Effect of uncompensated resistance on the cyclic voltammetric response of an electrochemically reversible surface-attached redox couple: Uniform current and potential across the electrode surface

Journal of Electroanalytical Chemistry 624 (2008) 45–51

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Effect of uncompensated resistance on the cyclic voltammetric response of an electrochemically reversible surface-attached redox couple: Uniform current and potential across the electrode surface Stephen W. Feldberg * Chemistry Department, Brookhaven National Laboratory, Upton, NY 11973, United States

a r t i c l e

i n f o

Article history: Received 29 April 2008 Received in revised form 15 July 2008 Accepted 19 July 2008 Available online 24 July 2008

a b s t r a c t The effect of uncompensated resistance, Ru, on the cyclic voltammetric response of an electrochemically reversible surface-attached redox couple is quantified using simulation. When current density and inter0 facial potential are uniform across the electrode surface, the shift in the peak potential, Epeak  E0 , the peak current, ipeak, and uncompensated resistance, Ru, can be correlated by a simple, general, empirical expression:

" #  0:764 Epeak  E00 2nF  00  : ffi 2  exp 2:61 Epeak  E  RT ipeak Ru

Keywords: Resistance Uncompensated Butler–Volmer Marcus–Hush Cyclic voltammetry Peak potential Peak current Simulation

0

0

When the value of (nF/RT)jEpeak  E0 j ) 0, (Epeak  E0 )/ipeakRu ) 2; when the value of (nF/ 0 0 RT)jEpeak  E0 j ) 1 the value of (Epeak  E0 )/ipeakRu ) 1.0; these limiting values are independent of the double layer capacitance, Cdl. For intermediate values this expression is adequately accurate as long as RTCdl/n2F2C total 6 0.1. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction The effect of uncompensated resistance, Ru, on the shape of a cyclic voltammogram has been thoroughly discussed for systems with soluble redox species and for electrode geometries which effect uniform current density and potential distribution over the electrode surface (see e.g., [1,2] and references therein). In the present work, I consider the effect of uncompensated resistance and double layer capacitance on the cyclic voltammetric response for an electrochemically reversible couple that is irreversibly attached to the electrode surface: z

Red Ox Oxzads þ ne ¼ Redads :

0022-0728/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2008.07.020

2. Analysis The ideal cyclic voltammetric response for a surface-attached redox couple, assuming reversible n-electron transfer and Ru = 0, is given by the well known expression (see e.g., [2]) with the added capacitive component [8]:

ð1Þ

Interest in the electrochemical behavior of surface-attached species was undoubtedly stimulated by Chidsey’s now classic 1991 paper [3] – the objective of that paper was the measurement of the potential dependence of the heterogeneous electron transfer rate constant over a wide potential range. A plethora of related studies has followed (see e.g., [4] and references therein). The computation of uncompensated-resistance effects becomes considerably more complicated when the uncompensated resistance is a function of the location on the electrode surface as is * Tel.: +1 631 344 4480; fax: +1 631 344 2887. E-mail address: [email protected]

the case with a small disk electrode [5,6]. The result is non-uniform distribution of the interfacial potential and current density. Svir and coworkers have addressed this more complicated problem [7]; it will not be addressed in the present work.

2 2

i¼

an F vCtotal RT

 nF ðEapp  E00 Þ RT   2 þ avC dl ; nF 1 þ exp ðEapp  E00 Þ RT exp



ð2Þ

where a (cm2) is the electrode area, v (=dEapp/dt) (V s1) is the scan rate, Eapp (V) is the potential applied to the working electrode rela0 tive to the reference electrode, E0 (V) is the formal potential for the redox reaction, Ctotal (=COx + CRed) (mol cm2) is the total coverage of the surface-attached redox species, Ox and Red, COx and CRed (mol cm2) are the individual surface coverages of Ox and Red, Cdl (farad cm2) is the double layer capacitance assumed to be constant over the potential range of interest, F, R, and T have their usual

