Impedance of a Disk Electrode with Reactions Involving an Adsorbed

Journal of The Electrochemical Society, 156 共1兲 C28-C38 共2009兲

C28

0013-4651/2008/156共1兲/C28/11/$23.00 © The Electrochemical Society

Impedance of a Disk Electrode with Reactions Involving an Adsorbed Intermediate: Local and Global Analysis Shao-Ling Wu,a,* Mark E. Orazem,a,**,z Bernard Tribollet,b,*** and Vincent Vivierb,*** a

Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611, USA UPR 15 du CNRS, Laboratoire Interfaces et Systèmes Electrochimiques, Université Pierre et Marie Curie, 75252 Paris Cedex 05, France

b

The influence of nonuniform current and potential distributions associated with a disk electrode geometry on the impedance response was explored by numerical simulation for faradaic reactions coupled by an adsorbed intermediate. Consistent with the literature, the associated frequency or time-constant dispersion influenced the impedance response at frequencies above a characteristic value due to the frequency dependence of the radial distribution of the interfacial potential. The time-constant dispersion was also found to influence the impedance response at low frequencies due to the potential dependence of the fractional surface coverage of the adsorbed intermediate. The geometry effects were reflected in values for the local ohmic impedance, which had complex behavior at both high and low frequencies. The dispersion of time constant was described in terms of a local constantphase element 共CPE兲 that represented the impedance response at low frequencies as well as at high frequencies. A graphical method is described for finding values for the CPE exponent. © 2008 The Electrochemical Society. 关DOI: 10.1149/1.3009226兴 All rights reserved. Manuscript submitted June 16, 2008; revised manuscript received September 30, 2008. Published November 13, 2008. This was Paper 898 presented at the Phoenix, Arizona Meeting of the Society, May 18-22, 2008.

Impedance models are usually developed based on the assumption of a uniformly active electrode surface. Electrochemical systems, however, rarely show ideal behavior because the distributions of current and potential in the electrolyte adjacent to the electrode are constrained by the electrode geometry. Newman1,2 has analyzed the current distribution on a disk electrode embedded in an insulating plane. The primary current distribution, which considers only the ohmic potential drop in the electrolyte, shows an infinite current density at the edge of the disk electrode and half the value of average current density at the center of the disk. Nisancioglu and Newman3 have investigated the transient response of a disk electrode with a single faradaic reaction subject to a step change in applied current. The model did not account for mass-transfer effects, and the analytical solution to the Laplace’s equation was obtained using a transformation to rotational elliptic coordinates and a series expansion in terms of Legendre polynomials. Geometry-induced current and potential distributions cause a frequency dispersion, which may be considered to be a time-constant dispersion, that distorts the impedance response.4 This nonideal impedance response for a reactive system can be described in terms of a constant-phase element5 共CPE兲 as Z = Re +

Rt 1 + 共j␻兲␣QRt

关1兴

where Re is the electrolyte resistance, Rt is the charge-transfer resistance, j = 冑−1, ␻ is the angular frequency, and the parameters ␣ and Q are constant. When ␣ = 1, Q has units of capacitance, e.g., ␮F/cm2, and resembles a capacitor. Usually, the electrochemical interface of a real cell is not ideal and behaves like a CPE in which the exponent ␣ is between 0.5 and 1. The existence of CPE indicates a time-constant dispersion reflecting the variation of reactivity on the electrode surface. One of the causes of this dispersion may be a nonuniform current or potential distribution. The influence of electrode geometry on the impedance response has been studied numerically and experimentally by use of local electrochemical impedance spectroscopy 共LEIS兲 measurements on a disk electrode.6-9 Huang et al.6 showed that the relaxation of the potential distribution on a disk electrode gave rise to

* Electrochemical Society Student Member. ** Electrochemical Society Fellow. *** Electrochemical Society Active Member. z

E-mail: [email protected]

an ohmic impedance which had nonzero imaginary components at frequencies above a characteristic value. Jorcin et al.10 showed by use of LEIS that the geometry of a disk electrode made of a magnesium alloy can induce a CPE behavior, and this CPE behavior may be associated with a radial distribution of local resistance. Through numerical simulations, Huang et al. found that the global impedance response of a disk electrode in an insulating plane has quasi-CPE behavior at high frequencies for blocking electrodes6,7 and electrodes subject to a single faradaic reaction.8 The calculated local impedance and local ohmic impedance exhibited time-constant dispersion induced by the disk geometry, and the complex behavior of local ohmic impedance was found to exist only at high frequencies. Huang et al.8 showed that for single faradaic reactions, geometry effects play a role only at frequencies above a critical value, which is determined by the disk capacitance, disk radius, and electrolyte conductivity. The object of the present work is to investigate whether the geometry effects may also play a role at lower frequencies for more complicated reaction mechanisms that involve adsorbed intermediates on the electrode surface. Mathematical Development A mathematical development incorporating the kinetics of reactions involving an adsorbed intermediate and the potential distribution constrained by the electrode geometry is presented in this section. Reaction kinetics.— The electrode kinetics can be described by linear or Tafel kinetics. At a low surface overpotential, the reaction rate is linearly dependent on the surface overpotential. In this linear regime, the electrode current is associated with both the cathodic and anodic reactions; however, the electrode coverage by intermediate is not a function of surface overpotential. The surface coverage remains uniform along the electrode and depends only on parameters such as rate constants and transfer coefficients. Under the assumption of linear kinetics, inductive and capacitive loops are not evident at low frequencies. When the surface overpotential is large, the electrode behavior obeys Tafel kinetics and the surface concentration is a function of potential. Under these conditions, inductive loops may be observed at low frequencies. In order to investigate the influence of adsorbed intermediates on the impedance response, the electrode kinetics was assumed to be in the anodic Tafel regime. Two electrochemical reactions were considered which comprise successive charge-transfer steps involving an intermediate species adsorbed on the electrode surface, i.e.

