Khet ) k0 rd/D is the dimensionless heterogeneous rate constant. Interestingly, eq 22b is nonlinear and therefore is expected to introduce nonlinear effects. Yet, this effect disappears under steady conditions, and when the rate of ET is sufficient for (hη/Khet)(∂C/ ∂η) f 0. The case of a perfect steady state under a Nernstian regime (viz., when (hη/Khet)(∂C/∂η) f 0) is examined in the next section to mathematically justify the above considerations, which led us to derive eq 19. Steady State Regimes. When full steady state is achieved, eq 20 is rewritten as
[
0 ) ∆ξ,η
]
π ∂2C ∂2C ∂C + cos2 η + Fξ,η 2 2 ∂η2 ∂ξ ∂ξ
( )
(23)
This equation has an obvious solution (C ) aη + b, where a and b are constants), which respects all boundary conditions in eqs 22c and 22d, viz., ∂C/∂ξ ) 0 and ∂2C/∂η2 ) 0. When (hη/ Khet) f 0, this solution is compatible with eq 22b when (hη/ Khet)(∂C/∂η) f 0 since this becomes C(ξ, 0) ) 1/[1 + exp(-θ*)]. Hence, upon considering the boundary condition in eq 22e, one immediately obtains C(ξ,η) ) Γ(1-η) and ∂C/∂η ) -Γ, where Γ is constant across the electrode surface. These were the key results which led us to establish eq 19 and therefore validate the classical views about ohmic drop distortion of steady state voltammograms at microdisk electrodes. Nonsteady State Regimes. Because of the nonlinearity of eq 22b, to the best of our knowledge it seems impossible to achieve an analytical derivation of the problem at hand as soon as the system departs from a pure steady state regime. Hence we will resort to simulations and take advantage of the great simplification of all numerical difficulties when the simulations are performed in the conformal space defined in eqs 7a,b and shown in Figure 1b. Though, for keeping an extended generality to the results of simulations, one had better perform them using dimensionless formulations. Thus, in addition to all dimensionless values introduced above, we define the dimensionless scan rate, σ ) nFvrd2 ⁄ RgTD
(24)
the dimensionless current, Analytical Chemistry, Vol. 80, No. 21, November 1, 2008
7951
f(θ) )
i(E) 4nFDc0rd
(25)
and the dimensionless current density normalized with respect to the disk steady-state current and disk surface area,
J(R) ) -
(
nFDc0 ∂C rd ∂Z
)(
)
πrd2 π ∂C π ∂C ) - hη )4nFDc0rd 4 ∂Z 4 ∂η (26)
Finally, though we have established above that a simulation approach is not necessary for the pure steady state conditions, we will also examine it in the following. Indeed, this is in this case where the simulations are the most delicate owing to the fact that in the real space the current density at the electrode edge is infinite. Therefore, though not necessary, such simulations will serve as an internal test for the accuracy and precision of our numerical solutions. DETAILS OF NUMERICAL SIMULATIONS The problem defined by eqs 20 and 22 was solved numerically using the alternating direction implicit (ADI) method coupled with the Newton’s method for the solution of nonlinear algebraic equation systems. Because of splitting iterations in the spatial directions in the ADI method and the form of the boundary conditions (eqs 22a-22e), the resulting algebraic systems are linear across the ξ-direction (i.e., for implicit discretization of ξ-derivatives only) and nonlinear across the η-direction (i.e., for implicit discretization of η-derivatives only). At each time step, the resulting algebraic system along the η-direction was iteratively solved by means of the Newton’s method until the concentration field and its gradient at the electrode surface satisfied eq 22b. Uniform time and spatial (in transformed space)4a,b grids were used for computations. A typical grid was Nξ × Nη ) 100 × 100 with 1000 time steps for each volt (or its dimensionless equivalent) that ensured the numerical error in peak current of less than 0.3% for all considered cases. RESULTS AND DISCUSSIONS FOR FAST ET It is necessary to emphasize that from the derived result in eq 17, it is easily seen that the resistance effect depends on the magnitude of the term (nDC0)/γ which is particular to each given experiment. The equivalent dimensionless quantity is ν ) (n2F2D c0)/ (γ RgT). The larger the quantity, the larger the resistance effect observed in the current response. Note also that when the ν value is close to zero (i.e., when either γ f ∞ or c0 f 0), the last term in eq 17 vanishes and classical voltammetric responses should be observed. Finally, we will pursue here upon considering that the rate of electron transfer is fast. This amounts to consider that (hη/Khet)(∂C/∂η) f 0. However, we keep the general formulation in eq 22b without imposing (hη/Khet) ) 0. As mentioned above, the simulation results will be presented in the dimensionless form. However in some instances we shall give the corresponding dimensioned values in order to give a feeling of the “real” situations. For this particular sake, we will thus consider the following real parameters, n ) 1, D ) 10-5 cm2/ s, rd ) 10 µm, k0 ) 1 cm/s, R ) 0.5, which represent a typical fast system and an usual microdisk electrode; for simplification, 7952
Analytical Chemistry, Vol. 80, No. 21, November 1, 2008
Figure 2. Computed steady-state voltammograms accounting for the uncompensated solution resistance. Curves correspond to (1) ν ) 0 (no resistance); (2) ν ) 3.76; (3) ν ) 9.40; (4) ν ) 18.80; and (5) ν ) 37.59. Symbols denote data obtained through eq 27 for ν ) 37.59.