46

S.W. Feldberg / Journal of Electroanalytical Chemistry 624 (2008) 45–51

Nomenclature electrode area, cm2 activity of species Ox (see Eq. (3)), mol cm2 activity of species Red (see Eq. (3)), mol cm2 distance between reference and working electrodes, cm double layer capacitance, farad cm2 potential applied to the working electrode relative to the reference electrode, V 0 DEapp Eapp  iRu  E0 (see Eq. (8)), V 0 formal potential, V E0 anodic or cathodic peak potential, V Epeak Epeak,anodic value of Eapp for anodic peak, V Epeak,cathodic value of Eapp for the cathodic peak, V DEpeak Epeak,anodicEpeak,cathodic, V 0 0 0 Epeak  E0 Epeak,anodic  E0 or Epeak,cathodic  E0 , V potential of zero charge, V Epzc DEpzc DEpzc = Eapp  iRu  Epzc (see Eq. (9)), V reversing potential for cyclic voltammetric scan, V Erev starting potential for cyclic voltammetric scan, V Estart e elementary charge (1.6022  1019 C), C F Faraday’s constant, 96485 C mol1 i current, A ipeak,anodic anodic peak current, A ipeak,cathodic cathodic peak current, A Didl change in capacitive current at Estart or Erev (see Eq. (15)) and accompanying discussion), A Dipeak ipeak,anodic  ipeak,cathodic, A Boltzmann’s constant, 1.38065  1023 J K1 kB ks standard rate constant for electron transfer associated with a surface-attached species, s1 a aOx aRed d Cdl Eapp

connotations, and n is the number of electrons transferred in the reversible redox reaction (a Glossary of the nomenclature is provided for the reader’s convenience). An implicit assumption in the derivation of Eq. (2) is that the activity of each surface-attached moiety is proportional to its surface coverage [2], i.e.:

aOx

COx

¼ cOx

and

aRed

CRed

¼ cRed ;

an2 F 2 vCtotal þ avC dl : 4RT

number of electrons transferred in a reversible electron transfer zox ox ; Redads Ox and Red redox moieties attached to the elecOxzads trode surface (see Eq. (1)). zox ox total charge associated with species Oxzads and Redads ,C qtotal uncompensated resistance, X Ru R gas constant, 8.3145 J K1 mol1 r0 radius of disk electrode, cm T temperature, K t time, s v cyclic voltammetric scan rate, V s1 Dv change in scan rate at Estart or Erev (see Eq. (15)) and accompanying discussion), V s1 zOx ; zRed charges on Ox and Red redox moieties (see Eq. (1)) a transfer coefficient (see Eqs. (34)–(38)) potential step increment in simulation, V dEapp COx concentration of surface-attached Ox, mol cm2 CRed concentration of surface-attached Red, mol cm2 Ctotal Ctotal = COx + CRed, mol cm2 cOx activity coefficient of surface-attached Ox moiety (see Eq. (3)). cRed activity coefficient of surface-attached Red moiety (see Eq. (3)) s time measured from either the time of initiation of the CV or the time of reversal, s sss time to achieve steady-state charging of the double layer capacitance (see Eqs. (15) and (16)), s x x = COx/CRed (see Eq. (7))

zOx x þ zRed þ aDEpzc C dl 1þx ðzRed þ nÞx þ zRed ¼ aF Ctotal þ aDEpzc C dl 1þx   nx ¼ aF Ctotal þ zRed þ aDEpzc C dl 1þx

qtotal ¼ aF Ctotal

ð3Þ

where aOx and aRed (mol cm2 ) are the activities of the surface-attached species and cOx and cRed are the activity coefficients. Eq. (2) is valid only if FjipeakjRu/RT is always small enough to ignore, e.g., FjipeakjRu/RT < 0.04 or equivalently, jipeakjRu < 0.001 V when 0 T  300 K; ipeak, the peak current which obtains when Eapp  E0 = 0, is defined by (see Eq. (2)):

ipeak ¼

n

or alternatively

zOx x þ zRed þ aDEpzc C dl 1þx zOx x þ zOx  n ¼ aF Ctotal þ aDEpzc C dl 1þx   n ¼ aF Ctotal zOx  þ aDEpzc C dl ; 1þx

qtotal ¼ aF Ctotal

ð4Þ

Of particular interest in the present work is the effect of the uncompensated resistance, Ru, on peak potentials. When Ru – 0 the actual interfacial potential will be Eapp  iRu. The value of Ru is assumed to be constant, independent of time and position on the electrode surface, so the current density and interfacial potential will necessarily be uniform over the entire electrode surface. The operative expression describing a system with reversible electron transfer and uncompensated resistance is easily deduced. The total charge, qtotal, associated with the surface-atzRed ox and Redads (see Eq. (1)) can be expressed tached redox species Oxzads as:

ð5Þ

ð6Þ

where

x¼









COx nF nF ¼ exp DEapp ; ðEapp  iRu  E00 Þ ¼ exp RT RT CRed

DEapp ¼ Eapp  iRu  E00 ;

ð7Þ ð8Þ

and analogously

DEpzc ¼ Eapp  iRu  Epzc ;

ð9Þ

where Epzc is the potential of zero charge. The derivative of x with respect to t is (see Eq. (7)):

dx nF x ¼ RT dt



m  Ru

 di : dt

ð10Þ

47

S.W. Feldberg / Journal of Electroanalytical Chemistry 624 (2008) 45–51

!