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Journal of The Electrochemical Society, 156 共1兲 C28-C38 共2009兲 + M → Xads + e−

关2兴

+ → P2+ + e− Xads

关3兴

C29 ~ Φ =0

and

The reactant could be a metal M which dissolves through an ad+ , and then further reacts to form the final sorbed intermediate Xads 2+ product P . The reactions were assumed to be irreversible, and diffusion processes were considered negligible. Similar mechanisms were proposed by Epelboin and Keddam11 for calculating the impedance of iron dissolution through two steps involving an adsorbed FeOH intermediate, and by Péter et al.12 for the impedance model of the dissolution of aluminum in three consecutive steps with two adsorbed intermediates. Under the assumption of Tafel kinetics, the steady-state current densities for Reactions 2 and 3 are expressed by ¯ −⌽ ¯ 兲兴 ¯i = K 共1 − ¯␥兲exp关b 共V M M M 0

关4兴

¯ −⌽ ¯ 兲兴 ¯i = K ¯␥ exp关b 共V X X X 0

关5兴

and

where KM and KX are effective rate constants for Reactions 2 and 3 with units of current density, ␥ is the fractional surface coverage by + , V is the potential of the electrode, ⌽0 is the pointermediate Xads tential in the solution adjacent to the electrode, and bM and bX are ␣MF/RT and ␣XF/RT where ␣M and ␣X are transfer coefficients for Reactions 2 and 3, respectively. The bar notation on the variables represents the steady-state condition. The variation of surface coverage is given by ␥˙ =

d␥ iM − iX = dt ⌫F

关6兴

+ where ⌫ is the maximum surface concentration of Xads . The steadystate condition is defined by d␥/dt = 0; thus, the expression of steady-state surface coverage can be calculated from Eq. 4 and 5 as

¯␥ =

¯ −⌽ ¯ 兲兴 KM exp关bM共V 0 ¯ −⌽ ¯ 兲兴 + K exp关b 共V ¯ −⌽ ¯ 兲兴 KM exp关bM共V 0 X X 0

关7兴

The impedance responses for coupled reactions involving an intermediate adsorbed species under potentiostatic control have been given by Armstrong and Edmondson13 and Cao.14 Following the method developed by Epelboin et al.,15 the faradaic impedance ZF at a given potential is given by 1 1 A = + ZF Rt j␻ + B

关8兴

where the charge-transfer resistance Rt is defined by 1 1 1 = + = bM兩i¯M兩 + bX兩i¯X兩 Rt Rt,M Rt,X

关9兴

and parameters A and B are potential-dependent variables given by A=

共⳵¯iM + ⳵¯iX兲 ⳵␥˙ ⳵¯iF ⳵␥˙ = ⳵¯␥ ⳵¯V ⳵¯␥ ⳵¯V

ze(r)

关10兴

ze(r)

ze(r)

~ Φ 0 (r ) zF(r)

~ i (r)

zF(r) C0

~ i (r)

zF(r) C0

~ i (r)

zF(r) C0

~ i (r)

C0

~ V

Figure 1. Schematic representation of a 2D impedance distribution for a disk electrode where ze represents the local ohmic impedance, C0 represents the interfacial capacitance, and zF represents the local faradaic impedance.

B=−

¯ −⌽ ¯ 兲兴 + K exp关b 共V ¯ −⌽ ¯ 兲兴 ⳵␥˙ KX exp关bX共V 0 M M 0 = ⌫F ⳵¯␥ 关11兴

While B is always positive, the sign of A varies with the potential across the electrode–electrolyte interface, and the feature of the impedance plane changes according to the sign of A. When A is positive, the impedance plot in the Nyquist plane of ZF in parallel with the double-layer capacitance 共C0兲 shows a highfrequency capacitive loop corresponding to Rt in parallel with C0, i.e., Rt储C0, with a low-frequency inductive loop corresponding to the second term of Eq. 8. When A is negative, the impedance plot shows a high-frequency capacitive loop 共Rt储C0兲 with a low-frequency capacitive loop, also corresponding to the second term of Eq. 8. When A is equal to zero, the two terms of the numerator in Eq. 10 cancel, i.e., ⳵¯iF /⳵¯␥ = 0. In this case, the reaction current density is not dependent on the surface coverage, and therefore, the impedance plot shows a single capacitive loop corresponding to Rt储C0 with no lowfrequency loop. The nature of the electrode–electrolyte interface can be illustrated in terms of a two-dimensional 共2D兲 distribution of localized equivalent circuits, as shown in Fig. 1. Each local equivalent circuit corresponds to a local impedance at different radial positions on the electrode surface, and the different accessibility toward the electrode surface at different radial positions results in nonuniform current and potential distributions. The block in parallel with a constant capacitance C0 represents the local faradaic impedance zF, similar to that represented by Eq. 8, which accounts for reactions coupled by the adsorbed reaction intermediate. The local ohmic impedance was also represented by a block in order to reflect the complex character contributing to the local impedance response.6 Potential distribution.— The potential distribution implied by Fig. 1 results from the geometry of a disk electrode with radius r0 embedded in an insulating plane. The counter electrode was assumed to be placed infinitely far from the disk electrode. The potential in the solution can be solved by using Laplace’s equation in cylindrical coordinates, i.e. ⵜ 2⌽ = 0

−1 −1 ¯ −⌽ ¯ 兲兴 − K exp关b 共V ¯ −⌽ ¯ 兲兴其 − Rt,X 兲兵KX exp关bX共V 共Rt,M 0 M M 0 = ⌫F

and

ze(r)

关12兴

The system is assumed to have cylindrical symmetry such that the potential in solution is dependent only on the radial position 共r兲 along the electrode surface and the normal distance 共y兲. In response to an alternating current with a particular angular frequency ␻ 共␻ = 2␲f兲, the potential can be separated into steady and timedependent parts as



