Determination of diffusion coefficients of the electrode reaction products by the double potential step chronoamperometry at small disk electrodes

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Journal of Electroanalytical Chemistry 493 (2000) 93–99

Determination of diffusion coefficients of the electrode reactionproducts by the double potential step chronoamperometry at small

disk electrodes

Haruko Ikeuchi *, Mitsuhiro Kanakubo 1

Department of Chemistry, Faculty of Science and Technology, Sophia Uni6ersity, 7-1, Kioicho, Chiyoda-ku, Tokyo 102-8554, Japan

Received 20 June 2000; received in revised form 27 July 2000; accepted 31 July 2000

Abstract

We propose an easy method for determining the diffusion coefficients of electrode reaction products by double potential stepchronoamperometry at small disk electrodes. The necessary theoretical relationship between current and time for this method wasobtained by digital simulation of a hopscotch algorithm. We quantified the digital results and showed the routine of themeasurement. The method was verified successfully by an experiment in which the diffusion coefficients of both the elements ofthe [Fe(CN)6]4−/[Fe(CN)6]3− redox couple were determined by the usual potential step chronoamperometry and by this method.© 2000 Elsevier Science B.V. All rights reserved.

Keywords: Diffusion coefficient; Digital simulation; Double potential step chronoamperometry; Hexacyanoferrate(II) ion; Hexacyanoferrate(III)ion

1. Introduction

Diffusion coefficients (Ds) of species that have thesame chemical compositions and similar sizes but dif-ferent charges will give valuable information about thetransport mechanisms in solutions. These species oftenappear as redox couples of electrode reactions. There-for we commonly measure D of the reactant of theelectrode reaction by using an electrochemical methodsuch as chronoamperometry [1–3], chronopotentiome-try [4], normal pulse polarography [5,6], or steady-stateamperometry at ultra-micro electrodes [7]. The D of theelectrode reaction product can also be measured bythese electrochemical methods after the bulk of thesolution has been electrolyzed [8]. But only one and notboth species of this kind can be stable for a long timein the usual ambience, except in some special cases, forexample the [Fe(CN)6]4−/[Fe(CN)6]3− redox couple. So

that, if we could measure the Ds of the electrodereaction products in a short time of several seconds,that would be very helpful. The data of this kind areimportant not only in solution chemistry but also arenecessary for rigorous analysis of the electrode kinetics.

For this purpose, a method that uses the scanningelectrochemical microscope has been proposed [9,10],but this method demands expensive instruments andsophisticated techniques. Michael and Wightmanshowed by digital simulation how the shape of thecyclic voltammogram at the micro-disk electrodechanges when the D of the electrode reaction productdiffers as much as several times from the D of thereactant [11]. This fact implies that cyclic voltammetryby the micro-disk electrode could be used for the samepurpose. But the difference in the diffusion coefficientsof the usual redox couple is not so large that it cancause a significant difference on the cyclic voltam-mogram. It might be difficult to precisely measure theD of the electrode reaction product, even if the phe-nomena could be quantified.

The technique of double potential step (DPS)chronoamperometry has been developed and used forthe determination of the rate constants of homogeneous

* Corresponding author. Tel.: +81-3-32383370; fax: +81-3-32383361.

E-mail address: [email protected] (H. Ikeuchi).1 Present address: Tohoku National Industrial Research Institute,

4-2-1 Nigatake, Miyagino-ku, Sendai 983-8551, Japan.

0022-0728/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved.PII: S 0 022 -0728 (00 )00327 -2

H. Ikeuchi, M. Kanakubo / Journal of Electroanalytical Chemistry 493 (2000) 93–9994

chemical reactions that couple with electrode reactions[12–17]. If we apply this method to an electrode reac-tion not coupled with a homogeneous chemical reac-tion, by using a small disk electrode of appropriate size,we can determine the D of the electrode reaction prod-ucts. Actually, Park et al. determined the Ds of themetal atoms in mercury by DPS chronoamperometry athanging mercury drop electrodes (HMDEs) [18]. Theyhad mathematically derived an equation for the cur-rent–time relationship for DPS chronoamperograms ata spherical electrode. Though they used an HMDE ofas large a diameter as 1 mm, a smaller electrode willwork better for more precise measurements, becauseradial diffusion becomes more effective the smaller theelectrode. From this point of view, a disk electrode ispreferable because preparation of a disk electrode ofdesirable size is rather easy, while that of a small idealsphere HMDE is almost impossible. However, we havenot been able to use disk electrodes for this measure-ment, since no theoretical equation that represents thecurrent–time relationship has yet been derived.