Note also (Eqs. (8) and (9)) that

dqtotal n2 F 2 dDEapp x dDEpzc Ctotal þ ¼a C dl dt RT dt ð1 þ xÞ2 dt !   2 2 di n F x ¼ a m  Ru Ctotal þ C dl dt RT ð1 þ xÞ2 0 1   nF  B 2 2 exp ðEapp  iRu  E00 Þ C di RT Bn F C C  ¼ a m  Ru B  2 þ C dl C: @ RT total A dt nF 00 1 þ exp ðEapp  iRu  E Þ RT

i¼

dDEapp dDEpzc dEapp di di ¼ ¼  R u ¼ m  Ru : dt dt dt dt dt

ð11Þ

The current is deduced by taking the derivative of qtotal (Eq. (6)) with respect to t:

ð12Þ The value of i is obtained numerically as described in the Appendix. The results can be discussed in terms of five dimensionless parameters: DEpeak =Dipeak Ru ð¼ ðEpeak  E00 Þ=ipeak Ru ÞÞ, (nF/RT)DEpeak, RTCdl/ n2F2 Ctotal, aFjvjRuCdl/RT and aF3jvjCtotalRu/(RT)2. The terms DEpeak (V) and Dipeak (A) are defined by:

2RT nF 2RT 00 P E þ 4ajvjRu C dl þ nF

2RT ðif Estart < E00 Þ; nF 2RT 6 E00  4ajvjRu C dl  ðif Estart > E00 Þ nF

Estart 6 E00  4ajvjRu C dl 

and Erev P E00 þ 4ajvjRu C dl þ

Estart

and Erev

ð17Þ

DEpeak ¼ Epeak;anodic  Epeak;cathodic ¼ 2ðEpeak;anodic  E00 Þ ¼ 2ðEpeak;cathodic  E00 Þ

ð13Þ

and

Dipeak ¼ ipeak;anodic  ipeak;cathodic :

ð14Þ

Simulations in Fig. 1 show the effects of increasing the uncompensated resistance and the interfacial capacitance as well as the sharpening effect of a reversible two-electron transfer. Simulation details

A

B

1e-6

5e-7

i (A)

i (A)

5e-7

1e-6

0 -5e-7

0 -5e-7

-3

-1e-6

Cdl = 1.0 x 10 farads n= 1 -1.0

-0.5

-1e-6 0.0

0.5

Cdl = 1.0 x 10-4 farads n= 1 -1.0

1.0

-0.5

E - E 0' (V)

C

D

2e-6

1e-6

0.5

1.0

0.5

1.0

4e-6 2e-6

0

i (A)

i (A)

0.0

E - E 0' (V)

0

-1e-6

-2e-6

Cdl = 1.0 x 10-3 farads -2e-6 n = 1

-4e-6 n = 2

-1.0

-0.5

Cdl = 1.0 x 10-3 farads

0.0

E - E 0' (V)

0.5

1.0

-1.0

-0.5

0.0

E - E 0' (V)

Fig. 1. Simulations of a surface-attached redox system with uncompensated resistance and double layer capacitance. For all panels: a = 1.0 cm2, Ctotal = 1.0  109 mol cm2, v = 0.001 V s1, Estart = 1.0 V, jdEapp j ¼ 1:0  104 V. For (A) Cdl = 1.0  105 farad cm2, n = 1, Ru = 0, 1.0  103, 1.0  104, 1.0  105, 1.0  106 X; (B) Cdl = 1.0  104 farad cm2, n = 1, Ru = 0, 1.0  103, 1.0  104, 1.0  105, 1.0  106 X; (C) Cdl = 1.0  103 farad cm2, n = 1, Ru = 0, 1.0  103, 1.0  104, 1.0  105, 1.0  106 X; (D) Cdl = 0, 1.0  103 farad cm2, n = 2, Ru = 0, 1.0  103, 1.0  104, 1.0  105, 1.0  106 X.