C30

¯ + Re兵⌽ ˜ ej␻t其 ⌽=⌽

关13兴

¯ is the steady-state solution for potential, and ⌽ ˜ is the where ⌽ complex oscillating component which is a function of position only. Similarly, the applied potential can be expressed as ˜ ej␻t其 V = ¯V + Re兵V

关14兴

Therefore, Laplace’s equation becomes

冉冊冉冊

˜ 1 ⳵ ⳵⌽ r r ⳵r ⳵r

+

˜ ⳵ 2⌽ ⳵y2

关15兴

=0

The boundary conditions at insulators and far from the electrode surface are given by

冏冏 ˜ ⳵⌽ ⳵y

=0

at r ⬎ r0

Table I. The values of kinetic parameters used for the simulations. Symbol

Meaning

Value

Units

bM bX KM KX ¯V

␣MF/RT ␣XF/RT Effective reaction constant for Reaction 2 Effective reaction constant for Reaction 3 Steady-state electrode potential

40 10 77.2 0.19

V−1 V−1 A cm−2 A cm−2

˜V

For 具A典 ⬎ 0 共0.011 ⍀−1 cm−2 s−1兲 For 具A典 = 0 共−6.6 ⫻ 10−4 ⍀−1 cm−2 s−1兲 For 具A典 ⬍ 0 共−0.83 ⍀−1 cm−2 s−1兲 Perturbation of electrode potential

−0.15 −0.10 0.10 0.01

V V V V

关16兴

y=0

J共r兲 = JM共r兲 + JX共r兲 =

and ˜ =0 ⌽

as r2 + y 2 → ⬁

关17兴

The current density at the electrode surface can be expressed as i = C0

冏冏

⳵共V − ⌽0兲 ⳵⌽ + iM + iX = − ␬ ⳵t ⳵y

关18兴 y=0

where C0 is the interfacial capacitance and ␬ is the electrolyte conductivity. The current at the electrode surface can be written by use of the reaction kinetics developed in Eq. 4 and 5, and expressed in frequency domain as

冏冏

˜ ˜ −⌽ ˜ 兲 + J 共V ˜ −⌽ ˜ 兲 + J 共V ˜ −⌽ ˜ 兲 = − r ⳵⌽ Kj共V 0 1 0 2 0 0 ⳵y

关19兴 y=0

where ˜V represents the imposed perturbation in electrode potential referenced to an electrode at infinity and K is the dimensionless frequency K=

␻C0r0 ␬

关20兴

The parameters J1 and J2 are dimensionless functions dependent on angular frequency and radial position on the electrode surface, i.e.

J1共r,␻兲 = JM共r兲 −

兩i¯M共r兲兩关JM共r兲 − JX共r兲兴 1 − ¯␥共r兲兩i¯M共r兲兩兩i¯X共r兲兩 ⌫Fj␻ + + ¯␥共r兲 1 − ¯␥共r兲

关21兴

and

J2共r,␻兲 = JX共r兲 +

兩i¯X共r兲兩关JM共r兲 − JX共r兲兴 ¯␥共r兲兩i¯X共r兲兩兩i¯M共r兲兩 + ⌫Fj␻ + ¯␥共r兲 1 − ¯␥共r兲

关22兴

The parameters JM and JX are defined to be dimensionless current densities for Reactions 2 and 3, respectively, and are functions of radial position on the electrode surface, as given by JM共r兲 =

bM兩i¯M共r兲兩r0 ␬

关23兴

JX共r兲 =

bX兩i¯X共r兲兩r0 ␬

关24兴

and

respectively. The sum of JM and JX represents the dimensionless current density which flows through the charge-transfer steps

r0 ¯ −⌽ ¯ 兲兴兵KMbM共1 − ¯␥兲exp关bM共V 0 ␬

¯ −⌽ ¯ 兲兴其 + KXbX¯␥ exp关bX共V 0

关25兴

The relationship between the parameter J and the charge-transfer and ohmic resistances can be established using the high-frequency limit for the ohmic resistance to a disk electrode obtained by Newman1 Re =

␲r0 4␬

关26兴

where Re has units of ⍀ cm2. The parameter J can therefore be expressed in terms of the ohmic resistance Re and charge-transfer resistance Rt as8 J=

4 Re ␲ Rt

关27兴

Large values of J are seen when the ohmic resistance is much larger than the charge-transfer resistance, and small values of J are seen when the charge-transfer resistance dominates. The definition of parameter J in Eq. 27 is the reciprocal of the Wagner number,16 which is a dimensionless quantity that measures the uniformity of the current distribution in an electrolytic cell. Simulations were performed to investigate the electrochemical impedance behavior for different potentiostatic situations when the surface-average value of 具A典 is positive, negative, and zero. The values of kinetic parameters used for the simulations are given in Table I. The values in Table I for bM and bX are close to the values of 38.4 and 7 V−1 reported by Keddam et al.17 for the dissolution of iron in acidic media. The calculated result for each simulation corresponds to a particular value of J, representing different contributions from the ohmic and the charge-transfer resistances. The equations were solved by using the finite-element package COMSOL Multiphysics with the conductive media dc module in a 2D axial symmetric coordinate system. A quarter-circle was constructed with an axis of symmetry at r = 0 and the electrode positioned at y = 0. The domain size shown in Fig. 2 was 2000 times larger than the disk electrode dimension in order to meet the assumption that the counter electrode was located infinitely far from the electrode surface. Results As indicated by Eq. 7, the steady-state fraction of surface cover¯ . While ¯V can be age varies with the interfacial potential ¯V − ⌽ 0 assumed uniform for a conductive electrode, the potential at the ¯ is a function of radial posioutside of the diffuse double layer ⌽ 0 ¯ ¯ tion. Thus, V − ⌽0 and ¯␥ are functions of radial position. The distribution of the normalized steady-state fractional surface coverage is presented in Fig. 3a and b for positive and negative average val¯典 ues of 具A典, respectively. The value of average surface coverage 具␥ is 0.069 for curve 1 and 0.98 for curve 9. The parameter A is itself