we posit that E0 ) 0 V and other parameters (such as γ, c0, and ν) will be given in the text below where appropriate. Steady State Regime. Figure 2 shows computed steady-state dimensionless voltammograms obtained for different ν values. The leftmost curve is the classical voltammogram which is observed when the solution resistance is negligible (i.e., ν ) 0 which corresponds to infinite solution conductivity). All others from the left to right correspond to increasingly resistive conditions according to the following ν values: 3.76, 9.40, 18.80, 37.59. These ν values for the concentration c0 ) 1 mM and the set of parameters specified above translate into the following conductivity values: 10-5, 4 × 10-6, 2 × 10-6, 10-6 Ω-1 cm-1. Note that in the case when other parameters are specified, ν depends on the c0/γ ratio only, which means that curves given in Figure 2 correspond to a variety of c0 and γ values linked by the same ratio value. It is clear from the results given in Figure 2 that the more resistive the solution the more sluggish is the voltammogram. This corresponds to the usual observation for steady state voltammetry under very resistive conditions. More quantitatively, as predicted by eq 19, each voltammogram in Figure 2 derives from that without ohmic drop distortion (curve labeled 1) by simply adding iRe to the potential scale. Indeed, in the case of Nernstian ET under steady-state, the voltammetric wave may be obtained simply by solving the following transcendental equation with respect to the dimensionless current f: f(θ) )
1 1 + exp[-(θ - νf(θ))]
(27)
which follows from the discussion in Steady State Regimes subsection. This procedure is exemplified in Figure 2 by the symbols for ν ) 37.59. The relative difference between the numerically computed results and results of this analytical procedure is less than 0.2%. Note, however, that this slight but reliable difference is conditioned by the fact that in our simulations the value of the rate constant (k0 ) 1 cm/ s) though high was not sufficient to account for the high current density at the electrode edge (see below). Hence, in this specific region, simulation results differed systematically from the predictions
Figure 3. Dimensionless current density distribution along the radial coordinate. Distributions correspond to specified values of the dimensionless current.
of eq 27, which considers purely Nernstian conditions over the whole electrode (i.e., k0 f ∞). Accordingly, it was checked that increasing the rate constant to the value k0 ) 100 cm/s (data not shown) led to a decrease of this slight discrepancy, the relative error being then less than 0.04%. Figure 3 displays the variations of the current density for each situation considered in Figure 2 when the overall current equals 10%, 50%, and 90% of the steady-state limiting current, viz., when f(θ) is 0.1, 0.5, or 0.9. A striking feature is that for each f(θ) value, all the J(R) curves obtained when ν varies are undistinguishable. Their relative difference is less than ∼2.5% except near the very edge of the electrode (0.9 e R e 1). Indeed, as explained above, near the electrode edge the current density grows limitlessly so that the finite (Khet/hη) values used in our simulations may not be sufficiently large for a Nernstian condition to apply here. Hence, the nonlinearity of eq 22b introduces a systematic slight distortion which increases when ν is decreased. Anyway, the plot in Figure 3 confirms that under steady-state voltammetry, the current density is independent of the c0/γ ratio and demonstrates that even under these harshest conditions our numerical approach is both extremely precise and accurate. Nonsteady State Regimes. Figure 4 depicts a series of transient current-potential curves simulated for a range of parameter ν (see Figure 4 caption for the values). The dimensionless scan rates are σ ) 38.96 (Figure 4a) and σ ) 389.63 (Figure 4b), which is equivalent to v ) 10 V/s and v ) 100 V/s, respectively, for the fixed earlier set of real parameters. For the largest values of ν, the overall resistance effect is similar to that observed in steady-state voltammetry, i.e., it reveals itself as increasing the sluggishness of voltammograms with increasing of solution resistance. This sluggishness leads, of course, to flattening of the current peak. However, as expected each voltammetric curve converges toward the same diffusion limit at high potentials. Indeed, when the potential increases, the ohmic drop limitation is not enough to prevent each point of the electrode to reach an effective potential located on the wave plateau. Yet this occurs at larger and larger potentials when the overall cell resistance increases.
