On the Extraction of Double-Layer Capacitances for Nonideal

Article pubs.acs.org/IECR

On the Extraction of Double-Layer Capacitances for Nonideal Capacitive Behaviors Muzaffer Ö zcan,*,† Il̇ yas Dehri,‡ and Mehmet Erbil‡ †

Department of Science and Technology Education and ‡Department of Chemistry, Cukurova University, 01330 Adana, Turkey ABSTRACT: The equation for ideal capacitive behavior of a metal/solution interface was modified in such a way that it allows direct extraction of double-layer capacitance (Cdl) values as a function of the frequency from complex plane plots exhibiting nonideal behavior. Experimental impedance data verified by the Kramers−Kronig (K−K) transforms were presented for a mild steel/hydrochloric acid interface in the absence and presence of an organic additive. The results revealed that the modified equation works well for real systems. The rod was placed in a polyester resin in which only 0.5 cm2 of the surface area was subjected to the solution. The electrolyte was a 0.5 M hydrochloric acid (HCl) solution without and with various concentrations (0.1, 0.5, 1, and 5 mM) of 2-(2hydroxybenzylideneamino)pyridin-3-ol (HBAP). The solutions were prepared using analytical-reagent-grade HCl in distilled water. Before each measurement, the electrode surface was polished with a sequence of Emery papers up to 1200 grade to a mirror finish, degreased in acetone, rinsed with distilled water, dried with soft paper, and then immediately placed in an open glass cell containing 100 mL of solution. All of the measurements were carried out at 25 ± 1 °C in solutions under unstirred conditions after 30 min of exposure time. Impedance measurements were performed in the 100 kHz to 10 mHz frequency range using a CHI 604B AC electrochemical analyzer. The data were obtained at the open-circuit potential (Eocp) of the working electrode and measured against an Ag/ AgCl (3 M KCl) reference electrode, with a 5 mV sine wave as the excitation signal. A platinum counter electrode with 1 cm2 surface area was also used. The K−K transforms and determination of the Cdl values by fitting the experimental results to an appropriate equivalent circuit were carried out using ZView software from Scribner Associates, Inc.

1. INTRODUCTION Though the application of impedance in various areas of electrochemistry is increasing, certain aspects are still vague, and contradictory results can be found in the literature, particularly for the extraction of double-layer capacitance (Cdl) values. A number of papers exist on the analysis of impedance data with a variety of plot methods.1−3 In general, analysis of the impedance data is performed with the complex plane plot’s imaginary component (ZIm) as a function of the real component (ZRe) for a range of frequencies. As is known, the imaginary component is related to the capacitance of a capacitor. Therefore, it is possible to determine the charge that the metal/solution interface stores, namely, Cdl for ideal capacitive behavior, by setting the derivative of the imaginary component (with respect to ω) equal to zero:4 Cdl(id) =

1 ωR p

(1)

where ω is the angular frequency and Rp the polarization resistance. Because ω is the angular frequency at which the imaginary component reaches its maximum ( f max), it is not possible to use eq 1 at other frequencies. Nevertheless, the charge storage at the metal/solution interface occurs at every frequency. It is also not appropriate to use the same equation for real systems exhibiting nonideal behavior. It is critical to be vigilant with regard to the validity of the impedance data before evaluating the experimental results. It can be misleading, however, to look only at the results to ascertain whether the data are sufficiently valid or not. A general mathematical procedure is available for verification of the impedance data, i.e., the Kramers−Kronig (K−K) transforms.5−9 In parallel with the discussions above, the objective of this paper is to modify eq 1 in such a way that it would be possible to directly extract Cdl values as a function of the frequency from complex plane plots exhibiting nonideal behavior.

3. RESULTS AND DISCUSSION Figure 1 shows the impedance results of the mild steel/HCl interface as a function of the concentration of HBAP in the form of complex plane plots that are similar in shape, except for the diameters. The complex plane plots seem to meet the expectations from theory; however, a careful inspection reveals that none of the diagrams are perfect semicircles with some data scatter at the low-frequency region, usually observed at that region.10,11 Although the appearance at the metal/solution interface is similar to the charge storage in a capacitor, the solution side is

2. EXPERIMENTAL DETAILS The working electrode was a cylindrical rod cut from a mild steel rod having the following compositions (wt %): C, 0.17; Mn, 1.40; S, 0.045; P, 0.045; N, 0.009; Fe, remaining amount. © 2012 American Chemical Society

Received: Revised: Accepted: Published: 14061

June 18, 2012 August 16, 2012 September 28, 2012 September 28, 2012 dx.doi.org/10.1021/ie301609j | Ind. Eng. Chem. Res. 2012, 51, 14061−14064

Industrial & Engineering Chemistry Research

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the charge storage at the interface occurs at every frequency. Therefore, the frequency until which charge storage is continued and charge transfer starts with deactivation of the frequency-related polarity change should be determined to clarify the interactions at the metal/solution interface. For this, the dependence of the capacitance on the frequency can be used, which is obtained from the total impedance of the circuit that represents the metal/solution interface, and is given by4 Cdl(re) =

1 ⎛ Z Im ⎞ ⎜ ⎟ ωR p ⎝ Z Re − R s ⎠

(2)

where all of the terms have the usual meanings. It is possible with eq 2 to determine Cdl at every frequency in the studied frequency range because ZIm and ZRe can be measured simultaneously as a function of the frequency. This possibility makes it possible to estimate the rate (υ) at which the capacitance changes with the frequency. This rate gives information about the charging and discharging time of the metal/solution interface. The above-mentioned rate values for the systems under investigation are given in Table 1. Figure 1. Complex plane plots for mild steel in a 0.5 M HCl solution as a function of the HBAP concentration: R1, solution resistance; CPE1, constant phase element; R2, polarization resistance.