The Shoup–Szabo equation [19] has been commonlyregarded as the most precise theoretical equation of thecurrent–time curve at a disk electrode for potential step(SPS) chronoamperometry. Shoup and Szabo devisedthis equation by combining the long time and the short

time mathematical solutions derived by Aoki et al. [20]while taking into account the results of digital simula-tion obtained by themselves and others [2,21]. It wouldbe much more difficult to mathematically solve thediffusion equation of the double potential step tech-nique since the Shoup–Szabo equation is not a puremathematical solution. In this situation, the techniqueof digital simulation is the most suitable for thisproblem.

Here, we perform digital simulation to obtain arelationship between the current and time functions forDPS chronoamperometry, and on the basis of theresults we quantify the dependence of current functionon the ratio of the Ds of the redox couple. We proposean easy method for determining the D of the electrodereaction product by showing the routine of the mea-surement. Computer-processed potentiostats that arenow quite common in laboratories will help in theexperiments and in data analysis.

We verify the method by comparing the D valuesmeasured by SPS and DPS chronoamperometry for the[Fe(CN)6]4−/[Fe(CN)6]3− redox couple.

2. Digital simulation

When a diffusion controlled electrode reaction of Asuch as Eq. (1) occurs at the first potential step, E1, thespecies B generated at the electrode diffuses into thesolution around the electrode, while we observe currentI1.

A=B+ne− (1)

where n is the charge number of the electrode reaction:positive for oxidation and negative for reduction. Atthe second potential step, E2, the species B diffusestowards the electrode and the reverse electrode reactionoccurs. If E1 and E2 are properly set, the observedcurrents at these potentials, I1 and I2, become diffusion-controlled currents. The scheme is sketched in Fig. 1.

The profile of I1 at a microdisk electrode is wellreproduced by the equation:

I1/pnFDAacA=1+p−1/2(a2/DAt)1/2

+0.2732 exp{−0.3911(a2/DAt)1/2}(2)

which was presented by Shoup and Szabo [19]. Here Fis the Faraday constant, DA and cA are the diffusioncoefficient and the bulk concentration of the species A,respectively, a is the radius of the disk electrode and tis the electrolysis time. This equation is plotted in Fig.2 along with the plot of each term. When (a2/DAt)1/2 ismuch larger than 1, then t is short compared to a2/DA,the second term, the linear diffusion term, is predomi-nant in the right side of Eq. (2). As is well known, if the

Fig. 1. (a) A cyclic voltammogram of [Fe(CN)6]4− in 1 mol dm−3

KCl aqueous solution at a Pt Disk electrode of 0.241 mm radius andscan speed of 50 mV s−1 at 25°C. (b) Schematics of the potentialsand the currents of double potential step chronoamperometry.

H. Ikeuchi, M. Kanakubo / Journal of Electroanalytical Chemistry 493 (2000) 93–99 95

Fig. 2. Plot of Eq. (2). The thick solid line represents the profile ofEq. (2). The thin solid lines, 1, 2 and 3 represent the first, second andthird terms of the equation, respectively.

#ci

#t=Di

�#2ci

#r2 +1r#ci

#r+#2ci

#z2

�i=A, B (3)

in the cylindrical coordinates (r, z) of which the originlies on the center of the electrode disk.

This equation is solved for the electrolysis of thesolution for which the bulk concentration of A is cA,0

and that of B is zero. The digital simulation wasperformed in two parts; they are for the first and thesecond potential steps, respectively.

The initial and boundary conditions for both thepotential steps are as follows. Here, t is the durationtime of the first potential step (see Fig. 1).

For the first potential step, 05 t5t,

cA(r, z, 0)=cA,0

cB(r, z, 0)=0

cA(r, 0, t)=0 for 05r5a

For the second potential step, t]t.

cA(r, z, t)=cA,t(r, z)

cB(r, z, t)=cB,t(r, z)

The concentrations, cA,t(r, z) and cB,t(r, z), are givenas the results of the calculation of the first potentialstep at time t.

cB(r, 0, t)=0 for 05r5a

For both the potential steps,

�#cA

#z�

z=0

= −�#cB

#z�

z=0

for 05r5a

�#ci

#z�

z=0

=0 i=A, B for r\a

cA(r��, z��, t)=cA,0

cB(r��, z��, t)=0

The quantity we observe is the diffusion current at eachpotential step; that is given by