48

S.W. Feldberg / Journal of Electroanalytical Chemistry 624 (2008) 45–51

8e-7

b

6e-7

c

a

4e-7

i (A)

2e-7 0 -2e-7 -4e-7

a

c -6e-7

b

-8e-7 -0.50

-0.25

0.00

0.25

0.50

0'

E - E (V) Fig. 2. Comparison of quasireversibility produced by slow heterogeneous kinetics or by uncompensated resistance: For simulations all curves a = 1.0 cm2, jdEapp j ¼ 1:0  103 V, v = 0.001 V s1, Cdl = 1.0  105 farad cm2, n = 1 For curve (a): ks = 1.0 cm s1, a = 0.5 and Ru = 5  105 X; for curve (b): Butler–Volmer kinetics (see Appendix Eqs. (37) and (38)) with ks = 1.0  107 cm s1, a = 0.5 and Ru = 0; for curve (c) Marcus–Hush kinetics (see Appendix Eqs. (39) and (40)) with ks = 1.0  107 cm s1, k = 1.0 eV and Ru = 0.

are given in the figure legend. An important feature of the voltammograms in Fig. 1 is the time required for the capacitive current component, idl, to achieve its steady-state value (see e.g.,) [1,6]:

  Didl ¼ aC dl Dv 1  exp 

s aRu C dl

 ;

ð15Þ

where s is the time measured from either the time of initiation of the CV or the time of reversal and is the change in the scan rate at the time of initiation when Dv = v, or at the time of (first) reversal when Dv = v  (v) = 2v . The time required to achieve steady-state, sss, can be approximated by:

sss P 4aRu C dl :

ð16Þ

The values of Estart and Erev that will ensure that steady-state charging of the double layer capacitance is effectively achieved before the faradaic process becomes significant are defined by: The analysis I present here assumes that the conditions of Eq. (17) have been met.

The focus of the present work is on the effects of the uncompensated resistance on reversible electron transfer. Aficionados of cyclic voltammetry will immediately recognize that the shapes of the widely separated peaks produced by large values of iRu (Fig. 2a) cannot be explained by slow heterogeneous electron transfer (using either the Butler–Volmer formalism (Fig. 2b) or Marcus– Hush (Fig. 2c) [3] (see Appendix). For very large values of aFjv jRuCdl/RT and/or aF3jvjCtotal Ru/(RT)2 virtually all of the ohmic potential drop occurs across the resistance Ru. The observed response will then be ohmic – manifested by a virtually linear relationship between current and voltage. For very small peak separations, however, the peak shapes for electron transfer control and uncompensated-resistance control are almost identical – an analogous congruence was recognized long ago for solution based species [9]. This will be discussed in more detail in the Section 3. Peak shapes predicted by the Butler–Volmer formalism (with a = 0.5) and the Marcus–Hush formalism become identical as DEpeak ) 0 (see e.g., [4]). Cyclic voltammetric responses for a reversible surface-bound redox system with finite Ru can be characterized by a virtually universal working curve plotted as: DEpeak/DipeakRu vs log10[(nF/ RT)DEpeak] (Fig. 3). The dependence upon the capacitive parameter, RTCdl/(n2F2Ctotal) is quite weak. Note that the ratio of the (steadystate) capacitive current and the faradaic current peak for Ru = 0 is:

iC dl amC dl 4RTC dl ¼ : ¼ ipeakRu ¼0 amn2 F 2 Ctotal n2 F 2 Ctotal 4RT

ð18Þ

Thus, when RTCdl/(n2F2Ctotal) = 0.1 or 1.0 (Fig. 3), the capacitive contribution to the current is significant with icdl/ipeak = 0.4 or 4.0.

Table 1 Demonstration that upper limit is independent of Cdl RTC dl n2 F 2 Ctotal

Cdl (farad)

Ru (X)

DEpeak Dipeak Ru

0 0.1 1 10 100 1000

0 3.755  104 3.755  103 3.755  102 3.755  101 3.755

1000 100 100 10 1 0.1

1.998 1.998 2.000 1.999 1.999 2.001

Fixed parameter values used in simulation: a = 1.0 cm2, Estart = Erev = 1.0 V, jdEapp j 1.0  106 V, v = 0.001 V s1, Ctotal = 1.0  109 mol cm2, n = 1.

4e-8

1.8

c

3e-8

1.6

2e-8 2

2

RTCdl /n F Γ 0 0.10 1.00

1.4 1.2

total

c

O Δ

0

a

-1e-8

1.0 -3

-2

-1

0

1

2

3

log 10[(nF /RT) ΔE peak] log 10[(2nF /RT)(Epeak- E )] Fig. 3. Universal working curve: a plot of DEpeak/DipeakRu vs log10[(nF/RT)DEpeak] or equivalently (Epeak  E00 ) vs log10[2nF(Epeak  E00 )/RT]. Data are obtained from simulations presented in Fig. 1 for different values of RTCdl/(n2F2Ctotal) = 0 (d), 0.1 (O) and 1.0 (D). The solid line is a least-squares fit of Eq. (21) to data for RTCdl/ (n2F2Ctotal) = 0. The values of Estart and Erev are chosen so that the conditions of Eq. (17) are met.

b

-2e-8 -3e-8

0'

b

a

1e-8

symbol

i (A)

ΔEpeak /(ΔipeakR u ) (E peak-E 0')/i peak Ru

2.0

c

c

-4e-8 -8

-6

-4

-2

0

2

4

6

8

E - E 0' (V) Fig. 4. Comparison of simulated uncompensated-resistance effects for Ru = 2  108 X, aF3jvjCtotalRu/(RT)2 = 2.92  104 with (a) Cdl = 0; (b) Cdl = 1.0  106 farad cm2; and (c) Cdl = 1.0  105 farad cm2. Other parameter values are the same as those used in Fig. 2.

