{Staircase voltammetry of dissolved redox couple in a thin layer twin electrode cell}


http://dx.doi.org/10.5599/jese.713    11 

J. Electrochem. Sci. Eng. 10(1) (2020) 11-19; http://dx.doi.org/10.5599/jese.713  

 
Open Access: ISSN 1847-9286 

www.jESE-online.org 
Original scientific paper 

Staircase voltammetry of dissolved redox couple in a thin layer 
twin electrode cell 

Milivoj Lovrić 

Divkovićeva 13, Zagreb 10090, Croatia  

Corresponding author: mlovric@irb.hr  

Received: August 10, 2018; Accepted: September 18, 2019 
 

Abstract 
A model of reversible and quasi-reversible electrode reaction of dissolved redox couple is 
developed for the staircase voltammetry and the twin electrode thin layer cell. It is assumed 
that the neutral molecule is oxidized to a monovalent cation. The calculations were 
performed for the absence of supporting electrolyte and for its various concentrations. The 
influence of migration of cations and of IRΩ drop on the peak currents and peak potentials 
was investigated. Also, the kinetically controlled electrode reactions were simulated. These 
reactions can be distinguished from the reactions influenced by the resistance in the 
solution. 

Keywords 
Steady state voltammetry; two electrodes cell; migration; theory 

 

Introduction 

There is a considerable interest in the investigation of the influence of supporting electrolyte on 

the mass transfer [1-4] and the charge transfer [5] on microelectrodes [6-8] and rotating disk 

electrodes [9,10]. These measurements can be also performed in the thin layer cells [11-15], 

particularly in those that operate without a reference electrode [16-18]. These twin electrode cells 

are useful for the analysis of solutions of redox couples [19-21]. In this paper, a diffusion and 

migration in the micrometer-type cell are simulated for the staircase voltammetry that appears in 

the modern digital instrumentation [22-27]. 

The model 

An electrode reaction of a neutral compound and its cation in the twin electrode thin layer cell is 

investigated. 

Red ↔ Ox+ + e-   (1) 

http://dx.doi.org/10.5599/jese.713
http://dx.doi.org/10.5599/jese.713
http://www.jese-online.org/
mailto:mlovric@irb.hr


J. Electrochem. Sci. Eng. 10(1) (2019) 11-19 STAIRCASE VOLTAMMETRY 

12  

Initially, a solution contains equal concentrations of the compound and the salt of its cation. A 

certain inert electrolyte MX may be also present in the solution. The solvent is polar and supports 

the dissociation of ionic compounds OxA and MX. The model applies, for instance, to the mixture of 

ferrocene and ferrocenium tetrafluoroborate [28]. As the difference between the potentials of two 

electrodes increases, the first electrode acts as an anode and the second one as a cathode. At the 

anode the compound is oxidized and at the cathode the cation is reduced. The current depends on 

the concentrations of ionic species Ox+, A-, M+ and X-. For the sake of simplicity, it is assumed that 

all ionic mobilities are equal: 

uion = FD / RT  (2) 

In eq. (2), D is a common diffusion coefficient. The electric potential gradient in the film depends 

on the ohmic resistance: 

 / x = IR / l (3) 

where I is a current and l is the film thickness. The resistance is estimated using the relationship 

between the equivalent conductivity at infinite dilution and the concentration of ions: 

−= 
1

1 -1
Ω

0

dR S x  (4) 

( ) + −= + + ++ -2 * * * *Ox M X /AF D c c c c RT  (5) 

where S is the electrode surface area and *Yc is initial concentration of ions. Also, an electroneutrality 

in the solution is assumed. 