Ij=2pnFD& a

0

�#ci

#z�

z=0

r dr i=A, B for j=1, 2

The technique of the digital simulation is conven-tional and is the same as that used by Shoup and Szabo[19]. The space adjacent to the disk electrode surface isdivided into concentric rings and cylinders as shown inFig. 3, where L is the number of rings and a cylinder onthe disk, and is related to a by:

L=aDr

+0.5 (4)

Fig. 3. Discretization of the space adjacent to the electrode, for digitalsimulation.

diffusion is completely linear, the diffusion current atthe second potential step is solely controlled by DA, andnot by D of B, DB [22,23]. Therefore if the pulse widthof the first potential step, t, is so small that I1 can beregarded as almost a linear diffusion current, I2 iscontrolled mainly by DA, and a little by DB. If t is longenough, I2 is controlled by not only DA but also DB. Ifthe equation that represents the I2– t curve is known,we can then extract DB from it.

We applied the digital simulation on DPS chronoam-perometry at a disk electrode for deriving a theoreticalrelationship between the currents and electrolysis timefunctions. The digital simulation was performed by thesimple but efficient hopscotch algorithm that was origi-nally proposed by Gouray [24] and has been applied tochronoamperometry at a disk electrode and some otherelectrochemical problems by Shoup and Szabo[19,25,26].

2.1. Procedure of the digital simulation

The partial differential equation is formulated as:


The electrolysis time, t, is also divided by finite timeDt. t is represented by:

t=kDt (5)

where k is the number of iterations.We chose the iteration numbers of the first potential

step, k1, so that the first term of Eq. (2) becomescomparable with the second term. Thus the durationtime of the first potential step, t is given by:

t=k1Dt (6)

From Eqs. (4) and (6) with the aid of the equation:

dA=DADt(Dr)2 (7)

we can obtain

� a2

DAt

�1/2

=(L−0.5)(dAk1)1/2 (8)

from which we can calculate the variable (a2/DAt)1/2.We chose three parameter sets, as shown in Table 1,

and performed the digital simulation for eight differentratios of DB:DA from 0.4 to 2.0 for each parameter set.The values of the variable, (a2/DAt)1/2, are also listed inTable 1, and are indicated in Fig. 2.

The output for the first potential step is the currentfunction at each iteration, I1(k), and the concentrationfunction of each ring and cylinder at the end of the firstpotential step, c(i, j, k1). One can calculate the currentfunction at the second potential step, I2(k), by using thevalues of the concentration functions c(i, j, k1) as theinitial condition.

In order to relate the results of the digital simulationto the experimental DPS chronoamperogram, we intro-duce the ratios, x and y :

x=�t−t

t�1/2

=�k−k1

k�1/2

(9)

y=I2(t)

I1(t−t)=

I2(k)I1(k−k1)

(10)

where, t\t and k\k1.Programs for digital simulation were compiled by

Lahey Fortran, and the calculation was performed bypersonal computer, AV1/55CD-95 (Sharp Co., CPU:Pentium 75 MHz, 16 MB memory).

2.2. Results of the digital simulation

The results of the digital simulation for the DPSchronoamperogram cannot be unified into one equa-tion, because I2 is a function not only of DB, t, cA.0 anda but also of DA and t. Therefore, the results arerepresented by the group of curves for three differentdigital simulation parameter sets. For example the re-sults for parameter set no. 1 are shown in Fig. 4. Thecurves are characterized by the ratios, DB/DA, and thewider they are separated, the smaller are the values of(a2/DAt)1/2. We can determine DB/DA then DB when DA

is known, by superimposing the experimental I2(t)/I1(t−t) versus {(t−t)/t}1/2 curves on the correspond-

Table 1Sets of parameters for the digital simulation

(a2/DAt)1/2dAk1LParameter set no.

1 50 4000 0.2 1.7540002 0.275 2.485000 0.2 3.151003

Fig. 4. The results of the digital simulation for parameter set number1 represented by y versus x curves.

Fig. 5. Relationship between y and DB/DA at different x values. Filledcircles are the values of y on the curves at the definite x values readfrom the curves of Fig. 4(a), and the solid lines are fitted quadraticcurves.