values of ¯V. The corresponding surface-averaged value of 具J典 to the applied potential at steady state was calculated from Eq. 25 and is given in Fig. 3d. As ¯V increases, the value of 具J典 increases and the sign of 具A典 changes from positive to negative. The sign of 具A典 determines the shape of low-frequency features of the impedance plot. In order to understand the impedance behavior under different potentiostatic conditions, three cases for 具A典 ⬎ 0, 具A典 = 0, and 具A典 ⬍ 0 共points 4, 5, and 6 in Fig. 3c兲 are discussed. The maximum variability of surface coverage was shown in Fig. 3a and b to occur for 具A典 = 0. Interestingly, the potential yielding the maximum variability of surface coverage does not coincide with the potential yielding the most nonuniform distribution of current or potential. The adsorption isotherm given in Fig. 4a shows an inflection at 具A典 = 0, representing the stronger dependence of the surface coverage on the interfacial potential, whereas at 具A典 = −0.83 ⍀−1 cm−2 s−1, the isotherm crosses the largest potential interval, indicating the most nonuniform potential distribution on the electrode surface. The distribution of current can also be observed in Fig. 4b, which shows a more nonuniform distribution of current when the applied potential is increased. For 具A典 = −0.83 ⍀−1 cm−2 s−1, the parameter J has the largest value, meaning, as shown in Eq. 27, that the ohmic resistance is much larger than the charge-transfer resistance. Hence, the current distribution is more nonuniform.18 The notation used in the present work follows that introduced in Table I of Huang et al.6 A global impedance, designated by an upper case Z, represents an averaged response of the electrode surface; the local impedance variables, designated by lower case z, are dependent on radial position along the electrode surface and are calculated from local current densities and potentials defined at different locations. A local interfacial impedance z0 involves a local potential drop ˜V-⌽ ˜ 共r兲 across the diffusion double layer; a local ohmic impedance 0 ˜ 共r兲 from the outer limit of the double z involves a potential drop ⌽

Figure 2. The domain used for the finite-element simulations. The solid lines represent steady-state isopotential planes, and the dashed lines represent steady-state trajectories for flow of current.

a function of radial position due to its dependence on interfacial potential. The range of the value of A from electrode center to electrode periphery is 0.0109 to 0.00509 ⍀−1 cm−2 s−1 for curve 4, 0.0106 to − 0.0665 ⍀−1 cm−2 s−1 for curve 5, and −0.109 to − 7.60 ⍀−1 cm−2 s−1 for curve 6. The variation for local value A along the electrode surface is larger as the applied potential is increased. The relationship between the surface-averaged values of 具A典 reported in Fig. 3a and b and the applied steady-state electrode potential is presented in Fig. 3c. The coverage by intermediate is most nonuniform at ¯V = −0.1 V, where the corresponding average value of 具A典 is equal to zero. It becomes more uniform at larger or smaller 1.3

(a)

4 5

1.0

1 2

0.9

3 4

‹A› / Ω-1cm-2s-1

1.1

5 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

r / r0

(b)

0

7 8 9 9

1.0

8 7 6

5 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

r / r0 9

6

7

8

6

10

-500

8

7

0.02

0.01

-1000

2

1

3

5

4 5

0.00

4

2

-0.01

-0.02 -0.30

-0.25

-0.20

-0.15

-0.10

1

-0.05

0.1

9

-0.2

Figure 3. Radial distribution of the normalized steady-state fractional surface coverage on a disk electrode: 共a兲 for positive surface-averaged values of 具A典; 共b兲 for negative surface-averaged values of 具A典; 共c兲 the relationship between the surface-averaged values of 具A典 reported in parts 共a兲 and 共b兲 and the applied steadystate electrode potential; and 共d兲 the corresponding surface-averaged values of 具J典 to the applied steady-state electrode potential.

3

1

-1500

-2000

6

0.9

0

(c)

1.2

‹γ›

3

γ /

1.1

2

e

˜ 共⬁兲 = 0, and a layer to an electrode located far from the electrode ⌽ ˜ local impedance z involves the electrode potential V with respect to

-1 -2 -1 ‹ A› = 0 Ω cm s (V = − 0.10 V) -1 ‹A› = − 0.83 Ω cm-2s-1 (V = 0.10 V) ‹A› = − 5.6 Ω-1cm-2s-1 (V = 0.25 V) ‹A› = − 44 Ω-1cm-2s-1 (V = 0.45 V) ‹A› = − 1900 Ω-1cm-2s-1 (V = 0.95 V)

5

‹J›

γ /

‹γ›

1.2

1.3

‹A› = 0.0012 Ω-1cm-2s-1 (V = − 0.28 V) ‹A› = 0.0031 Ω-1cm-2s-1 (V = − 0.25 V) ‹A› = 0.0083 Ω-1cm-2s-1 (V = − 0.20 V) ‹A› = 0.011 Ω-1cm-2s-1 (V = − 0.15 V) ‹A› = 0 Ω-1cm-2s-1 (V = − 0.10 V)

1

C31

0.0

0.2

0.4

V / Volts

0.6

0.8

1.0

-0.4

(d)

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

V / Volts



C32 1.0 0.8 0.7 0.6 0.5

0.08

0.4

‹A› = 0.011 Ω-1cm-2s-1 ‹J› = 2.0 ‹ γ › = 0.37

0.2 0.1 0.0 -0.30

-0.25

-0.20

-0.15

-5

K=10

0.24

0.28

0.32

0.04 0.02

0.40

0.44

0.04

‹A › = 0 K=10

K=1

0.02 0.00 0.24

0.26

0.28

0.30

0.32

0.34

0.36

0.38

Zrκ / r0π

(b)

