J. Electroanal. Chem., 77 (1977) 225--235 225 © Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands

C O N V O L U T I O N P O T E N T I A L SWEEP V O L T A M M E T R Y

P A R T VI. E X P E R I M E N T A L E V A L U A T I O N IN THE K I L O V O L T PER SECOND SWEEP R A T E R A N G E

J.M. SAVt~ANT and D. TESSIER

Laboratoire d'Electrochimie, Universitd de Paris VIL 2, Place Jussieu, 75221 Paris Cedex 05 (France)

(Received 23rd March 1976; in revised form 21st April 1976)

ABSTRACT

Convolution potential sweep voltammetry is experimentally evaluated up to a sweep rate of 2278 V s - 1 using a diffusion controlled system: the reduction of fluorenone in DMF. The effects of ohmic drop double layer charging and bandpass limitations of the in- strument are discussed. The corresponding corrections are determined experimentally in the range 22.7--2278 V s -1 . It is shown from the logarithmic analysis of the convoluted faradaic current that reliable results can be obtained up to this sweep rate after the above corrections have been carried out.

INTRODUCTION

Sweep rate is one of the mos t i m p o r t a n t exper imenta l parameters in the ap- pl icat ion of CPSV to e lec t rochemical kinetic studies. Indeed , the accuracy in mechanism de te rmina t ion and the upper limit o f the measurable rate cons tants are largely depending u p o n the ex ten t of the explorable sweep rate range [1] . This is t rue in the case of rate de termining charge transfers [2] as well as for systems in which the kinetics o f the associated chemical react ions p lay the major role [3 - -5 ] .

As discussed previously [1] , a reasonable lower limit is a b o u t 0.1 V s - 1 due, below this value, to the in ter ference of natural convec t ion and of uneasi ly correctable spherici ty effects.

The upper limit of the sweep rates tha t have been used so far is a b o u t 300 V s - 1 . This l imi ta t ion results f rom the value o f the shor tes t sampling t ime (30 ps for an 8 bit accuracy) of the digital ization sys tem (A to D conver te r + m e m o r y ) taking into accoun t tha t the i - -E curve must be def ined by a suffi- cient n u m b e r of points in order to obta in a sa t is factory accuracy in fur ther t rea tments , part icular ly convolu t ion . Digitalization devices n o w commerc ia l ly available allow a gain o f a b o u t one order o f magni tude in the sampling rate.

226

From this view-point one may thus expect to reach sweep rates on the order of 3000 V s -1.

However, several phenomena that had little influence below 300 V s -1 may become significant in the kV s -1 range and require to be appropriately cor- rected for in data processing. The first of these is the influence of double layer charging on the effect of ohmic drop on the faradaic current.

Assuming the double layer capacitance Cd to be independent of the course of the faradaic reaction, the correction for this effect is obtained [ 1 ] by:

(a) modifying the potential scale according to:

E' = E + R u i

where E is the applied potential, E' the effective potential difference across the double layer and the faradaic impedance, i the total current flowing through the working electrode. Ru = R ° --/3R~ is the residual resistance re- maining from the solution resistance comprised between the working and ref- erence electrode after partial compensation (Re = sampling resistance,/3 = feed- back ratio, R ° = solution resistance comprised between the working and ref- erence electrode).

(b) Extracting the faradaic current if from the total current according to:

if = i + C d ( d E / d t ) + R u C d ( d i / d t )

In the case of triangular scanning with a sweep rate v:

if = i - - Cd v + R u C d ( d i / d t )

where v is to be taken as positive in the cathodic part of the sweep and nega- tive in the anodic one. The corrected convoluted current vs. potential curve ( I - - E ' ) is then computed by convolution of the faradaic current if thus ob- tained with the function (Trt) -1/2 and plotting the resulting values versus the corrected potential E'.

Reference e{ect rode

La

Fig. 1. Equivalent circuit of the working electrode accounting for the effects of double layer charging and amplifier bandwidth on the faradaic current.

227

Another important effect results from the bandpass limitations of the con- trol and measuring instrument, i.e., the potentiostat and current measuring amplifiers. It has been shown previously [6,7] that this effect can be repre- sented with a good accuracy by an inductance La introduced in series with the residual solution resistance Ru in the equivalent circuit of the working electrode as shown in Fig. 1.