Table 2The values of t, for the different digital simulation parameter setswhen a=0.101 mm

Parameter set no.Species 1010DA/m2 s−1 t/s

1 5.0396.609[Fe(CN)6]4−

2 2.2251.5593

1[Fe(CN)6]3− 4.3017.7442 1.899

1.3313


Table 3Parameters of the quadratic Eq. (11) fitted to the y versus (DB/DA) curve for different values of x a

p qParameter set no. sx s2

0.11 1.101(10) −0.378(19) 0.0857(76) 0.00070.858(6) −0.275(11)0.2 0.0602(44) 0.0002

0.3 0.673(4) −0.214(8) 0.0462(33) 0.00010.518(3) −0.166(6) 0.0357(26) B0.00010.4

1.049(7) −0.265(13)2 0.0561(54)0.1 0.00030.845(3) −0.189(7) 0.0381(28)0.2 0.00010.684(3) −0.149(5)0.3 0.0293(21) B0.0001

0.4 0.543(2) −0.118(4) 0.0228(16) B0.0001

1.022(6) −0.217(11)3 0.0440(43)0.1 0.00020.2 0.837(3) −0.157(6) 0.0303(24) 0.00010.3 0.686(2) −0.125(4) 0.0236(18) B0.0001

0.551(2) −0.100(3)0.4 0.0185(14) B0.0001

a The standard error in the least significant digits is given in parentheses.

ing group of y versus x curves. For convenience, thevalues of y on the curves of different values of DB/DA ata definite value of x, for example 0.1, 0.2, 0.3 and 0.4are plotted against DB/DA, and the plots are fitted by aquadratic curve. One example is shown in Fig. 5, andthe values of the parameters of the quadratic equation(Eq. (11)) are given in Table 3 along with the varianceof the fitting for the three cases of the digitalsimulation.

y=p+q�DB

DA

�+s

�DB

DA

�2

(11)

3. Experimental

Potassium hexacyanoferrate(III), potassium hexa-cyanoferrate(II) trihydrate (Tokyo Chemical IndustryCo. Ltd., reagent grade, B99% up) and potassiumchloride (Wako Pure Chemical Industries, reagentgrade, B99.5%) were used as purchased.

A potentiostat, digital universal signal processingunit HECS 326 with head box 326-2 Fuso ElectroChemical System Co. was remodeled to an auto-range-change system of 16 ranges from 32.25 nA to 1.024mA. Test electrodes were platinum disk electrodes em-bedded in glass. Their radii were measured with ameasuring optical microscope. Two electrodes of 0.101and 0.241 mm radii were used in SPS chronoamper-ometry and the former one in DPS chronoamperome-try. The electrode was polished with aluminum oxide ofaverage grain size of 0.03 mm on a turntable for severalminutes before every series of measurements. A refer-ence electrode, Ag � AgCl � 3 M NaCl (BAS Co.), andplatinum wire counter electrode were used.

The procedure for the determination of DB is asfollows.1. Determine DA by the usual SPS chronoamperome-

try at the potential of the first step.

2. Choose a disk electrode of suitable size. The radiusa is known.

3. Choose a parameter set of the digital simulation.The values of L, k1 and dA are known.

4. Calculate t from

t=a2dAk1

DA(L−0.5)2 (12)

5. Acquire the DPS chronoamperogram for which thet value is calculated in procedure (4), by using thedisk electrode chosen in procedure (2).

6. Calculate the value of I2(t)/I1(t−t) at the timewhen the value of {(t−t/t)}1/2 is for example 0.1,0.2, 0.3 and so on.

7. Insert the value of I2(t)/I1(t−t) obtained in proce-dure (6) to y, and the corresponding parametervalues (Table 3) to p, q and s in Eq. (11) and thencalculate DB/DA. The value of DB can be obtainedsince DA is known by procedure (1).

In procedure (1), we used the SPS technique that hadbeen developed for the determination of reliable andprecise diffusion coefficients [27,28]. In this technique,the product of the concentration of the diffusing species(c) and the charge number of the electrode reaction (n)is determined together with the diffusion coefficientfrom the chronoamperogram, and the value is used forthe assessment of the reliability of the diffusion coeffi-cient obtained. If the determined value (ncobs) agreeswith the known value (nccalc), the diffusion coefficientobtained with it should be correct.

The chronoamperograms were recorded over a timeperiod of about 0.1–10 s. The applied potentials are asfollows: for the measurement of [Fe(CN)6]4−, E0=80mV, E1=400, 450 or 500 mV, and for [Fe(CN)6]3−,E0=450 or 500 mV, E1=50, 75, 100 or 125 mV.