Figure 4. Computed transient voltammograms accounting for the uncompensated solution resistance for (1) ν ) 0 (no resistance); (2) ν ) 0.38; (3) ν ) 3.76; (4) ν ) 15.04; (5) ν ) 37.59 obtained for σ ) 38.96 (a) and σ ) 389.63 (b). The b in part b indicate the potential (time) range shown in Figure 5.
Figure 5. Current density distributions for ν ) 37.59 and σ ) 389.63 for the time range comprised between τ ) 0.31 and τ ) 0.39.
When the resistance is extremely large (curves 3-5 in Figure 4), a peculiar feature is observed beyond the above one. Indeed, it is observed that the current-potential curve before the peak is almost Analytical Chemistry, Vol. 80, No. 21, November 1, 2008
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linear as soon as the current departs from the very foot of the wave. This is coupled with the fact that all these linear segments have the same potential intercept. This in fact features a purely ohmic behavior, i.e., i ≈ E/Re, which onset corresponds to that of the Faradaic process. Indeed, under such highly resistive conditions, the Faradaic reaction acts in a diodelike fashion. Any significant current cannot flow before the electrode potential allows it. However, when this condition is fulfilled, the Faradaic impedance is too small compared to the resistance of the solution to effectively control the current flow. This is then essentially regulated by the solution resistance as for a diode in series with a large resistance. This occurs up to when the Faradaic impedance remains smaller than the cell resistance. Yet, when the Faradaic reaction proceeds along with time, the diffusion depletion of the solution makes the Faradaic impedance become increasingly large, so it eventually exceeds the cell resistance. When this range is reached, the Faradaic reaction regains control of the current flow. This change of behavior (viz., from a diodelike behavior to a classical Faradaic one) is reflected by the presence of a current peak with a strange appearance compared to usual voltammetric peaks. Evidently, the above rationale of the effects noted is based on a macroscopic view of the phenomena which occur truly at a microscopic level. Yet, it is evident that this transition occurs at different voltammetric times (i.e., at different potentials) for each current tube linking each microscopic domain of the microelectrode to the reference equipotential, so that one has truly an equivalent circuit consisting of an infinite number of parallel branches containing each a microscopic diodelike system in series with a microscopic resistance whose value varies depending on the position of the active infinitesimal element on the electrode surface (compare eq 13). It is then expected that one should observe a series of microscopic transitions between the diodelike behavior toward a classical Faradaic one occurring across the electrode at different times. This microscopic behavior is precisely observed when the current density distribution across the electrode is plotted as a function of time for the range of potential in which the “voltammetric peak” is observed. This is performed in Figure 5 for the voltammogram labeled 5 in Figure 4b for a series of fixed times, viz., for a series of potentials distributed over the range marked by two b on this curve. The microscopic control of current transitions is clearly evidenced by the uneven current density variations across the microelectrode. Figure 5 clearly reveals the redistribution (seen as a moving wave) of the current density along the disk surface just after the current peak. In fact, this series of snapshots for the current density shows that beyond a certain potential, the flux tubes at the edge of the electrode are not able to conduct the required (by the electrode polarization) amount of material, forcing the “excess” of the active material to spill toward the center of the electrode. When considering the current density distribution across the electrode as in Figure 5, this appears as a sharp density wave moving from the edge to center of the electrode. Examining the current density wave depicted in Figure 5, one may be surprised by the sharpness of this transition even though the solution resistance is substantial in this case. In order to 7954
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Figure 6. Distributions of the resistance-induced potential correction di(R) dRe(R) ) (E - E0) - E* as a function of time and radial position on the disk surface. Distributions correspond to those in Figure 5.