0.08 0.06

0.36

Zrκ / r0π 0.06

0.05

‹A› = 0.011 Ω cm ‹J› = 2.0 ‹ γ › = 0.37 -1

-2 -1

s

‹A› = 0 ‹J› = 3.1 ‹ γ › = 0.48

0.00 -0.30

-0.25

0.04

-0.20

-0.15

-Zjκ / r0π

-2

0.00

-Zjκ / r0π

‹A› = − 0.83 Ω-1cm-2s-1 ‹J› = 7.8 ‹ γ › = 0.78

0.10

K=10

0.02

-0.10

0.14 0.12

K=1

(a)

(V - Φ0) / Volts

(a)

i / A cm

0.04

-0.02

0.3

(b)

‹A› = 0.011 Ω-1cm-2s-1

0.06

-Zjκ / r0π

0.9

γ

0.10

‹A› = − 0.83 Ω-1cm-2s-1 ‹J› = 7.8 ‹ γ › = 0.78 ‹A› = 0 ‹J› = 3.1 ‹ γ › = 0.48

0.03 0.02

K=10 -5

K=10

0.01

-0.10

0.00 0.24

K=1

0.26

0.28

Figure 4. The variation of 共a兲 steady-state surface coverage and 共b兲 steadystate current density with the interfacial potential. Dashed squares are used to identify the range of current and surface coverage corresponding to the simu¯V = −0.15 V lations performed at 共具A典 = 0.011 ⍀−1 cm−2 s−1兲, −1 −2 −1 ¯V = −0.1 V ¯V = 0.1 V 共具A典 = 0 ⍀ cm s 兲, and 共具A典 = −0.83 ⍀−1 cm−2 s−1兲. The position r = 0 corresponds to the lower-left corner of each box.

a distant electrode. The local impedance can be represented by the sum of the local interfacial impedance and the local ohmic impedance, i.e. 关28兴

The discussion presented below follows the influence of kinetic parameters on the global, local interfacial, local ohmic, and local impedances. Global impedance.— The global impedance represents an averaged response of the electrode. The calculated results of global impedance for the cases with 具A典 ⬎ 0 共0.011 ⍀−1 cm−2 s−1兲, 具A典 = 0, and 具A典 ⬍ 0 共−0.83 ⍀−1 cm−2 s−1兲 are presented in Nyquist format in Fig. 5a-c, respectively. The impedance is made dimensionless according to Z␬/r0␲ in which the unit of impedance Z is scaled by unit area 共⍀ cm2兲. The solid lines in Fig. 5 represent the simulation results by solving Laplace’s equation, coupling the boundary condi-

0.30

0.32

0.34

Zrκ / r0π

(c)

(V - Φ0) / Volts

z = z0 + ze

‹A› = − 0.83 Ω-1cm-2s-1

Figure 5. Calculated Nyquist representation of the global impedance response for a disk electrode considering the influence of electrode geometry 共solid lines兲 and in the absence of geometry effect 共dashed lines兲: 共a兲具A典 ⬎ 0 共0.011 ⍀−1 cm−2 s−1兲; 共b兲具A典 = 0; and 共c兲具A典 ⬍ 0 共−0.83 ⍀−1 cm−2 s−1兲.

tions that account for the time-constant dispersion associated with the electrode geometry. The dashed curves represent the global impedances calculated by use of a mathematical expression Z = Re +

1 1/ZF + j␻C0

关29兴

associated with the ohmic resistance in solution and a combination of faradaic reaction and the electric double layer at the electrode. The ohmic resistance developed by Newman1 is given in Eq. 26. The faradaic impedance is calculated from Eq. 8 in terms of the surface-average parameters given from Eq. 9-11 and therefore did not account for the influence of electrode geometry. The geometry of the disk electrode is shown to distort the global impedance response. The geometry-induced distortion of impedance response and corresponding depressions of semicircles at high and low frequencies are more obvious in Fig. 5c where 具A典 ⬍ 0. The charge-transfer resistance Rt for the coupled reactions can be evaluated from the diameter of the high-frequency loop of the global impedance, and the low-frequency loop yields the resistance R␥ associated with the concentration of adsorbed species. A comparison



1.3

‹A › = 0

‹A › > 0

0.12

Rγ,eff/Rγ

1.2 1.1

0.16

‹A › < 0

Rt,eff/Rt

r/r0=0.96

0.08 0.04

K=10

Rγ,eff/Rγ

-5

K=10

-0.04 0.00

0.1

r/r0=0.8

r/r0=0.5

0.00

1.0 0.9

r/r0=0

K=1

-z0,jκ / r0π

Rt,eff/Rt or Rγ,eff/Rγ

1.4

C33

1

‹ J›

10

(a)

= −0.83 ⍀−1 cm−2 s−1兲.

Local interfacial impedance.— Nyquist plots for the calculated local interfacial impedance are presented in Fig. 7a-c for 具A典 ⬎ 0, 具A典 = 0, and 具A典 ⬎ 0, respectively, with normalized radial position r/r0 as a parameter. The impedance diagrams are superposed at high frequencies, showing that current flows mainly through the doublelayer capacitance, which was assumed to be uniform at the electrode surface. The shape of the low-frequency faradaic loop is dependent on the sign of parameter 具A典. The local interfacial impedance has a low-frequency inductive loop at all positions on the electrode when 具A典 ⬎ 0 and shows low-frequency capacitive features when 具A典 ⬍ 0. For 具A典 = 0, although only a single capacitive loop is observed in the global impedance 共Fig. 5b兲, two time constants are seen in the local interfacial impedance. At the periphery of the electrode, a lowfrequency capacitive loop is seen representing a local negative value of A. Near the electrode center, a low-frequency inductive loop is observed, indicating the local positive value of A. The different lowfrequency features seen at different radial positions demonstrates that the global impedance is an average representation of the electrode surface. For the three potentiostatic conditions, the values of the local interfacial impedance are larger at the electrode center and smaller