Table 1. Rate Values at Which the Capacitance Changes with the Frequency as a Function of the HBAP Concentration

dispersed in a larger volume. The equivalent circuit with a series of capacitors in which the charge storage capacity is decreasing away from the surface is the most appropriate one to represent the dispersion on the solution side in a larger volume (Figure 2).

a

C (mM)

υ (μF decade−1)a

C (mM)

υ (μF decade−1)a

0.0 0.1 0.5

48.25 14.16 7.86

1.0 5.0

7.71 7.73

Extracted from the slope of a semilogarithmic graph, Cdl vs log f.

It is clear that the rate values decreased as the concentration of HBAP increased, and accordingly the required time for the interface to be charged or discharged became longer. It is also possible with eq 2 to determine Cdl at only f max for the systems that deviate from ideal behavior, as is the case here because the last term in eq 2 is always less than 1 at that frequency. The above-mentioned Cdl values for the systems under investigation are given in Table 2. To determine how eq Table 2. Double-Layer Capacitance Values at f max as a Function of the HBAP Concentration CPEdl molecule blank HBAP

C (mM)

Cdl(×10 s Ω−1 cm−2)a

Y0(×10 sn Ω−1 cm−2)

n(0−1)

Cdl(×106 s Ω−1 cm−2)b

0.1 0.5 1.0 5.0

69.6 21.7 11.6 10.3 10.2

198.0 49.0 28.3 24.8 23.9

0.799 0.865 0.848 0.859 0.841

80.1 26.7 14.7 13.5 12.8

6

6

Figure 2. (a) Metal/solution interface, (b) an ideal capacitor, and (c) an equivalent circuit model representing the metal/solution interface: Rs, solution resistance; Ceq, corresponding to Cdl; Rp, polarization resistance.

Cdl values were extracted at f max with eq 2. bCdl values were extracted from CPE parameters Y0 and n.

As is known, in capacitors connected in series, the equivalent capacitance (Ceq), corresponding to Cdl, is less than any of the series capacitors’ individual capacitances. Therefore, the capacity to store the charge of the metal/solution interface is always less than that of an equivalent ideal two-plate capacitor. The capacitance at the frequency at which ZIm reaches its maximum represents a value after which charge transfer starts. After that frequency, the Faradic current is effective. However,

2 works for real systems, the complex plane plots were also fitted to the equivalent circuit given as an inset in Figure 1, which is generally used to model the mild steel/acid interface.12−16 A comparison of the results in Table 2 (columns 3 and 4) shows that none of the results agree. At first glance, this may be perceived as a failure of eq 2 for real systems. However, this is clearly due to the determination of Y0 with fitting rather than

a

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4. CONCLUSIONS 1. A circuit with a series of capacitors in which the capacity to store charge decreases away from the surface was used to model the metal/solution interface for representing the dispersion of the solution side in a larger volume. 2. The rate (υ) at which the capacitance changes with the frequency decreased as the concentration of HBAP increased. 3. The agreement between the Cdl values obtained from two different methods at f max indicates that eq 2 can directly be used for real systems exhibiting nonideal behavior without the need for any fitting and conversion processes. 4. The experimental and transformed impedance data for both real and imaginary components were observed to be close to each other, with AE values of less than 2.1%.

Cdl. Therefore, a conversion should be performed with the following equation17,18 before comparison: Cdl = Y0(ωm″)n − 1

(3)

After conversion, an excellent agreement between the Cdl values (columns 3 and 6) was observed, which is an indication of the suitability of using eq 2 for real systems exhibiting nonideal behavior. The validity of the data was checked with the K−K transforms to determine whether they represent the systems under investigation. As a representative example, results of the transformations of the complex plane plot 5 in Figure 1 are given in Figure 3. In these figures, plot a compares the result of

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AUTHOR INFORMATION

Corresponding Author

*Tel: +90 (322) 3386084, ext. 2789. Fax: +90 (322) 3386830. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors thank the Cukurova University Research Fund for financial support. The authors are also grateful to Prof. Dr. Osman Serindağ for permission to use his research laboratory for the experiments and Assistant Prof. Dr. Gökhan Gece for improving the English of the paper.

■ Figure 3. K−K transforms of impedance data in Figure 1, curve 5: (a) I → R; (b) R → I (, experimental data; ◇, transformed data).

the imaginary-to-real transform (I → R) to the real component of the experimental impedance, and plot b compares the result of the real-to-imaginary transform (R → I) to the imaginary component of the experimental impedance. There is a very good agreement between the experimental and transformed data for both transforms over the entire frequency range. For each transformation, the values of the average error (AE) were determined as described in ref 6 and are given in Table 3. As observed, complex plane plots transformed as in Figure 1 exhibit AE values of less than 2.1% for both I → R and R → I, which reveals that the experimental data are valid. Table 3. AE Values of the Transformations for Experimental Impedance Data in the Presence of HBAP

a

C (mM)

I → Ra

R → Ib

0.1 0.5 1.0 5.0

2.070 0.646 0.609 0.982

1.440 0.387 0.917 1.752

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