In the Laboratory
A Simple Numerical Procedure for the Simulation of “Lifelike” Linear-Sweep Voltammograms
W
Benedetto Bozzini INFM – Dipartimento di Ingegneria dell’ Innovazione, Università di Lecce, V. Arnesano, I-73100 Lecce, Italy;
[email protected] The well-known linear-sweep voltammetric (LSV) method consists in applying a potential ramp using a potentiostat and recording the current response (1). Although the method is reasonably easy to employ and therefore widely used and rich in pedagogical value, mechanistic conclusions from an actual plot tend to be rather fuzzy owing to the presence of several concurrent phenomena. The present paper addresses the following problem: how can an undergraduate student in physical chemistry or a research student in electrochemistry perform a “lifelike” numerical simulation of an LSV curve that resembles an actual situation observed in the laboratory (i.e., includes masstransport and ohmic contributions in addition to activation)? If an answer to this question is available, the student can get a feeling of how chemical quantities (such as transfer coefficients or exchange current densities) and operating parameters (hydrodynamic conditions, working electrode–to–reference electrode distances) will affect the outcome of a measurement. In addition, the reverse procedure is possible, and the predictive algorithm can be run as the model routine in a nonlinear regression program to estimate electrochemical parameters. This paper should be of interest to those involved in teaching electrochemical kinetics and may be of particular interest to those involved in metal deposition. The method to be described has also been applied in a research paper on electrodeposition engineering (2). Curiously, advances in research on numerical analysis of LSV curves have never been reported in monographs. There are analytical expressions describing several kinds of kinetic control and instrumental effects, but no mention is made of numerical analysis of actual experimental data. As for the phenomenological study of overall curves and the analytic study of single effects, even though some very fine texts have appeared recently (3, 4 ), some classic texts are still unrivaled for cathodic studies (5–7) or anodic studies (8). The problem of databases for anodic LSV curves has recently been raised (9), but no attempt at a quantitative study of the overall curves is mentioned. The author of ref 9 simply proposes to use current density (c.d.) and voltage values corresponding to a set of “critical points” of the LSV curve as electrochemical parameters. In addition, almost all of the simulation work in electrochemical engineering is carried out by using Butler– Volmer and Tafel expressions for electrode kinetics (as reviewed in ref 10). This Journal has published several papers on electrochemical kinetics in which mention is made of LSV curves, but all of these fail to discuss to what extent and under which assumptions the proposed expressions represent a realistic experimental situation (11–19).
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Description of the Approach
Contributions to Electrode Overpotential Individual contributions to kinetic control are well known and thoroughly described in the references mentioned in the previous section, the main ones being listed below for symbolic consistency. The overpotential related to activation control can be expressed with the Butler–Volmer equation (1):
j = j 0 exp
1 – α z F ηa αz F η a – exp RT RT
(1)
where j is c.d., j0 is the exchange c.d., α is the transfer coefficient, z is the ionic charge of the electroactive species, F denotes Faraday’s constant, ηa is the activation overpotential (i.e., the fraction of the applied overpotential ηLSV that is actually employed for the electron tunnelling process), R is the gas constant, and T is the absolute temperature. Overpotential related to mass transport control (ηtr denoting the mass-transport overpotential) can be described by the limiting c.d. (eq 2):
j η tr = RT ln 1 – j zF lim
(2)
where: j lim is the limiting c.d., which can be simply related to hydrodynamic conditions through eq 3 (3):
jlim =
zFDC b δ
(3)
where Cb and D are the bulk concentration and diffusion coefficient of the electroactive species and δ is the Nernst boundary layer thickness. Ohmic drops are described by eq 4:
η ohm =
L ⋅ j = RjA Λ Σ i Ci
(4)
i
where L is the working electrode–to–reference electrode distance, Λi and Ci are the molar conductivity and concentration of the ionic species i, R is the electrolyte resistance between the reference and the working electrode, and A is the cross section of the electrolytic cell (assumed equal to the electrode surface area).