La = (Rc + R ° ) / ~ + (R e + R°) /Co '

where ~ and ~ ' are the gain-bandwidth product pulsations of the potentio- star and the current measurer, and Rc the solution resistances comprised be- tween the counter and reference electrodes. The correction of both effects, double layer charging and bandpass limitations of the instrument results im- mediately from this representation:

E' = E + R u i + L a ( d i / d t ) (1)

if = i + C d ( d E / d t ) + R u C d ( d i / d t ) + L a C d ( d 2 i / d t 2) (2)

In the case of a triangular sweep:

if = i - - CdV + R u C d ( d i / d t ) + LaCd(d2i/dt 2) (3)

A dimensionless formulation can be obtained by introducing the following variables and parameters:

= n F v t / R T , qdf = i f / n F S c ° D 1 / 2 ( n F v / R T ) l / 2 , q~ = i / n F S c O D 1 / 2 ( n F v / R T ) l / 2 ,

p = ( n F / R T ) R , n F S c O D 1 / 2 ( n F v / R T ) I / 2 , 7 = C d ( n F v / R T ) 1 / 2 / ( n F / R T ) n F S c ° D 1 / 2 ,

p = ( n F / R T ) L a n F S c ° D 1 / 2 ( n F v / R T ) 3 / 2 (The symbolism is the same as in ref. 7.) Then:

( n F / R T ) E ' = ( n F / R T ) E + p@ + p d q s / d r

~ f = ~ - - 7 + p y ( d q 2 / d r ) + pT(d2~2 /dr 2)

qs~, dq'/dT and d2~/dT 2 do not vary very much when raising the sweep rate. From the fact that p and 3' are proportional to v 1/2 and p to v 3/2 it results therefore that the double layer charging and bandpass limitation corrections increase with the sweep rate and the second one more rapidly than the first one.

S E L E C T I O N OF A TEST SYSTEM

The simplest way of testing the validity of the above correction procedures is to carry out the experiments on a system whose kinetics is diffusion con- trolled even at the highest sweep rate used. The reduction of 1.9 mM fluorenone in DMF with 0.1 M NEt4C10 4 was chosen for this purpose. The standard rate constant k s of fluorenone is likely to be of the same order of magnitude as that of anthracene in the same solvent, i.e. about 5 cm s -1 [8].

228

Whether such a system will deviate or not from a diffusion control led behav- iour at the highest sweep rate used here, i.e. 2500 V s -1 , can be predicted as follows.

Assuming that the kinetics are of the Volmer-Butler type: it = F S h s exp- [ - - ( (~F /RT) (E ' - - E°)] ((CA)0 -- (CB)0 e x p [ F / R T ( E ' - - E°)] } (E ° = standard potential , a = transfer coefficient, (CA)o and (%)0 = concentrat ions of fluo- renone and f luorenone anion radical, respectively, at the electrode), it has been shown that the relative kinetic control by charge transfer and diffusion is determined by the value of the dimensionless parameter:

A = k s ( D F v / R T ) - 1 / 2

and the characteristics of the vol tammetric i f - -E ' wave for various values of A have been numerically computed [9]. We have re-calculated these curves using the same numerical method as Matsuda and Ayabe [9], assuming that a = 0.5, and then pursued the formal analysis by computing the convolut ion integral:

t I = 71 " - 1 / 2 f i~(t - - 7 ) - 1 / 2 dr / (4)

0

as a funct ion of E' -- E ° and of the parameter A so as to obtain the formal convoluted wave for a series of A values. Combined logarithmic analysis of the initial i f - -E ' and convoluted waves may then be performed according to three possible behaviours [2] :

(a) reversible, diffusion control led (R):

E' = E ° + ( R T / F ) ln[(I 1 - -I) /I ] (5)

(b) quasi-reversible (QR):

E' = E ° + ( R T / a F ) In k s / D 1/2 + ( R T / a F ) ln( I 1 - - I [ 1 + e x p ( F / R T ) ( E - - E ° ) ] } / i f

(c) irreversible (IR):

E = E ° + ( R T / a F ) In k s / D 1/2 + ( R T / a F ) ln(I l - - I ) / i f

For a given value of A these three logarithmic analyses give rise to curves which are never far f rom being straight lines. The characteristic quant i ty show- ing if the particular behaviours R and IR are reached within the experimental uncertainty is the slope. For the R case, the slope is 58.5 mV per decade at 22°C. It is 117 mV for the IR case when a = 0.5. The quasi-reversible behav- iour corresponds to the general case and is therefore always followed (the slope is 117 mV per decade at 22°C for a -- 0.5).

The slopes of the three logarithmic analyses are represented as functions of A in Fig. 2.

For the f luorenone/ f luorenone anion system in DMF taking the same k and D values as for anthracene [8], it is found that A at 22°C and for v = 2500 V s -1 is equal to about 6. As shown in Fig. 2 this corresponds to a deviation of less than 2 mV per decade on the slope of the R analyses. It can therefore be

229

1 2 0 t'M o,i

l o o

"o

~ 60

if)

QR

-& -~ 6 5 toga

Fig. 2. S lope of the logarithmic analyses as a funct ion of the kinetic parameter A for a = 0.5. R = reversible, QR = quasi-reversible, IR = irreversible.

predicted that no experimentally significant deviation from the diffusion con- trolled behaviour is to be found at the highest sweep rate.