49

S.W. Feldberg / Journal of Electroanalytical Chemistry 624 (2008) 45–51

The plots in Fig. 3 show that the ratio DEpeak/DipeakRu reaches an upper limit of 2.00 as Ru ) 0(or (nF/RT)DEpeak ) 0). Particularly interesting is that the upper limit (2.00) appears to be truly independent of the double layer capacitance, Cdl, over a very large range RTCdl/(n2F2 Ctotal) as shown in Table 1. There is a subtler indication that DEpeak/DipeakRu reaches a lower limit of 1.0 as Ru ) 1 (or (nF/RT)DEpeak ) 1). However, it is possible to offer a graphical argument that when nFDEpeak/RT ) 1, DEpeak/DipeakRu ) 1.0. Consider the voltammograms shown in Fig. 4 which show simulations for Ru = 2  108 X, aF3jvjCtotal Ru/(RT)2 = 2.92  104 with (a) Cdl = 0; (b) Cdl = 1.0  106 farad cm2; and (c) Cdl = 1.0  105 farad cm2. Other parameter values are the same as those used in Fig. 2. The values of DEpeak/DipeakRu are virtually unchanged. Inspection of the responses reveals that regardless of the value of the capacitance (and as long as the ‘‘RC” time constant for the capacitive charging is sufficiently short – see Eqs. (15)–(17) and accompanying discussion), the current response curves in the vicinity of the peak are well represented by a straight line with a slope 1/Ru and an intercept with the zero-current line approximately (but 0 not exactly) where E  E0 = 0 (the variation in interfacial potential required to drive the redox process is very small compared to the very large potential drop associated with iRu). Thus, for sufficiently 0 large values of jE  E0 j [10]:

ð19Þ

and, therefore,

ð20Þ

The ‘‘peak” occurs because COx or CRed has been depleted. A very good least-squares fit to simulations of a system with Cdl = 0 is obtained with the empirical expression (solid line, Fig. 3):

"



DEpeak nF ¼ 2  exp 2:610 DEpeak RT Dipeak Ru

0:764 #

ð21Þ

"

ð22Þ

Eq. (21) converges, as it must, to the upper and lower limits of 2.00 and 1.00 for DEpeak/(DipeakRu).

3. Conclusions

ð25Þ 0

Epeak  E00 ipeak Ru

ffi

F jEpeak E

00

2 or

j=RT1

DEpeak Dipeak Ru

ffi

F DEpeak =RT1

2

ð26Þ

Because ipeak is virtually proportional to Ctotal under these conditions (see Eq. (4); ignore the Cdl) Eq. (26) can also be written as

4RTðEpeak  E00 Þ F 2 vCtotal Ru

ffi

F jEpeak E00 j=RT1

2 or

2RT DEpeak F 2 vCtotal Ru

2

ffi

F DEpeak =RT1

ð27Þ Epeak  E or DEpeak will also be a function of Ctotal (as well as of v) in contradistinction to the system where Ru is negligible and deviation from reversible behavior is caused by slow electron transfer. Thus, changing the value of Ctotal can reveal the presence of ohmic effects assuming that the system is behaving ideally (see Eq. (3) and accompanying discussion). Without this simple (but critical!) experimental ploy, it is virtually impossible to distinguish ‘‘Ru-control” from ‘‘ks-control” when FDEpeak/RT  1 as shown in Fig. 5. For the voltammograms shown in Fig. 5A (left ordinate) the input val-

1e-6 4e-8

A 0

i (A)

ð23Þ

2e-8 Cathodic sweep

-1e-6

ð24Þ

This lower limit is also independent of the electrode capacitance, Cdl. A relatively simple empirical expression (see Eqs. (21) and (22)) correlates the peak shifts and uncompensated resistance for RTCdl/(n2F2Ctotal) = 0. Because of the weak dependence on the dou-

i

Particularly interesting is that this limiting value is completely independent of the electrode capacitance, Cdl. For very large values 0 of nFjEpeak  E0 j/RT

  dEpeak ¼ Ru : dipeak ðnF=RTÞjEpeak E0 0j1

1:50:

Note that the parameter (Epeak  E0 )ks/v is independent of Ctotal. (The relationship expressed in Eq. (25) can be extracted from Fig. 4 in Ref. [12]). Compare this behavior to that of a system with finite (small) Ru and infinitely large ks (see Eqs. (21) and (22)). When 0 FjEpeak  E0 j/RT  1 or nFjDEpeaj/RT:

For cyclic voltammetric responses of reversible surface-attached redox species a curious and surprising result of this analysis is  the demonstration that for very small values of    nF Epeak  E00 =RT:

  dEpeak ¼ 2Ru : dipeak ðnF=RTÞjEpeak E00 j1

ffi

FðEpeak E00 Þ=RT1

00

or (see Eqs. (13) and (14))

#  0:764 Epeak  E00 2nF  00  : ¼ 2  exp 2:610 Epeak  E  RT ipeak Ru

ks v

0:750 or

ffi

FðEpeak E00 Þ=RT1

ks

Epeak  E00 DEpeak ¼ ffi 1:0: ipeak Ru Dipeak Ru

DEpeak

ks v

-i (A)

E  E00 Ru

ðEpeak  E00 Þ

Ru

iffi

ble layer capacitance, the expression is adequate as long as RTCdl/ (n2F2Ctotal) 6 0.1. The key question that should accompany interpretation of any 0 cyclic voltammetric response is: Are non-zero values of jEpeak  E0 j caused by uncompensated resistance or by slow electron transfer, i.e., by a small rate constant ks (s1) [11]? For large   heterogeneous  00  values of nF Epeak  E =RT produced by large Ru (with) or by small ks (with Ru = 0 and implementing either the Marcus–Hush or Butler–Volmer formalisms – see Appendix) the distinction is visually obvious (see Fig. 2). 0 For small values of nFjEpeak  E0 j/RT the responses predicted by the Marcus–Hush formalism and Butler–Volmer formalism (with a = 0.5) become identical (see Appendix). Furthermore, the distinction between ‘‘Ru-control” and ‘‘ks-control”) is subtle (just as it is 0 for a soluble redox species [9]). For nFjEpeak  E0 j/RT  1 (assuming a = 1/2):

0 -2e-6

B Anodic sweep

-0.15

-0.10

-0.05

0.00

0.05

0.10

-2e-8 0.15

E - E 0' (V) Fig. 5. Demonstration of similarity of the cyclic voltammetric response for ‘‘Rucontrol” and ks-control”. (A) (left ordinate): Cyclic voltammograms for () Ru = 0, ks = 1; (d) Ru = 2663.325 X, ks = 1; (s) Ru = 0, ks = 0.15 s1. For all voltammograms Cdl = 2  105 farad cm2 and other parameter values the same as for Fig. 2. (B) (right ordinate): iRu  iks .

50

S.W. Feldberg / Journal of Electroanalytical Chemistry 624 (2008) 45–51

ues of Ru and ks are set to: 0 and 1 to produce the reversible response with DEpeak = 0.0 V, 2663.3 X and 1 or 0 X and 0.15 s1to produce the quasireversible response with DEpeak = 0.010 V as deduced from Eqs. (25)–(27). For all voltammograms Cdl = 2  105 farad cm2 and other parameter values are the same as for Fig. 2. Fig. 5B (right ordinate) shows the difference, iRu  iks , between the quasireversible responses for Ru = 2663.3, ks = 1 and Ru = 0, ks = 0.15 s1 is less than 1% of the peak current – i.e., the responses are effectively indistinguishable. The analysis presented here is based on the presumption that the uncompensated resistance is constant and that the current density and potential are uniformly distributed over the electrode surface - this will not be the case for an ultramicroelectrode (ume) (e.g., a small disk of radius r0). Then, the uncompensated resistance is a function of r/r0 [5] and, therefore, the current density and electrode potential are also functions of r/r0. The analysis of this problem is currently being addressed by Svir and coworkers [7]. I expect that when FDEpeak/RT  1 the behavior of a ume disk will be adequately described by the present treatment defining Ru = q /4r0 ([5]) where q (ohm cm) is the resistivity of the solution – assuming that the distance between the reference and ume electrodes is a factor of ten or more larger than r0. For larger values of FDEpeak/RT the relationships developed by Svir et al. will be considerably more complicated than those described here. A final caveat – throughout this analysis I have made the simplistic assumption that the double layer capacitance is potential independent. The actual double layer capacitance can (and quite likely will) vary with potential – and in a different way for all the myriad combinations of electrode material (and electrode pretreatment) and solution composition (when the redox moieties are covalently tethered to the electrode in conjunction with covalently attached electroinactive species the double layer capacitances tend to be smaller with a weaker potential dependence than generally observed on bare metallic electrodes [3]). The potential-dependence of the capacitance, if known, could be included in the simulation. However, determining the capacitance in the presence of the surface-attached redox species is not trivial [14,15]. Acknowledgements Michael Mirkin, Irina Svir, Christian Amatore and Sasha Oleinick are thanked for their discussion and insights. John Miller and the Chemistry Department, Brookhaven National Laboratory are thanked for their support of a Guest Appointment. The author gratefully acknowledges support of the US Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, under Contract DE-AC02-98-CH10886.