0.12

0.16

-z0,jκ / r0π

0.20

r/r0=0

0.08

r/r0=0.5

0.06

K=1

r/r0=0.8 r/r0=0.96

0.04 K=10

0.02 0.00 0.00

0.04

0.08

0.12

0.05

0.20

r/r0=0

0.04 0.03

r/r0=0.5 r/r0=0.8

K=10

r/r0=0.96

0.02 0.01 0.00

(c)

0.16

z0,rκ / r0π

(b)

-z0,jκ / r0π

Fig. 3d. The ratios Rt,eff /Rt and R␥,eff /R␥ approach unity as 具J典 → 0. The value of Rt,eff /Rt increases when 具J典 increases, which is in agreement with the result presented by Huang et al.8 that the influence of time-constant dispersion is greater when 具J典 is large. The value of R␥,eff /R␥ is largest for 具J典 close to 3.09 where 具A典 = 0. The surface coverage by the reaction intermediate has the greatest nonuniformity at 具A典 = 0, as shown in Fig. 3a and b, and hence a significant error in R␥,eff is seen when 具A典 approaches zero. The results for global impedance can be understood through examination of the local impedance distributions. In the following sections, the calculated results for local, local interfacial, and local ohmic impedance are presented and compared for three different potentiostatic conditions: ¯V = −0.15 V 共具A典 = 0.011 ⍀−1 cm−2 s−1兲, ¯V = −0.1 V 共具A典 = 0 ⍀−1 cm−2 s−1兲, and ¯V = 0.1 V 共具A典

0.08

z0,rκ / r0π

0.10

Figure 6. The value of Rt,eff /Rt and R␥,eff /R␥ evaluated from the global impedance as a function of 具J典.

between the effective kinetic parameters Rt,eff and R␥,eff, which account for electrode geometry, to the respective values that assume a uniform electrode, is presented in Fig. 6 as a function of 具J典. The relationship between 具J典 and the electrode potential ¯V is given in

0.04

-5

K=10

0.00

0.02

0.04

z0,rκ / r0π

0.06

0.08

Figure 7. Calculated Nyquist representation of the local interfacial impedance response of a disk electrode with normalized radial position r/r0 as a parameter: 共a兲具A典 ⬎ 0 共0.011 ⍀−1 cm−2 s−1兲; 共b兲具A典 = 0; and 共c兲具A典 ⬍ 0 共−0.83 ⍀−1 cm−2 s−1兲.

at the periphery, indicating a greater accessibility near the edges of the electrode. The applied electrode potential is the largest in the case 具A典 ⬍ 0, driving larger current densities through the electrode– electrolyte interface; thus, the interfacial impedance has the smallest value when 具A典 is negative. Local ohmic impedance.— The calculated local ohmic impedances for 具A典 ⬎ 0, 具A典 = 0, and 具A典 ⬍ 0 in Nyquist format are shown in Fig. 8a-c respectively, with normalized radial position as a parameter. As discussed by Huang et al.,6-8 the resistance in the electrolyte is not a pure resistance but acts as an impedance with complex features. The high-frequency loops at K ⬎ 10−2 are in agreement with the previous studies for a blocking electrode and a disk electrode subject to a single faradaic reaction, in which the geometry-induced current and potential distributions are observed.8



C34

-ze,jκ / r0π

0.1

r/r0=0 K=10

r/r0=0.5 r/r0=0.8

0.0

K=100

K=100 -0.1 0.0

r/r0=0.96

K=10 0.1

0.2

0.3

0.4

0.5

0.6

ze,rκ / r0π

(a)

-ze,jκ / r0π

0.1

r/r0=0 r/r0=0.5

K=10 0.0

-0.1 0.0

0.1

r/r0=0.8

K=100

K=100

0.2

0.3

0.4

0.5

r/r0=0.96 0.6

ze,rκ / r0π

(b)

-ze,jκ / r0π

0.1

r/r0=0.5 r/r0=0.8

0.0

K=100 -0.1 0.0

(c)

r/r0=0

ing in the local interfacial impedance. A high-frequency inductive loop is observed at all radial positions, and in addition, capacitive or inductive loops are observed at low frequencies. As it represents a summation of the local interfacial and local ohmic impedances, the high-frequency inductive behavior in the local impedance plot must be seen as well in the local ohmic impedance. The features at lower frequencies are strongly dependent on the radial position. The changes in sign of the imaginary part of the local impedance are evident in Fig. 11, where the absolute value of the imaginary part of the local impedance is presented as a function of the dimensionless frequency K. Changes in sign are evident in the frequency range 1 ⬍ K ⬍ 100, which accounts for the appearance of the highfrequency inductive loop in the Nyquist plot. This result is consistent with the results obtained by Huang et al.8 for a single faradaic reaction on a disk electrode. A second change in sign is observed at frequencies near 10−2 ⬍ K ⬍ 10−3, which can be attributed to the role of the adsorbed intermediate. For the case 具A典 ⬍ 0, yet another crossover is observed at even lower frequencies 共10−4 ⬍ K ⬍ 10−6兲. This effect is seen because the complex behavior of the local ohmic impedance is more significant when a higher electrode potential is applied, driving more current.

0.1

0.2

0.3

r/r0=0.96

K=100

K=100 0.4

0.5

0.6

ze,rκ / r0π

Figure 8. Calculated Nyquist representation of the local ohmic impedance response of a disk electrode with normalized radial position r/r0 as a parameter: 共a兲具A典 ⬎ 0 共0.011 ⍀−1 cm−2 s−1兲; 共b兲具A典 = 0; and 共c兲具A典 ⬍ 0 共−0.83 ⍀−1 cm−2 s−1兲.