rationalize this peculiar observation let us consider the distribution of the resistance-induced potential correction di(R) dRe(R) ) (E - E0) - E* shown in Figure 6 for the same time points as in Figure 5. We note here that the smaller the di(R) dRe(R) value the higher the actual potential (E*) of the electrode (see eq 3) for a given time (viz., a given value of (E - E0)). It is clear from Figure 6 that the effective potential distribution across the microelectrode also experiences a “wave-type” behavior within the same time window, yet, the propagating wave is both smoother and more illustrative of the microscopic behaviors which occur both across the electrode and along time. This is an expected issue owing to the progressive change of control (i.e., between the diodelike behavior to a diffusion-limited Faradaic control) as discussed in macroscopic terms above. Since the elementary resistances are smaller near the microelectrode edge, these sections experience the transition sooner than the central ones. This leads to the appearance of a smooth step increase of ohmic drop when the distance from the electrode axis decreases. Eventually this reaches the center of the microelectrode and collapses. Beyond this point, the system soon regains a classical diffusion-limited behavior over the whole electrode surface as evidenced by the current potential curve 5 in Figure 4b beyond the upper potential marker. This expected smooth behavior acquires a more sharp form when reflected in the current density distribution as in Figure 5. This comparatively sharp redistribution of the current density becomes clear after recalling that the ET rate exponentially depends on the potential (eqs 2a and 2b), i.e., the smooth potential transition becomes exponentially modulated, leading to the sharp wave in Figure 5. In agreement with this view, it is necessary to note that such sharp wave redistribution is observed only at sufficiently high scan rates. At lower v (viz., at lower σ), the redistribution of the current density is much more smooth and the propagating wave has a much lower amplitude. Finally, to conclude this section, we must recall that in this work we have thoroughly neglected the possibility that under the high resistive conditions considered the observed phenomena may also be limited by time-constants. In practice, the time-constant will smooth the phenomena described above but will not change the general principle of what occurs at the microscopic level as it
has been described above. Indeed, since from eq 13, the elementary resistance is smaller near the electrode edge and the local time constant will be proportionally smaller also. In other words, the local infinitesimal capacitances of the electrode will be charged sooner near the microelectrode edge than near its center. This will then make the outer zones of the microelectrode perform more quickly at high potentials than those near the center, in a similar way as that observed in Figure 6.
ACKNOWLEDGMENT The authors thank Dr. S. W. Feldberg for initiating our interest to this problem. A.O. thanks the Ministry of Education and Science of Ukraine and INTAS for a postdoctoral stay in Paris by YSF (Grant 06-1000019-6451). In Paris, this work was supported in part by CNRS (UMR 8640 “PASTEUR” and LIA “XiamENS”), Ecole Normale Supe´rieure (ENS), Universite´ Pierre et Marie Curie (UPMC), and by the French Ministry of Research. I.S. thanks CNRS for the Research Director position in UMR 8640.