If the electrode reaction (1) is fast and reversible, the diffusion and migration of electroactive 

species are defined by the following differential equations and the initial and boundary conditions: 

cRed/ t = D2cRed/ x2 (6) 

cOx/ t = D2cOx / x2 – uion( / x)(cOx / x) (7) 

t = 0, 0  x  l:   =* *Ox Redc c , E = 0 (8) 

t > 0, x=0:   = −  
* * 0
Ox, =0 Red, =0 1exp ( ) /x xc c F E E RT  (9) 

E1 – E0 = E / 2 – IR / 2 (10) 

D(cRed / x)x=0 = l / FS (11) 

D(cOx / x)x=0 = - l / FS + uion( / x)cOx, x=0 (12) 

t > 0, x = l:   cOx, x=l = cRed, x=l exp[F (E2 - E0) / RT] (13) 

E2 - E0 = -E / 2 + IR / 2 (14) 

D(cRed / x)x=1 = l / FS (15) 

D(cOx / x)x=l = - l / FS + uion( / x)cOx, x=l (16) 

E is a potential difference between two electrodes. Equations (6) and (7) are solved by the finite 

difference method [29]. By the combination of conditions (9) – (12) a non-linear equation for current 

is obtained. It appears in the form: 

(a + bx) exp(dx) = p – qx – rx2  (17) 

All parameters are positive and a < p. So, the solution of this equation can be found numerically, 

as the cross section of the exponential curve and the parabola.  

The dimensionless current  is defined as a ratio of the current and the limiting current that 

appears under steady state conditions: 



M. Lovrić J. Electrochem. Sci. Eng. 10(1) (2019) 11-19 

http://dx.doi.org/10.5599/jese.713  13 

*
ss Red2 /I FSDc l=  (18) 

In staircase voltammetry the scan rate is defined as a ratio of the potential increment and the 

duration of step: 

 = dE /  (19) 

It was assumed that dE = 1 mV if v  1 V/s and that  = 1 ms if v > 1 V/s. In the calculation the 

dimensionless diffusion coefficient Dt / x2= 0.4 and the time increment t = 10-4 s were used. 

Also, the distance between electrodes was divided in 100 space increments. 

Results and discussion 

The relationship between dimensionless currents and the potential difference in the twin 

electrode thin layer cell in the absence of supporting electrolyte is shown in Figure 1. All curves tend 

to the limit that indicates the establishment of steady state in the cell. At higher scan rates 

voltammograms exhibit maxima, while at lower scan rates they resemble polarographic wave.  
 

 
E / V 

Fig. 1 Dimensionless staircase voltammograms of electrode reaction (1). 
[MX]* = 0 and v / Vs-1 = 2 (1), 1 (2), 0.5 (3), 0.2 (4) and 0.01 (5). 

These characteristics are caused by the development of concentration gradients that are shown 

in Figure 2. At the potential difference of maximum, the diffusion layers are extended to the middle 

of cell, but they do not overlap. One can notice that the gradients of cation are lower than the 

gradients of compound, which is caused by the migration of cations.  

Under steady state the gradient of compound is constant, and its concentration is defined by the 

equation: =*Red Red/ 2 /c c x l . The concentration of cation is given by the equation: 

= − −*Ox Ox/ 2 exp( / ) / ( 1)c c x l e  (20) 

It is the solution of the differential equation: 

= − +*Ox Ox Oxd / d 2 / /c x c l c l  (21) 

And the boundary condition: 

=
*

Ox Ox

0

d
l

c x c l  (22) 

0.1 0.2 0.3 0.4 0.5

0.5

1

1.5

2



E / V

1

2

3

4

5


 

http://dx.doi.org/10.5599/jese.713


J. Electrochem. Sci. Eng. 10(1) (2019) 11-19 STAIRCASE VOLTAMMETRY 

14  

A 

 
x / L 

B 

 
x / L 

Fig. 2 Dimensionless concentrations of the reduced and oxidized components of electrode 
reaction (1). [MX]* = 0, v = 1 V/s and ΔE / V = 0.130 (A) and 1 (B). 