In the present case, however, low-frequency loops can also be observed. The size of these low-frequency loops increases with applied potential, i.e., for 具A典 ⬍ 0. The dependence of the local ohmic impedance with frequency is shown more clearly in the representation of the real and imaginary components as given in Fig. 9. For the real component, the values at high-frequency limit are independent of 具A典, whereas at the lowfrequency limit, the difference of real values between the electrode center and the periphery is larger for 具A典 ⬍ 0. The nonzero values in the imaginary component of the impedance at 10−7 ⬍ K ⬍ 10−2 indicate the complex behavior at low frequencies which does not appear in the system of a single faradaic reaction without adsorbed reaction intermediates. Complex features are more significant at low frequencies and less significant at high frequencies when the applied potential to the electrode increases, but the ranges of dimensionless frequency where the complex ohmic impedance are observed 共10−7 ⬍ K ⬍ 10−2 and 10−2 ⬍ K ⬍ 102兲 are the same for all values of 具A典. According to the expression for the dimensionless frequency K in Eq. 20, the frequency that applies in the impedance measurement is given by f=

1 K␬ 2␲ r0C0

关30兴

For an electrochemical system with conductivity ␬ = 0.01 S cm−1 共corresponding to a 0.1 M NaCl solution兲 and a double-layer capacitance C0 = 10 ␮F cm−2 at a disk electrode with radius r0 = 0.1 cm, the frequency range corresponding to the calculated dimensionless values 10−7 ⬍ K ⬍ 10−2 is 0.16 mHz ⬍ f ⬍ 16 Hz, which is well within the range of typical experimental measurements. Local impedance.— The calculated local impedance is shown in Nyquist format in Fig. 10 with radial position as a parameter. The local impedance shows distortion from the ideal semicircle appear-

Validation of Calculations The potential distribution was solved numerically by use of the finite-element method. The calculation was verified by refining the mesh until the solution reached a stable value. The number of elements generated at the electrode boundary was 200, and the total number of elements in the domain was about 32,000. For dimensionless frequencies K ⬍ 100, the differences between solution potential adjacent to the electrode with different mesh densities were less than 0.001%. The calculated current flux for different mesh densities had differences less than 0.5%, and the calculated local impedances had differences less than 0.6%. The numerical method was also applied to solve the system with a single faradaic reaction in order to compare to the results obtained by Huang et al.,8 who used a collocation method. At K ⬍ 100, the differences between calculated local impedance obtained from the two numerical methods were less than 0.5%. The role of electrode geometry in creating the low-frequency dispersion reported here was also verified by examining the impedance response of a recessed electrode. Frateur et al.19 have demonstrated that a uniform primary distribution can be achieved by use of a recessed electrode with a depth twice the electrode radius 共 P = p/r0 = 2兲. The impedance response presented in Fig. 12 was obtained for a recessed electrode with P = 4. The local impedance response at different radial positions shows no dispersion along the electrode surface. In addition, ideal features were observed in the global impedance plots. The elimination of the geometry effect by use of a recessed electrode demonstrates that the complex behavior for the local ohmic impedance at both low and high frequencies cannot be attributed to calculation artifacts. Discussion Geometry-induced current and potential distributions were shown to have influence on the impedance response only at dimensionless frequency K ⬎ 1 for a blocking electrode4,6 and a disk electrode subject to a single faradaic reaction.7,8 The results of the present work showed that for a more complicated system with reactions associated with an adsorbed intermediate, the impedance response is affected by the geometry of the disk electrode at low frequencies as well as at high frequencies. The concept that the nonideal impedance caused by geometryinduced current and potential distributions could be expressed in terms of CPE behavior at high frequencies has been discussed by Huang et al.7,8 The parameter ␣ in the CPE expression was obtained in their work from the slope of the magnitude of the imaginary part of the global impedance plotted as a function of frequency in logarithmic scales. This approach worked well for the high-frequency


Journal of The Electrochemical Society, 156 共1兲 C28-C38 共2009兲 0.6

0.04 r/r0=0

r/r0=0.8

r/r0=0.5

r/r0=0.96

0.4

0.02

-ze,jκ / r0π

ze,rκ / r0π

0.5

C35

0.3 0.2 0.1

0.00 -0.02 -0.04

r/r0=0

-0.06

r/r0=0.8

r/r0=0.5 r/r0=0.96

0.0 -7 -6 -5 -4 -3 -2 -1 0 1 2 10 10 10 10 10 10 10 10 10 10

K

(a)

-0.08 -7 -6 -5 -4 -3 -2 -1 0 1 2 10 10 10 10 10 10 10 10 10 10

0.6

0.04 r/r0=0

r/r0=0.8

r/r0=0.5

r/r0=0.96

0.4

0.02

-ze,jκ / r0π

ze,rκ / r0π

0.5

K

(b)

0.3 0.2 0.1

0.00 -0.02 -0.04

r/r0=0

-0.06

r/r0=0.8

r/r0=0.5 r/r0=0.96

0.0 -7 -6 -5 -4 -3 -2 -1 0 1 2 10 10 10 10 10 10 10 10 10 10

K

(c)

-0.08 -7 -6 -5 -4 -3 -2 -1 0 1 2 10 10 10 10 10 10 10 10 10 10

0.6

0.04

r/r0=0

r/r0=0.8

r/r0=0.5

r/r0=0.96

0.4 0.3 0.2

(e)

0.02

-ze,jκ / r0π

ze,rκ / r0π

0.5

K

(d)

0.00 -0.02 -0.04

r/r0=0 r/r0=0.5 r/r0=0.8

0.1

-0.06

0.0 -7 -6 -5 -4 -3 -2 -1 0 1 2 10 10 10 10 10 10 10 10 10 10

-0.08 -7 -6 -5 -4 -3 -2 -1 0 1 2 10 10 10 10 10 10 10 10 10 10

K

(f)

r/r0=0.96

K

Figure 9. Calculated real and imaginary parts of the local ohmic impedance response of a disk electrode as a function of dimensionless frequency K: 共a兲 real part for 具A典 = 0.011 ⍀−1 cm−2 s−1; 共b兲 imaginary part for 具A典 = 0.011 ⍀−1 cm−2 s−1; 共c兲 real part for 具A典 = 0; 共d兲 imaginary part for 具A典 = 0; 共e兲 real part for 具A典 = −0.83 ⍀−1 cm−2 s−1; and 共f兲 imaginary part for 具A典 = −0.83 ⍀−1 cm−2 s−1.

behavior. In the present study, however, the slope for the lowfrequency inductive or capacitive loops could not be clearly resolved because the range of frequency was too short. The impedance data within this frequency range was influenced by the highfrequency loop which obscured the slope.

