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Ohmic drop correction in electrochemical techniques. Multiple potential step chronoamperometry at the test bench
Energy Storage Materials xxx (xxxx) xxx
Contents lists available at ScienceDirect
Energy Storage Materials journal homepage: www.elsevier.com/locate/ensm
Ohmic drop correction in electrochemical techniques. Multiple potential step chronoamperometry at the test bench A R T I C L E I N F O
A B S T R A C T
Keywords Electrochemistry Cyclic voltammetry Linear sweep voltammetry Ohmic drop Multiple potential step chronoamperometry MUSCA
Ohmic drop correction is an important issue in the practice of cyclic and linear sweep voltammetry. While this problem has received satisfactory solutions over the last decades, a new approach, “multiple potential step chronoamperometry (MUSCA)” has been recently proposed. At the test bench, application of MUSCA is not only disappointing but the cure is clearly worse than the disease, opening the way to dangerous artefacts in the interpretation of experimental data.
1. Introduction Ohmic drop correction has been since a long time an important issue in the practice of electrochemical techniques such as cyclic voltammetry (CV) [1–3]. In standard three-electrode set ups (as represented in Fig. 15.6.1, p. 646 in Ref. [1], or Fig. 1.5, p. 11 in Ref. [2] or [3]), the potentiostat takes care of the ohmic drop in the resistance between the (high input impedance) reference electrode and the counter-electrode as well as the possible polarization of the latter. Ohmic drop in the resistance between the working and reference electrodes remains and may distort the CV current-potential response. The problem is all the more important as working electrodes are small-sized (a few millimeter diameters at most) for CV to be a practically non-destructive technique, allowing repetition of the same experiment a large number of times. The resistance around the working electrode is therefore large, which makes it important to find ways to compensate for the resulting ohmic drop or to correct the data accordingly. The problem has in fact been satisfactorily mastered for decades now through various complementary approaches. These include positive feedback compensation, mathematical or computational manipulation of data (see particularly the convolution method), which are easily available through textbooks and/or commercial instrumentation and simulation packages. Within this context, a new approach has been recently proposed [4]. It consists in “a multiple potential step chronoamperometry (MUSCA) technique”, which “allows for reconstruction of cyclic voltammograms with considerably lower ohmic drop contribution.” This clearly call for putting the “MUSCA” technique at the test bench. In this aim, we examine two simple systems. One involves a RC circuit with a constant capacitance in series with an uncompensated resistance. The other is a purely faradaic and involves a reversible redox couple adsorbed at the electrode surface (no diffusion) with an uncompensated resistance. In both cases, we compare the current-potential trace obtained by linear scan voltammetry (LSV) and the current-potential trace
generated by MUSCA method, as well as, in the case of faradaic current the current-potential LSV trace without uncompensated resistance. We first recall below the principle of the MUSCA technique: The investigated potential range (from Ei to Ef ) is discretized in potential steps; at each potential En ¼ Ei þ nΔE (n > 0) a potential step ΔE is applied and the current in ðtÞ is recorded as function of time until it ideally drops to zero. An averaged current is then calculated Z Δt in ðΔtÞ ¼ Δt1 in ðtÞdt where Δt is chosen so that the corresponding value 0
vap ¼ ΔE=Δt is comparable to the scan rate v used in a LSV experiment.
2. RC circuit It includes a constant capacitance Cd in series with an uncompensated resistance Ru . The LSV trace for a scan in the anodic direction where the electrode potential, E, is scanned according to E ¼ Ei þ vt is given by (v: scan rate, Ei : initial potential): E Ei i ¼ Cd v 1 exp vRu Cd In the MUSCA technique [4], where at each potential En ¼ Ei þ nΔE (n > 0), a potential step ΔE is applied and the current is recorded as function of time until it drops to zero. The ensuing current is: in ðtÞ ¼
ΔE t exp Ru Ru C d
which is then averaged as: in ðΔtÞ ¼
1 Δt
Z
Δt
in ðtÞdt ¼ Cd
0
ΔE Δt 1 exp Δt Ru Cd
Defining vap ¼ ΔE=Δt,
DOI of original article: https://doi.org/10.1016/j.ensm.2019.07.038. https://doi.org/10.1016/j.ensm.2019.07.029 Received 8 July 2019; Received in revised form 21 July 2019; Accepted 22 July 2019 Available online xxxx 2405-8297/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: C. Costentin, J.-M. Saveant, Ohmic drop correction in electrochemical techniques. Multiple potential step chronoamperometry at the test bench, Energy Storage Materials, https://doi.org/10.1016/j.ensm.2019.07.029