Journal of Chemical Education • Vol. 77 No. 1 January 2000 • JChemEd.chem.wisc.edu
In the Laboratory
Combination of Overpotential Contributions to Predict the Measured Electrode Overpotential The sum of true electrochemical contributions η a and ηtr can be denoted by ηelchem: η elchem = ηa + η tr
(5)
and the overall overpotential ηLSV can be written ηLSV = η elchem + ηohm
(6)
The problem in obtaining an LSV curve that accounts for all kinds of overpotential mentioned is computing a value of η a that defines a value of j compatible with mass-transport limitations and ohmic losses. Numerically, the problem is to find a single root for the transcendental eq 7 in a physically meaningful potential range; the practical equations are well behaved, as shown in the examples reported below. η a( j) = ηLSV – ηtr( j) – η ohm( j)
(7)
Equation 1 refers to a single faradic reaction but can be generalized to any number of concurrent reactions without additional problems. An example is reported below. Scan rates (SR) are considered to be in the range within which quasistationary conditions prevail (S R ≤ 2 mV s 1); that is, no dynamic effects can be expected (3).
Previous Approaches to Mixed Activation/Mass-Transport Control Two simple analytical relations were proposed in the literature to try to correct LSV plots for mass-transport effects, eq 8 (3) and eq 9 (20), without considering ohmic effects: 1=1 +1 j ja ja
(8)
Applications In this Section two specific applications are described: (i) electrodeposition of Cu with and without ohmic drops, under different hydrodynamic conditions, and (ii) electrodeposition of Co with concurrent H 2 evolution, masstransport limitations, and ohmic drop. A MATLAB program is available on the JCE Online Web site.W
Electrodeposition of Cu from a CuCN Bath LSV curves are simulated for the electrochemical process characterized by the parameters reported in Table 2, which are typical for a cyanoalkaline bath (CuCN 1 M, KCN 0.4 M, Table 1. Differences beween Estimates of j and LSV from the Literature and from Eq 7 for Simulated Cu Experiment ∆ Av ∆ Peak Comparison Parameter (%) (%) Eq 8 a vs eq 7
where ja is the c.d. under pure activation control, and
j j –j η LSV = RT ln 0 ⋅ lim j lim j αnF
below) quite accurately. Typical average and peak differences in overpotential and c.d. estimates obtained by comparing eq 7 with eqs 8 and 9 are reported in Table 1. Electrochemical constants refer to the case discussed in the first application in the next section. Apart from the serious distortions of the curves caused by exceedingly crude approximations in eqs 8 and 9, the form of these equations does not allow simple modifications that can take ohmic drops into account. Other approaches (21–23) are based on rather elaborate mathematical and statistical procedures and can be used fruitfully in research applications; but they provide little physical insight and are basically devoid of didactic value, and are therefore not considered here.
Eq 8 a vs eq 7
(9)
Both of these theoretical equations, 8 and 9, have proved inadequate (see Fig. 1), since they tend to underestimate mass-transport effects with respect to the present theoretical approach (eq 7), which can simulate experimental data (see
b
Eq 9 vs eq 7 Eq 9 b vs eq 7
j ηLSV
6
15
13
42
12
28
65
450
NOTE: Data are for overpotential and c.d. estimates for the simulation of Cu electrodeposition from a cyanoalkaline bath. aRef 3. bRef 20.
Table 2. Electrochemical Parameters for Simulated Cu Experiment Parameter
Value
T
298 K
R
8.31 J K 1 mol 1
F α
96, 500 C eq 1
z
1 eq mol 1
j0
1 mA cm 2
jlim Λ C
Figure 1. Predictions of linear-sweep voltammetric curves with eq 8 of ref 3 and eq 9 of ref 20 compared with the exact numerical approach proposed in this work (eq 7); electrochemical constants refer to the case of Cu electrodeposition from a cyanoalkaline bath.
j ηLSV
L A
0.6
∞, 100 mA cm 2 200 S cm2 mol 1 1M 0, 0.08, 0.4, 0.8 cm 4 cm 2
NOTE: Simulated Cu electrodeposition from a cyanoalkaline bath.
JChemEd.chem.wisc.edu • Vol. 77 No. 1 January 2000 • Journal of Chemical Education
101
In the Laboratory
Figure 2. Simulated linear-sweep voltammetric curves computed with eq 10, case of Cu deposition from a cyanoalkaline bath with different ohmic drops (R) and limiting current densities ( j lim). (a) R = 0 Ω, j lim = ∞; (b) R = 0 Ω, j lim = 100 mA cm2; (c) R = 0.1 Ω, j lim = 100 mA cm2; (d) R = 0.5 Ω, j lim = 100 mA cm2; (e) R = 1.0 Ω, j lim = 100 mA cm2.
pH 12). For this application, eq 7 takes on the particular form
j Fηa/RT Fηa/RT η a = η LSV – RT ln 1 – 0 e 1– α – e α j F lim R j 0 A e 1– α
Fηa/RT
Typical results are plotted in Figure 2.