Figure 3 shows logarithmic analyses of responses to low scan rates. If the step duration 𝜏 is so 

long that the steady state is established before the current is measured, the response is defined by 

the following equation: 

 = {exp(FE / 2RT) - 1] / [1 + exp(FE / 2RT] (23) 

This equation can be transformed into the logarithmic form: 

E1 – E0 = 0.059 log [1 + ) / (1 - )] (24) 

The straight line (1) in Figure 3 satisfies eq. (24), but the curves (2) and (3) have lower slopes at 

the smallest potential difference. The curve (3) corresponds to the curve (4) in Figure 1. One can 

notice that the latter current potential curve is steeper than the curve (5) in this figure. This is the 

reason of the lower slope in Figure 3. 

0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

x / L

c
 / c

*

cox

cred

0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

x / L

c
 / c

*

cox

cred

c 
/ 

c*
 

c 
/ 

c*
 



M. Lovrić J. Electrochem. Sci. Eng. 10(1) (2019) 11-19 

http://dx.doi.org/10.5599/jese.713  15 

 
log [(1 + ) / (1 - )] 

Fig. 3. Logarithmic analyses of the responses of electrode reaction (1). [MX]* = 0 and  
v / Vs-1 = 0.01 (1), 0.1 (2) and 0.2 (3) 

Under transient conditions the responses are characterized by maxima. Figure 4A shows the 

relationship between the logarithms of peak current and scan rate. Black circles can be connected 

by the straight line (1): 

log max = 0.425 log   + 0.238 (25) 

This result is in agreement with the observation that the maximum develops before the diffusion 

layers are overlapped. The potentials of maxima increase with the increasing scan rate, as can be 

seen in Fig. 4B. The asymptote (1) is defined by the equation: 

Emax / V = 0.144 log   + 0.086 (26) 

As the electrode reaction (1) is reversible, this relationship can be ascribed to the influence of IR 

drop. 

Figure 5 shows voltammograms in the presence of various concentrations of supporting 

electrolyte. They differ in the properties of maxima. The peak current is the highest if  

[MX]* / *Redc  = 10 and it stagnates if this ratio is higher than 1000. The potential difference of max-

imum is 0.068 V if [MX]*/ *Redc  > 100. These changes can be explained by the diminished IR drop. 

It seems that the contribution of migration is significant if the concentration of supporting 

electrolyte is low, but it vanishes in highly concentrated electrolytes.  

These effects were investigated further by the variation of scan rate in fully supported systems. The 

results are presented in Figure 4, as black squares. Firstly, the peak potentials are independent of scan 

rate. Their average value is 0.074 V, which is shown by the straight line (2) in Figure 4B. Furthermore, 

the peak currents can be approximated by the linear relationship (see the line (2) in Fig. 4A): 

log max = 0.454 log   + 0.255 (27) 

This slope is closer to 0.5 than the slope 0.425 that appears in the absence of supporting 

electrolyte. 

The difference between these two slopes is ascribed to the IR drop. However, in the thin layer 

cell one cannot expect that the current depends on the square root of scan rate, as in the case of 

semi-infinite diffusion. These calculations show that the electrode reaction (1) is controlled only by 

the diffusion if the concentration of supporting electrolyte is thousand times higher than the 

concentration of the electroactive redox couple. This is in agreement with literature data [1,6,10]. 

1 2 3 4

0.05

0.1

0.15

0.2

(E
1
 - 
E

0 
) / V

Log((1+ ) / (1- ))

1 2 3

(E
1
 -

 E
0
) 

/ 
V

 

http://dx.doi.org/10.5599/jese.713


J. Electrochem. Sci. Eng. 10(1) (2019) 11-19 STAIRCASE VOLTAMMETRY 

16  

A 

 
log ( / V s-1) 

B 

 
log ( / V s-1) 

Fig. 4. Dependence of the logarithm of the maximum current (A) and the potential difference of 

the maximum (B) on the logarithm of scan rate. [MX]*/ *Redc  = 0 (1) and 10
3 (2). The straight 

lines are linear approximations 

In the last part of this paper, the influence of the kinetics of electrode reaction is investigated. It 

is assumed that the supporting electrolyte is present in high concentration, so that the migration of 

electroactive cations can be neglected. Under these conditions eqs. (9) and (13) are replaced by the 

well-known Butler-Volmer equations. Some results are shown in Figure 6. Voltammograms depend 

on the normalized kinetic parameter =*S s( / ) / 100k k l , where ks is the standard rate constant of 

electrode reaction, and on the transfer coefficient α. If l = 50 m, the constant *Sk  = 10
-3 s-1 

corresponds to ks = 5×10-4 cm s-1.  