C. Costentin, J.-M. Saveant
in ¼ Cd vap 1 exp
Energy Storage Materials xxx (xxxx) xxx
ΔE vap Ru Cd
;
Hence: login ¼ log Cd vap þ log 1 exp
ΔE vap Ru Cd
as represented in Fig. 1 for various values of ΔE and fixed values of Ru and Cd . These variations are exactly the same as the ones observed experimentally in Fig. 3 of reference [4]. Two limiting behaviors are observed: when vap >>
ΔE ΔE ði:e: Δt << Ru Cd Þ : login log Ru C d Ru
when vap <<
ΔE ði:e: Δt >> Ru Cd Þ : login ¼ logCd þ logvap Ru C d
Fig. 2. Plot of LSV charging of a Ru Cd ¼ 2 s circuit at 25 (red), 50 (green) and 100 (blue) mV/s. Horizontal lines correspond to MUSCA in =Cd v with ΔE ¼ 25 mV and v ¼ ΔE=Δt ¼ 25 (red), 50 (green) and 100 (blue) mV/s. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Hence, for in not to be affected by “ohmic drop”, it is required that vap < (i.e. Δt >> Ru Cd ). < RΔE u Cd Comparing now the LSV trace with the “reconstructed” trace via MUSCA with a potential step and a measurement time such as vap ¼ v amounts to compare: EEi in ¼ Cd v 1 exp vRΔE with i ¼ Cd v 1 exp vR u Cd u Cd
0
the electrode and the reaction site is: E ¼ E Ru i We consider a fast electron transfer, obeying the Nernst law: ΓB FðE E0 Þ : ¼ exp RT ΓA 0 Since, Γ A þ Γ B ¼ Γ :
If E Ei >> vRu Cd , the LSV trace is obviously unaffected by ohmic drop. However, the putative interest of “MUSCA” would be to decrease the ohmic drop on the rising parts of the charging curves in Fig. 2. The results of the application of MUSCA are represented by the horizontal lines in Fig. 2 for values of the parameters similar to those presented in Ref. [4]. It clearly appears that MUSCA is much worse than the corresponding LSV experiments in which no ohmic drop correction has been attempted.
and the current is obtained from:
3. Surface wave
i ¼ FS
ΓB ¼
Γ0 0 F ðE Ru iE 0 Þ 1 þ exp RT
dΓ B : dt leading to:
The system consists now of a simple surface redox couple A ¼ B þ e (of standard potential E 0 ) and an uncompensated resistance, Ru . In LSV with no ohmic drop correction. At t ¼ 0, Γ A ¼ Γ 0 and Γ B ¼ 0 (Γ A and Γ B are the surface concentrations of the subscript species and Γ 0 the total surface concentration of reactants). 0 At t > 0, the applied potential is E ¼ Ei þ vt and the potential between
F ðEi þvtRu iE 0 Þ exp RT i F di v Ru ¼ 2 dt FSΓ 0 RT F ðEi þvtRu iE 0 Þ 1 þ exp RT and therefore to:
Fig. 3. LSV of a surface redox wave without (blue trace) and with (green trace) uncompensated resistance Ru ¼ 10 Ω at 0.1 V/s. E 0 ¼ 0 V, Γ 0 ¼ 107 mol/cm2, S ¼ 1 cm2, T ¼ 298.15 K. Red dots: MUSCA data points in ðΔtÞ obtained with ΔE ¼ 25 mV and Δt ¼ ΔE=v ¼ 0.25 s. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 1. login as function of logvap (mV/s) for ΔE ¼ 25 (red), 50 (green), 100 (blue) mV with Ru Cd ¼ 2 s and Cd ¼ 102 F. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) 2
C. Costentin, J.-M. Saveant
di þ0 dt B B F BFSΓ 0 RT @
Energy Storage Materials xxx (xxxx) xxx
i
1
¼
F ðEi þvtRu iE0 Þ RT
C C 2 CRu A F ðE þvtRu iE0 Þ
exp
1þexp
i
Z
v Ru
0
0
4. Conclusion It must therefore be concluded that the MUSCA technique is far to “allow for reconstruction of cyclic voltammograms with considerably lower ohmic drop contribution.” In fact, the cure is worse than the disease: MUSCA severely distorts the CV responses, opening the way to dangerous artefacts in the interpretation of experimental data.
Shown as the blue curve in Fig. 3. Application of MUSCA: At each potential En ¼ Ei þ nΔE (n > 0) a potential step ΔE is applied and the current is recorded as function of time until it drops to zero. For a given n value: Γ
At t ¼ 0, En ¼ Ei þ nΔE, Γ A ¼ Γ 0 Γ B and Γ B ¼
0
References
FðEn E 0 Þ RT
1þexp 0
At t 0, E ¼ Ei þ ðn þ 1ÞHðtÞΔE ¼ En þ ΔE HðtÞ; HðtÞ is the Heaviside step function. dΓ B and Γ B ¼ dt
[1] A.J. Bard, L.R. Faulkner, Electrochemical. Methods: Fundamentals and Applications, second ed., John Wiley & Sons, New York, NY, 2001. [2] J.-M. Saveant, Elements of Molecular and Biomolecular Electrochemistry: an Electrochemical Approach to Electron Transfer Chemistry, John Wiley & Sons, Hoboken, NJ, 2006. [3] C. Costentin, J.-M. Saveant, Elements of Molecular and Biomolecular Electrochemistry: an Electrochemical Approach to Electron Transfer Chemistry, second ed., John Wiley & Sons, Hoboken, NJ, 2019. [4] H. Shao, Z. Lin, K. Xu, P.-L. Taberna, P. Simon, Electrochemical study of pseudocapacitive behavior of Ti3C2Tx MXene material in aqueous electrolytes, Energy Storage Material (2019), https://doi.org/10.1016/j.ensm.2018.12.017.