Electrodeposition of Co from a Sulfamate Bath with Simultaneous H2 Evolution LSV curves were simulated for the parameters reported in Table 3 and compared with experimental results for a bath of Co(NH2SO3)2 (1 M, pH 6.5), where ohmic drops were minimized with a lateral channel probe (24 ).
Parameter
Value
T
300 K 1
1
R
8.31 J K mol
F α (Co)
96, 500 C eq 1
α (H)
0.430
z (Co)
2 eq mol 1
z (H)
1 eq mol 1
j 0(Co)
2 × 103 mA cm 2
j 0(H)
6 × 103 mA cm 2
j lim(Co)
85 mA cm 2
j lim(H)
0.115 mA cm 2
Eeq(H/ H+)
148.7 mV
Eeq(Co/Co 2+)
277 mV
0.775
R
0.005, 1.0 Ω
A
4 cm 2
NOTE: Simulated Co–H2 codeposition from a sulfamate bath.
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j Co f ln 1 – 0 Q – j lim Co 1 2
(10)
– e α
Table 3. Parameters for Simulated Co–H2 Experiment
In this case eq 7 becomes eq 11: η a = η LSV –
– Fηa/RT
Figure 3. Simulated linear-sweep voltammetric curve ( j tot) for codeposition of Co and H2 with parameters reported in Table 3 (R = 0.005 Ω ) and partial curves for H2 (jH) and Co (jCo) deposition.
f ln 1 –
j0 H Q – R A j 0 Co Q 1 + j 0 H Q 2 j lim H 2
(11)
where f = RT/F,
Q 1 = exp 1 – α Co
z Co F η a Co – RT
z Co F η a Co exp α Co RT Q 2 = exp 1 – α H
z H F ηa H RT
exp α H
(11.1)
–
z H F ηa H RT
(11.2)
A fine point is implied in relating ηa(Co) and ηa(H) (the effective overpotentials for deposition of Co and H2) to ηLSV and ηa; such quantities depend on the mixed potential ηmix prevailing at the cathode for the coexistence of two reactions. Since the reaction does not alter the composition of the cathodes, the same ηmix values can be used throughout the LSV run. The same is not valid for alloy electrodeposition; ηmix is defined by eq 12: j0(Co) Q1 = j0(H) Q2
(12)
it follows that ηa(Co) and ηa(H) are defined by eqs 13: ηa(Co) = ηmix + ηa – Eeq(Co)
(13.1)
ηa(H) = ηmix + ηa – Eeq(H)
(13.2)
where Eeq denotes equilibrium potential.
Journal of Chemical Education • Vol. 77 No. 1 January 2000 • JChemEd.chem.wisc.edu
In the Laboratory W
Supplemental Material
A simple MATLAB program for use with this approach to the numerical simulation of current density vs potential curves in electrochemical kinetics is available; see this issue of JCE Online. Literature Cited 1. 2. 3. 4. 5. Figure 4. Simulated linear-sweep voltammetric curves (Table 3, R = 0.005 Ω and R = 1.0 Ω ) and experimental curves for codeposition of Co and H2.
6. 7. 8.
Results of overall (jtot) and partial (jH, jCo) c.d.’s are shown in Figure 3. Simulated (jsim(1) R = 0.005 Ω , jsim(2) R = 1.0 Ω) and experimental ( jexp) results are compared in Figure 4. Conclusions This paper describes a simple approach to the numerical simulation of current density vs potential curves in electrochemical kinetics. It is designed for classroom exercises meant to provide a physical insight into the effects of electrochemical and operating variables. This approach is a unique teaching tool for the simulation of actual experimental conditions. It provides a working feeling for lifelike scenarios involved in data correction work for mass-transport and ohmic effects. In addition, a student can get a lively idea of how quantitatively important omissions of accurate corrections can be in experimental work. The method is also useful for research, as it supplies a physically straightforward option for data correction and parameter estimation in linear-sweep voltammetric work.
9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
22. 23. 24.
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