The main difference between Figure 6 and Figure 1 is the inflexion point at 0.2 V in Figure 6. All 

curves in Figure 1 are convex between zero and the maximum, while in Figure 6 the curves start as 

concave. 

 

-0.25 0 0.25 0.5 0.75 1

0.2

0.4

0.6

0.8

log 


max

log(v / Vs-1)

1

2

-0.25 0 0.25 0.5 0.75 1

0.1

0.15

0.2


E
max

 / V

log(v / Vs-1)

1

2

lo
g

 
m

a
x 


E

m
a

x 
/ 

V
 



M. Lovrić J. Electrochem. Sci. Eng. 10(1) (2019) 11-19 

http://dx.doi.org/10.5599/jese.713  17 

 
E / V 

Fig. 5. Influence of the concentration of supporting electrolyte on the voltammograms of 

electrode reaction (1). v = 1 V/s and [MX]*/ *Redc = 0 (1), 1 (----), 2 (-.-.-), 10 (-..-..-) and 5000 (5) 

 
E / V 

Fig. 6. Dimensionless staircase voltammograms of quasireversible electrode reaction (1) in a great excess of 

supporting electrolyte. *Sk = 10
-3 s-1, α = 0.5 and  / Vs-1 = 0.01 (1), 0.5 (2), 1 (3) and 2 (4) 

Figure 7 shows that this change of curvature has the consequence on logarithmic analyses. They 

all tend to the straight line with the slope of 0.118 V for α = 0.5. However, below 0.1 V all curves 

have slopes higher than the limiting one. The effect of too high scan rate is added to the kinetic 

effect. 

The peak currents and peak potentials of voltammograms that are shown in Figure 6 satisfy the 

following relationships: 

max = 0.47 log   + 0.10 (28) 

Emax = 0.118 log   + 0.383 (29) 

The latter one can be used for the estimation of the transfer coefficient together with the 

logarithmic analysis. 

 

0.1 0.2 0.3 0.4 0.5

0.5

1

1.5

2



E / V

1

5

0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

1



E / V

2

3

4


 


 

http://dx.doi.org/10.5599/jese.713


J. Electrochem. Sci. Eng. 10(1) (2019) 11-19 STAIRCASE VOLTAMMETRY 

18  

 
log [1 + ) / (1 - )] 

Fig. 7. Logarithmic analyses of responses of quasi-reversible electrode reaction (1). *Sk  = 10
-3 s-

1, α = 0.5 and  / Vs-1 = 0.01 (1), 0.2 (2) and 0.4 (3). 

Conclusions 

This simplified model shows that the staircase voltammetry can be applied to the electroactive 

redox couple in the twin electrode thin layer cell in the absence of supporting electrolyte. Under 

this condition the response is influenced by an IR drop. At high scan rates the voltammograms of 

reversible electrode reaction exhibit maxima. The potentials at which these maxima appear depend 

linearly on the logarithm of scan rate. In the presence of supporting electrolyte in very high 

concentration, the potentials of maxima are independent of scan rates. This is an evidence of 

reversible charge transfer. 

Presented calculation show that under steady state conditions the concentration profile of ionic 

product is not linear because of migration. All responses tend to the limiting current that is caused 

by these concentration profiles. 

If the electrode reaction is not reversible, but the IR drop can be neglected, the logarithmic 

analysis of the response recorded under steady state conditions is a curve that tends to the straight 

line with the slope equal to RT / αF. In the beginning of this curve, the slope is higher. The logarithmic 

analysis of the response of reversible electrode reaction is not a curve, but the straight line with the 

slope RT / F.  

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M. Lovrić J. Electrochem. Sci. Eng. 10(1) (2019) 11-19 

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