E X P E R I M E N T A L R E S U L T S A N D E V A L U A T I O N OF THE C O R R E C T I O N S

Chemicals, instrumentation and experimental procedures were the same as previously described [1 ,2] . The only changes were in the digitalization device

500

- 5 0 0

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,--=:==::-~ ................

.,,.." ",... ,,....."

3b ~o go 6'o ( F / R T ) ( v I - E i )

Fig. 3 . 1 . 9 mM f luorenone in DMF with 0.1 M Et4NCIO4. Initial cycl ic vo l tammograms. Sweep rate: ( ) 22 .7 , ( - - - - ) 225 , ( - - - ) 655 , ( . . . . . . ) 2278 V s - 1 .

2 3 0

(Matek, 8 bits accuracy and 3 ps minimal sampling time) and in the com- puter (Matek 1026). The temperature of experiment was 22°C ( (RT/F) In 10 = 58.5 mV). The reference electrode was an Ag/AgI electrode in DMF [2].

The results obtained and the magnitude of the various corrections are shown in Figs. 3--9 for four increasing values of the sweep rate: 22.7, 225 ,655 and 2278 V s -1 . The various quantities represented in the Figures are plotted ver- sus the dimensionless time variable (F/RT)(v t - - Ei) where Ei is the initial po- tential of the scan, the value of which was not the same for the various sweep rates. The potential of scan inversion was also different, being more negative as the sweep rate is raised according to the increasing shift of the voltammo- gram caused by ohmic drop.

Blank experiments showed that the double layer capacity did not vary sig- nificantly in the potential range of the voltammograms. Its value was therefore derived from the height of the current plateau at the foot of the faradaic wave for the highest sweep rates. It was thus found that Cd = 8.6 X 10 - s F. The self-inductance L a characterizing the instrument bandwidth was determined by measuring the period to of the permanent oscillations obtained for suffi- cient positive feedback at a potential where no faradaic current flows, accord- ing to the equation [ 7 ] :

to = 2 7r(LaCd) 1/2

It was thus found that La = 5.8 X 10 -4 H.

100

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..,,,'"'-.,,..% .. '..,..

... ._

/ / ~ "%. / / --~

5_,_,_,_.~1 . . . . . . . . . . . . . . . . . . - . ~ . _ .~ ................... - - - = ~ ' C . . . . . ~.- . . . . . . . .

- - - . . / . / "..

% • . ..,,.-"

"%.. . . . , . . . . . . -"

( F / R T ) ( v t -E~)

Fig. 4. 1 .9 m M f l u o r e n o n e in D M F w i t h 0 .1 M E t 4 N C I O 4. C o r r e c t i o n o f t h e p o t e n t i a l scale fo r o h m i c d r o p . S w e e p r a t e : ( ) 2 2 . 7 , ( ) 2 2 5 , ( - - - ) 6 5 5 , ( . . . . . . ) 2 2 7 8 V s - 1 .

231

Figure 3 shows the original voltammograms. The rate of positive feedback was not same for each sweep rate. Indeed for the smallest sweep rates an over- oscillatory regime could be used for the initial charging of the double layer without major interference with the faradaic portion of the voltammogram since the period of the overoscillations is there short as compared to the scan- ning time. This was no more possible at the highest sweep rates for which a critical or subcritical regime was used [6]. It follows that the residual resis- tance Ru was varied according to the sweep rate:

v/V s - l : 22 .7 ,225 ,655 , 2278

Ru/~ : 25, 37.5,150, 187

In these conditions the ohmic drop correction Rui of the potential scale (eqn. 1) is quite large for the highest sweep rate reaching more than 100 mV at 2278 V s -1 as can be seen in Fig. 4.

Figure 5 shows the correction of the potential scale for the bandpass limita- tions of the instrument. At the inversion potential the current varies steeply resulting in a poor accuracy on di/dt in this region. It follows that the poten- tial scale is not corrected accurately from the bandpass limitations in a very narrow potential region around the inversion point which is not of great sig-

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[ F / R T ) ( v t - El)

Fig. 5 . 1 . 9 mM f luorenone in DMF with 0.1 M Et4NCIO 4. Correction of the potent ia l scale for the ins t rument bandpass l imitations. Sweep rate: ( ) 22.7, ( - - ) 225, ( - - . - - ) 655, ( . . . . . . ) 2278 V s - 1 .