Another approach for graphical quantification for CPE behavior can be developed by exploring how the shape of a single impedance loop deviates from that of a perfect semicircle. The maximum magnitude of the imaginary part of impedance ⌬Zj and the difference



C36

-1

0.2

10

r/r0=0 r/r0=0.8

K=1

K=100 -5

K=10

K=10 0.1

0.2

0.3

0.4

0.5

0.6

zrκ / r0π

(a) 0.1

-zjκ / r0π

K=10

0.1

0.2

0.3

0.4

10

-5

0.5

-5

-0.1 0.0

K=1

r/r0=0.5

K=1

K=10

K=100 0.3

-4

10

-3

10

-2

10

-1

10

0

10

1

10

1

10

10

2

K

-2

10

0.4

r/r0=0.96

-4

0.5

0.6

zrκ / r0π

(c)

-5

10

r/r0=0

-5

K=10 K=10

0.2

-6

10

-1

r/r0=0.8

0.1

-7

10

10

0.0 K=100

r/r0=0.96

(a)

0.6

0.1 K=10

r/r0=0.8

-6

r/r0=0.96

zrκ / r0π

(b)

r/r0=0.5

r/r0=0.5

K=100 K=10

r/r0=0

-4

10 10

r/r0=0.8 K=100

-3

10

-5

r/r0=0

K=1

K=1

0.0

-0.1 0.0

-zjκ / r0π

| zjκ / r0π |

0.0

-0.1

-2

10

r/r0=0.96

| zjκ / r0π |

-zjκ / r0π

r/r0=0.5 K=1

0.1

Figure 10. Calculated Nyquist representation of the local impedance response of a disk electrode with normalized radial position r/r0 as a parameter: 共a兲具A典 ⬎ 0 共0.011 ⍀−1 cm−2 s−1兲; 共b兲具A典 = 0; and 共c兲具A典 ⬍ 0 共−0.83 ⍀−1 cm−2 s−1兲.

-3

10

r/r0=0

-4

10

r/r0=0.5 r/r0=0.8

-5

10

r/r0=0.96 -6

10

between high- and low-frequency asymptotes for the real part of the impedance ⌬Zr, shown in Fig. 13 for a single loop, correspond to the radius and the diameter for a semicircle, respectively. The absolute ratio of ⌬Zj and ⌬Zr is related to the CPE parameter ␣ by

冉冊

1 ␣␲ ⌬Zj = tan 2 ⌬Zr 4

冏冏冏冏冉冊 ⌬Zj ⌬Zr,hf

or

⌬Zj ␣␲ = tan ⌬Zr,lf 4

-5

10

-4

10

-3

10

-2

10

-1

10

high-frequency loops show depressed semicircles and the corresponding ␣ values are smaller than unity. In contrast to the results given by Huang et al.8 for a single faradaic reaction, the lowerfrequency half shows CPE behavior instead of an ideal impedance response because the characteristic frequency of the high-frequency

10

2

K -1

-2

10

-3

10

-4

10

r/r0=0 r/r0=0.5

-5

10

r/r0=0.8

关32兴

If the values of the two ratios are different, the impedance loop is not symmetric and does not correspond to a true CPE for which ␣ must be constant. The shape of the global impedance loops can be observed in Fig. 5. The calculated values of ␣ from Eq. 31 and 32 for the impedance loops are presented in Table II. For the three potentiostatic conditions, ¯V = −0.15 V 共具A典 = 0.011 ⍀−1 cm−2 s−1兲, ¯V = −0.1 V 共具A典 = 0 ⍀−1 cm−2 s−1兲, and ¯V = 0.1 V 共具A典 = −0.83 ⍀−1 cm−2 s−1兲, all

0

10

10

关31兴

If ␣ = 1, 兩⌬Zj /⌬Zr兩 = 0.5, which represents a perfect semicircle in the impedance plane. If an electrode exhibits a local CPE behavior, the value of 兩⌬Zj /⌬Zr兩 is less than 0.5 and a depressed semicircle in the impedance plot is observed. The distortion of the impedance response can be found by dividing the impedance loop into a higherfrequency half and a lower-frequency half, as shown in Fig. 13. The dependence of 兩⌬Zj /⌬Zr,hf兩 and 兩⌬Zj /⌬Zr,lf兩 to the parameter ␣ can be found from Eq. 31 to be

-6

10

(b)

| zjκ / r0π |

冏冏

-7

10

r/r0=0.96 -6

10 (c)

-7

10

-6

10

-5

10

-4

10

-3

10

-2

10

-1

10

0

10

1

10

2

10

K

Figure 11. Calculated representation of the imaginary component of the local impedance response on a disk electrode as a function of dimensionless frequency K: 共a兲具A典 ⬎ 0 共0.011 ⍀−1 cm−2 s−1兲; 共b兲具A典 = 0; and 共c兲具A典 ⬍ 0 共−0.83 ⍀−1 cm−2 s−1兲.

loop shifts to a higher value as the applied electrode potential increases and, thus, the CPE features induced by the electrode geometry can be seen in the lower-frequency half.



1.5

Local Interfacial Impedance

1.0

-zjκ / r0π

‹A › > 0

0.5

Local Impedance

Local Ohmic Impedance

0.0 -0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

zrκ / r0π

(a)

-zjκ / r0π

0.10

‹A›