CONCLUSIONS
APPENDIX I Here we derive an explicit expression for the integral:
The present microscopic investigation of the effects of solution resistance in transient and steady-state voltammetry at a microdisk electrode has demonstrated that the usual macroscopic view neglects extremely important phenomena which occur under very resistive conditions. This microscopic investigation could be achieved, viz., formulated and solved numerically, thanks to the use of a specific quasi-conformal map. Using this quasi-conformal map allows the analytical treatment of the elementary ohmic drop di(R)dRe(R) at the microdisk surface as well as precise numerical simulation results. The results obtained show drastic changes in voltammograms when resistive effects are significant which is conditioned by nonuniform distributions of resistance and current density along the electrode surface. Transient voltammetry in essentially resistive media reveals specific microscopic features which are not inherent to the same processes in nonresistive media, being related to a progressive transition from a diodelike behavior toward a classical diffusion-limited one with elapsing time. Since this transition occurs microscopically at different times across the electrode, this is reflected by the presence of redistribution potential waves, hence of current density, around the voltammetric peak. Under steady state regimes and for sufficiently fast electron transfer rates one does not observe such behavior because the elementary resistance and the elementary current are exactly reciprocal to each other. Therefore the ohmic drop is essentially constant across the whole microelectrode surface at any point of a voltammetric wave. This seems to support the classical macroscopic view and the classical usage for compensating the effect of ohmic drop. However, we must stress that this unique property is achieved only when the electron transfer rate constants are large enough for a pure Nernstian control to be observed in the absence of cell resistance. Though this situation has not been examined here owing to our main purpose in this work, it is expected that when the electron transfer rate constants are smaller, the electron transfer kinetics will limit the current density near the electrode edge as has been established previously for the band electrode.10 Hence, the perfect reciprocity between the elementary resistance and the elementary current observed here cannot apply anymore. Therefore, it is expected that a microscopic distribution of ohmic drop will be observed whenever a perfect Nernstian regime is achievable across the whole microelectrode surface. A hint of such a microscopic distribution has been evidenced here by noting that even upon using high-electron transfer rate constants (viz., 1 cm s-1) our simulations (Figure 3) showed that upon increasing the solution resistivity, the distributions of current densities slightly varied by a few percent when approaching the electrode edge.
∫
Cl
dL dS γ dS1
(A1.1)
encountered in eq 9. The quantities dS and dS1 are obtained by virtue of the formula for the surface area of a revolution surface: dS ) 2π
∫
R1+dR
R1
R dλ ) 2π
∫
R1+dR
R1
R(ξ, ηfix)lξ(ξ, ηfix)dξ
(A1.2)
where dλ is the differential element of the fragment of the isopotential line enclosed in the current tube; R1 and (R1 + dR) are the radial coordinates defining the given isopotential line segment; lξ is the metric coefficient (Lame coefficient) given in eq 10; ηfix is a constant η value corresponding to the considered isopotential line. dS1 is also given by eq A1.2 but with ηfix ) 0. Transforming coordinates in eq A1.1 according to eqs 7a,b and substituting eq A1.2 into eq A1.1, one can rewrite the latter as follows:
∫
Cl
dL 1 ) γ dS γ dS1
∫
1
0
lη|ξ
(∫ ∫ R(ξ, η)l dξ ξ1+dξ
ξ1+dξ
ξ1
)
R(ξ, 0)lξ|η)0dξ dη
ξ
ξ1
(A1.3) where lη is a Lame coefficient given by
lη )
π π 1 - sin ( ξ) cos ( η) 2 2 ( )
π
2
2
π 2 cos η 2
(A1.4)
2
In the limit dξ f 0, the integrals in eq A1.3 may be replaced by the integrand values multiplied by the length of the integration interval, and the substitution of the expressions for the metric coefficient and R(ξ,η) into eq A1.3 yields π π 1 - sin ( ξ) cos ( η) 2 2 × ∫ 2
∫
dL π ) Cl 2 dS γ dS1
2
1
( π2 η) π π cos( ξ) cos ( η) 2 2 × dη π π γ 1 - sin ( ξ) cos ( η) 2 2 1 π π π 1π cos( ξ) (A1.5) ) ∫ cos( ξ)dη) 2 γ 2 γ2 2
0
cos2
2
2
2
1
0
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APPENDIX II The derivation of eq 13 can be accomplished as follows. Take the inverse of the eq 12 and replace the integral by its definition when dξ f 0:
Re )
1
∑ dR1
(A2.2)
e
Equation 13 follows immediately from the comparison of eqs A2.1 and A2.2. 1
Re ) 2πrdγ
NOTE ADDED AFTER ASAP PUBLICATION The paper was posted on the Web on 10/01/08. Equation 22b required minor corrections, two Roman v’s in text were corrected to italics, and an italic v in text was changed to a Greek nu. The paper was reposted on 10/08/08.
0
1
) 2πrdγ )
∫ sin( π2 ξ)dξ 1
∑ sin( π2 ξ)dξ 1
∑
(A2.1)
1 1
( )
π 2πrdγ sin ξ dξ 2 All differential ring elements act in parallel and the overall resistance of the system of parallel resistances is given by:
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Received for review May 20, 2008. Accepted August 16, 2008. AC8010268 (10) Amatore, C. A.; Deakin, M. R.; Wightman, R. M. J. Electroanal. Chem. 1986, 207, 23–36.