232

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Fig. 6. 1.9 mM f luorenone in DMF with 0.1 M Et4NC104. Faradaic current as extracted from total current according to eqn. 3. Sweep rate: ( ) 22.7, ( - - - - ) 225 , ( - - . - - ) 655, ( . . . . . . ) 2278 V s - 1 .

nificance for further analysis of the corrected voltammograms. It is seen that in the other regions the effect of bandpass limitations on the potential scale, although not completely negligible at the highest sweep rate, is small com- pared to the effect of ohmic drop.

The faradaic current as extracted from the total current according to eqn. (3) is represented in Fig. 6. The two last terms of eqn. (3) are represented in Figs. 7 and 8 as fractions of the faradaic current itself. In examining these diagrams it is seen that two large increases in absolute value are observed in the regions where if becomes equal to zero. This is of course of little signifi- cance for the extraction of the faradaic current. Another large value of these ratios is obtained for the inversion potential which results in a systematic error in the convoluted current for times beyond the inversion point. It is also seen that the effect of bandpass limitations in the extraction of faradaic current is very small compared to the effect of double layer charging.

In order to test the overall efficiency of the above corrections the convolut- ed current I was then computed from the faradaic current (eqn. 4) and loga- rithmically analyzed according to eqn. 5 both for the reduction wave and the re-oxidation wave. The results (Fig. 9) show a very satisfactory agreement with the predicted behaviour as regards the linearity of the plot, the potential

2 3 3

0.5

~1:~ o "o

-0.5

I

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"-,, • I \

"x I I \ • ,.. I . L ' "

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: !

" i

3b 40 s'o 6"o (F/RT)(vt- Ei)

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Fig. 7 . 1 . 9 mM f luorenone in DMF with 0.1 M Et4NCIO 4. Relative effect of the double layer charging in the ex t rac t ion of faradaic current. Sweep rate: ( ) 22.7, ( - - ) 225, ( . . . . ) 655, ( . . . . . . ) 2278 V s - 1 .

o.1

-o.1

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~-~ .... _ _ _ ~ / y ~ _ "~,~ ~ ~ - ~ . ~

'"" """ \ "'" '"' .2" ".-.... .\ '.. "'....

I

\

3'0 4'0 ~o @o (F/RT) (v'e-E~)

Fig. 8 . 1 . 9 mM f luorenone in DMF with 0.1 M Et4NCIO 4. Relative effect of the ins t rument bandpass l imitat ions in the ex t rac t ion o f faradaic current . Sweep rate: ( - ) 22.7, ( - - - - ) 225, ( . - - ) 655, ( . . . . . . ) 2278 V s - 1 .

234

c

x+~

• ` 5 o

x`5

,5

-'0`5 0

I I I I I I

0 .75 - 0 .80

- E ' / V

0

"Ao

0

`5O

OZ~

A ~

`5

, I 0.85

Fig. 9. 1.9 mM fluorenone in DMF with 0.1 M Et4NC104. Logarithmic analysis of the convoluted current. Reduction wave, sweep rate: (A) 22.7, (×) 225, (+) 655, (•) 2278 V s -1 . Re-oxidation wave, sweep rate: (®) 21.8, (*) 217, (©) 695, {o) 2190 V s - 1 .

location and the slope which varies between 56 and 58 mV/decade. Only the curve obtained from the re-oxidation wave at 2278 V s -1 exhibits a slight deviation at the edges of the potential range probably due to the error in ex- traction of the faradaic current at the inversion point. The results were prac- tically the same if the bandpass limitation effect was ignored in the extraction of the faradaic current.

Our conclusion is therefore that convolution potential sweep vol tammetry can be used with confidence in the kV s -1 sweep rate range provided the ef- fects of ohmic drop, double layer charging and bandpass limitations of the instrument are properly taken into account.

ACKNOWLEDGEMENT

The work was supported in part by the CNRS (Equipe de Recherche Asso- cite No. 309: Electrochimie Organique).

235

REFERENCES

1 L. Nadjo, J.M. Sav6ant and D. Tessier, J. Electroanal. Chem., 52 (1974) 403.

2 J.M. Sav6ant and D. Tessier, J. Electroanal. Chem., 65 (1975) 57. 3 J.M. Sav6ant and D. Tessier, J. Electroanal. Chem., 61 (1975) 251. 4 L. Nadjo, J.M. Sav6ant and D. Tessier, J. Electroanal. Chem., 64 (1975) 143. 5 C.P. Andrieux, J.M. Sav6ant and D. Tessier, J. Electroanal. Chem., 63 (1975) 429. 6 D. Garreau and J.M. Sav6ant, J. Electroanal. Chem., 35 (1972) 309. 7 D. Garreau and J.M. Sav6ant, J. Electroanal. Chem., 50 (1974) 1. 8 H. Kojima and A.J. Bard, J. Amer. Chem. Soc., 97 (1975) 6317. 9 H. Matsuda and Y. Ayabe, Z. Elektrochem., 59 (1955) 494.