As we chose a platinum electrode of radius 0.101 mmfor DPS chronoamperometry, t calculated by means of


Table 4Diffusion coefficients in 1 mol dm−3 KCl aqueous solution at 25°C determined by SPS chronoamperometry a

ncobs/nccalc 1010D/m2 s−1 ReferenceSpecies E1/mV

Average Average

1.004(5) 6.62(5)[Fe(CN)6]4− 6.61(4)400 6.32(3) [1]1.012(9) 1.006(4) 6.61(10)450

500 1.007(7) 6.57(10)

[Fe(CN)6]3− 125 0.996(12) 7.67(19)0.997(8) 1.004(4) 7.68(12)100 7.74(6) 7.84(2) [2]1.011(7) 7.83(12)75 7.63(2) [1]1.016(11) 7.82(18)50

a The 95% confidence limit to the least significant digit is given in parentheses.

Eq. (12) for the parameter set nos. 1, 2 or 3 became thevalues shown in Table 2. The D values of [Fe(CN)6]4−

and [Fe(CN)6]3− in 1 mol dm−3 KCl aqueous solutionswere determined with DPS chronoamperometry by us-ing solutions that contained 1 mol m−3 [Fe(CN)6]3−

and [Fe(CN)6]4−, respectively. The applied potentialsare as follows: for the measurement of: [Fe(CN)6]4−,E0=500, E1=75, 100 or 125, E2=500 mV; and for[Fe(CN)6]3−, E0=80, E1=450, E2=80 mV.

4. Results and discussion

The values of D determined by SPS chronoamper-ometry are shown in Table 4. The deviations of theratio ncobs/nccalc and D by E1 were all within the limitsof experimental error as shown in Table 4. Also, theradius of the electrode did not affect these valuessignificantly. The ratios for both the species are close to1, and thus the obtained D values are reliable.

One example of the DPS chronoamperograms isshown in Fig. 6. The I2(t)/I1(t−t) versus {(t−t)/t}1/2

curves calculated from the observed DPS chronoamper-ograms for [Fe(CN)6]3− and [Fe(CN)6]4− are superim-posed on the theoretical y versus x curves in Fig. 7. Thecurve for [Fe(CN)6]4− lies between the curves for whichDB/DA values are 0.8 and 1.0, and that for [Fe(CN)6]3−

lies between those whose values are 1.0 and 1.25. Wecalculated DB/DA values by Eq. (11) from the values ofI2(t)/I1(t−t) at the values of {(t−t)/t}1/2 of 0.2 and0.3. The values of DB/DA were independent of theapplied potentials as in the case of SPS chronoamper-ometry. The results are shown in Table 5. We calcu-lated DBs by using DB/DA and the corresponding DA

values listed in Table 4. The errors in DB shown inTable 5 were estimated from the errors of DB/DA and ofDA. The diffusion coefficients determined by DPSchronoamperometry agree within 3% for [Fe(CN)6]4−

and within 2% for [Fe(CN)6]3− with those determinedby SPS chronoamperometry. We can conclude that themethod that has been developed here for the determina-

tion of the diffusion coefficient of electrode reactionproducts works successfully.

In this work, we used a disk electrode of 0.101 mmradius taking into account the precision of the sizemeasurement and the capacity of our potentiostat. Asmaller electrode would work more efficiently, because

Fig. 6. (a) An example of a DPS chronoamperogram of [Fe(CN)6]3−

in 1 mol dm−3 KCl aqueous solution at a Pt-disk electrode of 0.101mm radius with t of 4.302 s, at 25°C. (b) The residual current of (a).

Fig. 7. Fitting of the observed I2(t)/I1(t−t) versus {(t−t)/t}1/2

curves on the y versus x curves. Measured from a solution of[Fe(CN)6]3− (open circle), and [Fe(CN)6]4− (filled circle) at a Pt-diskelectrode of 0.101 mm radius, t of 4.302 s, at 25°C.


Table 5Diffusion coefficients determined by DPS chronoamperometry by three digital simulation parameter sets a

Parameter set no.Species B Number of runs DB/DA 1010DB/m2

s−1

Average

7 0.869(8)[Fe(CN)6]4+ 12 6 0.889(6) 0.882(5) 6.82(9)3 6 0.887(10)

9 1.15(2)[Fe(CN)6]3+ 19 1.16(3) 1.15(2) 7.61(15)29 1.15(4)3

a The 95% confidence limit in the least significant digit is given in parentheses.

a shorter t can make the first term of Eq. (1) smallercompared with the second term. For this purpose wehave to overcome the problem of precise measurementof the electrode size and the roughness of its surfacethat must be small enough compared with the thicknessof the diffusion layer formed in the time t. A smallerelectrode would make this technique usable for mea-surement of the diffusion coefficient of the electrodereaction products coupled with a homogeneous chemi-cal reaction.

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Determination of diffusion coefficients of the electrode reaction products by the double potential step chronoamperometry at small disk electrodes