Appendix. Numerical solutions numerical solution of Eq. (12) for reversible electron transfer with uncompensated resistance The derivative di/dt is defined as: 0

0

di i  i ii ¼ ffi ; dt t  t 0 Dt

ð28Þ

0

Then 0

1  nF 00  iR  E Þ ðE app u Bn2 F 2 C RT B C C  B  2 þ C dl C @ RT total A nF 00   ðEapp  iRu  E Þ 1 þ exp i0 RT 1: 0 i ¼ a m þ Ru    Dt nF 00 E exp  iR  E app u C 2 2 Ru B RT C Bn F Ctotal  1þ B  2 þ C dl C A Dt @ RT nF 00 1 þ exp ðEapp  iRu  E Þ RT exp



ð31Þ The value of i0 can be obtained recursively by initially assuming that the value of i on the RHS of Eq. (31) is i0 and then replacing that by the newly computed value of i). A single repetition is adequate and the reason for the rapid conversion will be discussed shortly. Accurate simulation will depend upon utilization of sufficiently small values of dEapp (see Eq. (29) and associated discussion). The numerical solution for Eq. (12) is easily obtained when the simulation is starting at a potential where the current amplitude is very small and iRu may be assumed to be zero. The rapid convergence is easily understood by exploring the sensitivity of the term exp[(nF/RT(Eapp  iRu  E00 )]/(1 + exp[(nF/ RT(Eapp  iRu  E00 )])2 to the value of its exponential argument. For convenience the term can be expressed as exp[u]/(1 + exp[(u])2 whose sensitivity to the value of u is the derivative:

d

exp½u ð1 þ exp uÞ2 du

! ¼

exp½u ð1 þ exp½uÞ2



exp½2u ð1 þ exp½uÞ3

¼

exp½u  exp½2u ð1 þ exp½uÞ3

:

ð32Þ The maximum or minimum value of this term is ±0.0962, which occurs when u = ±1.317. This can be deduced from the second derivative of Eq. (32):

d 0¼

! exp½u  exp½2u ð1 þ exp½uÞ3 du

¼

exp½u  4 exp½2u þ exp½3u ð1 þ exp½uÞ4

:

ð33Þ

Appendix. Numerical solution for quasireversible behavior with zero uncompensated resistance The operative expression for the current associated with interfacial electron transfer control is, for a one-electron process:

i ¼ Fðkox Cred  kred Cox Þ ¼ nF

dCox : dt

ð34Þ

The corresponding explicit numerical expression can be written as:

where

  dE  Dt ¼  app  v

1 0   nF 00   ðE exp iR E Þ 0 app u C Bn2 F 2 ii RT C B i¼a m Ctotal  þC dl C: Ru B    2 A @ RT Dt nF 1þexp ðEapp iRu E00 Þ RT ð30Þ

ð29Þ

and t is the time at which the previous current was computed, i’ is the previous current, and dEapp is the change in Eapp with each time increment, Dt. Combining Eqs. (12) and (29) gives:

DCox ¼ ðkox ðCtotal  Cox Þ  kred Cox Þ: Dt

ð35Þ

The more stable implicit expression is

DCox ¼ Dtðkox ðCtotal  ðCox þ DCox ÞÞ  kred ðCox þ DCox ÞÞ ¼

Dtðkox ðCtotal  Cox Þ  kred ðCox ÞÞ : 1 þ Dtðkox þ kred Þ

ð36Þ

S.W. Feldberg / Journal of Electroanalytical Chemistry 624 (2008) 45–51

The values of kox and kred are dependent upon potential and can be computed using the Butler–Volmer formalism [2,12] or a formalism based on Marcus–Hush theory [3,4]. With the Butler–Volmer expression [13]: BV

kox ¼ ks exp½ð1  aÞðE  E00 Þ

ð37Þ

and BV

kred ¼ ks exp½aðE  E00 Þ:

ð38Þ

The corresponding expressions for the Marcus–Hush formalism presented here (see [4]) were suggested by Yi-Ping Liu [16] and are mathematically identical to the expressions developed by Chidsey [3]:

2  

MH

kox ¼ ks

 e ðE  E00 Þ Z exp 2kB T

U

1

6 6 exp 6 4

e

2 3 e ðE  E00 Þ 7 kB T 7 7 k 5 4 kB T

2 cosh e

1

de

ð39Þ and

MH

kred ¼

  e ks exp  ðE  E00 Þ Z 2kB T

U

1

1

6 6 exp 6 4

e

e ðE  E00 Þ kB T k 4 kB T

2 cosh 2e

2 3 7 7 7 5 de; ð40Þ

where

"

U¼

Z

1

1

exp 

e2 4 kBkT

2 cosh e2

# de

operates well within the XP-Professional, and earlier, PC platforms, but which may have problems with Vista. The source code is available on request. Alternatively, identical results can be generated using DigiSim (Bioanalytical Systems) or DigiElch (ElchSoft.com; DigiElch SB, Version 3, copyright 2006)) [17–22]and (with a bit more effort) Bieniasz’s ELSIM [23].) DigiSim or DigiElch can be used to simulate a thin-layer system where solution-soluble redox species are confined within a thin layer. Laviron [12]. has pointed out that the cyclic voltammetric responses are identical for the thin-layer system (with redox concentrations cox and cred) and for the surface-attached system whenCox = dcox, Cred = dcred, ks (s1) = ks (cm s1)/d and (D redox is the diffusion coefficient of the redox species and d is the thickness of the thin-layer). References

2

2 

51

ð41Þ

and e(C) is the electronic charge, kB is Boltzmann’s constant and e is a dimensionless term corresponding to a continuum of energy levels in a metal electrode. For very small overpotentials [4] the Marcus–Hush and Butler– Volmer formalism become virtually identical when a = 1/2 (Eqs. (37) and (38)). With this formulation it is immediately obvious that MH MH kox =kred ¼ exp½ðe=kB TÞðE  E00 Þ. Computer codes for the above numerical computations were written using QuickBasic, a Microsoft DOS-based program that

[1] M. Orlik, J. Electroanal. Chem. 575 (2005) 281–286. [2] A.J. Bard, L.R. Faulkner, Electrochemical Methods: Fundamentals and Applications, second ed., John Wiley and Sons, New York, 2001. [3] C.E.D. Chidsey, Science 215 (1991) 919. [4] S.W. Feldberg, M.D. Newton, J.F. Smalley, in: A.J. Bard, I. Rubinstein (Eds.), Electroanalytical Chemistry, Marcel Dekker, New York, 2002. [5] J. Newman, J. Electrochem. Soc. 113 (1966) 501. [6] K.B. Oldham, Electrochem. Commun. 6 (2004) 210–214. [7] C. Amatore, A. Oleinick, I. Svir, Anal. Chem., submitted for publication. [8] The IUPAC convention will be followed: currents associated with oxidation are positive; currents associated with reduction are negative. [9] R.S. Nicholson, Anal. Chem. 37 (1965) 1351. [10] For the purposes of this demonstration the potential range will be unrealistically large. [11] Phase change can also effect apparent quasireversibility see e.g.: S.W. Feldberg, I. Rubinstein, J. Electroanal. Chem. 240 (1988) 1–15. but that was not considered in the present treatment. [12] E. Laviron, J. Electroanal. Chem. 101 (1979) 19. [13] Laviron [12] obtained an analytic, albeit cumbersome, solution based on the Butler–Volmer formalism. It was more convenient to use the numerical approach outlined here which is easily adapted for both the Butler–Volmer and Marcus–Hush formalisms. [14] P. Delahay, J. Phys. Chem. 70 (1966) 2373. [15] P. Delahay, G.G. Susbielles, J. Phys. Chem. 70 (1966) 3150. [16] Yi-Ping Liu, M.D. Newton, private communication. [17] M. Rudolph, J. Electroanal. Chem. 338 (1992) 85. [18] M. Rudolph, in: I. Rubinstein (Ed.), Physical Electrochemistry: Principles, Methods and Applications, Marcel Dekker, New York, 1995, p. 81. [19] M. Rudolph, J. Electroanal. Chem. 571 (2004) 289. [20] M. Rudolph, J. Comput. Chem. 26 (2005) 619. [21] M. Rudolph, J. Comput. Chem. 26 (2005) 633. [22] M. Rudolph, J. Comput. Chem. 26 (2005) 1193. [23] L.K. Bieniasz, Comput. Chem. 21 (1997) 1.