Γ0 0 F ðE Ru iE 0 Þ 1 þ exp RT
leading to: din þ dt
in
0 B B F Ru BFSΓ 0 RT @
exp
F ðEn þΔERu in E0 Þ RT
1¼
C C 2 C A F ðEn þΔERu in E0 Þ
1þexp
Γ0 Γ0 0 F ðEn þΔEE Þ F ðEn E 0 Þ 1 þ exp 1 þ exp RT RT
The resulting MUSCA data points are shown in Fig. 3 (red dots) for the same typical condition. It is clearly seen that MUSCA data do not provide the announced reconstruction of an “ohmic drop free” CV response.
F ðEi þvtE 0 Þ exp RT Fv i ¼ FSΓ 0 2 RT F ðEi þvtE 0 Þ 1 þ exp RT
0
in ðtÞ dt ¼ FS
Numerical computation of the above equation leads to in ðtÞ and then Z Δt in ðtÞdt. to in ðΔtÞ ¼ Δt1
RT
The current – potential trace is then obtained by numerical calculation. A typical example is given in Fig. 3 (green curve). In the absence of uncompensated resistance, the trace is given by:
EðtÞ ¼ E Ru in ; in ¼ FS
∞
δðtÞ Ru
Cyril Costentin*, Jean-Michel Saveant* Universite Paris Diderot, Sorbonne Paris Cite, Laboratoire d’Electrochimie Moleculaire, Unite Mixte de Recherche Universite - CNRS N 7591, B^ atiment Lavoisier, 15 rue Jean de Baïf, 75205 Paris Cedex 13, France
RT
(δðtÞ is the Dirac function).
Z
∞
At t ¼ ∞, in ¼ 0 and Γ B ðEn þ ΔEÞ Γ B ðEn Þ ¼ 0
in ðtÞ dt FS
*
Corresponding authors. E-mail addresses: [email protected] (C. Costentin), [email protected] (J.-M. Saveant).
3
Contents lists available at ScienceDirect
Energy Storage Materials journal homepage: www.elsevier.com/locate/ensm
Ohmic drop correction in electrochemical techniques. Multiple potential step chronoamperometry at the test bench A R T I C L E I N F O
A B S T R A C T
Keywords Electrochemistry Cyclic voltammetry Linear sweep voltammetry Ohmic drop Multiple potential step chronoamperometry MUSCA
Ohmic drop correction is an important issue in the practice of cyclic and linear sweep voltammetry. While this problem has received satisfactory solutions over the last decades, a new approach, “multiple potential step chronoamperometry (MUSCA)” has been recently proposed. At the test bench, application of MUSCA is not only disappointing but the cure is clearly worse than the disease, opening the way to dangerous artefacts in the interpretation of experimental data.
1. Introduction Ohmic drop correction has been since a long time an important issue in the practice of electrochemical techniques such as cyclic voltammetry (CV) [1–3]. In standard three-electrode set ups (as represented in Fig. 15.6.1, p. 646 in Ref. [1], or Fig. 1.5, p. 11 in Ref. [2] or [3]), the potentiostat takes care of the ohmic drop in the resistance between the (high input impedance) reference electrode and the counter-electrode as well as the possible polarization of the latter. Ohmic drop in the resistance between the working and reference electrodes remains and may distort the CV current-potential response. The problem is all the more important as working electrodes are small-sized (a few millimeter diameters at most) for CV to be a practically non-destructive technique, allowing repetition of the same experiment a large number of times. The resistance around the working electrode is therefore large, which makes it important to find ways to compensate for the resulting ohmic drop or to correct the data accordingly. The problem has in fact been satisfactorily mastered for decades now through various complementary approaches. These include positive feedback compensation, mathematical or computational manipulation of data (see particularly the convolution method), which are easily available through textbooks and/or commercial instrumentation and simulation packages. Within this context, a new approach has been recently proposed [4]. It consists in “a multiple potential step chronoamperometry (MUSCA) technique”, which “allows for reconstruction of cyclic voltammograms with considerably lower ohmic drop contribution.” This clearly call for putting the “MUSCA” technique at the test bench. In this aim, we examine two simple systems. One involves a RC circuit with a constant capacitance in series with an uncompensated resistance. The other is a purely faradaic and involves a reversible redox couple adsorbed at the electrode surface (no diffusion) with an uncompensated resistance. In both cases, we compare the current-potential trace obtained by linear scan voltammetry (LSV) and the current-potential trace
generated by MUSCA method, as well as, in the case of faradaic current the current-potential LSV trace without uncompensated resistance. We first recall below the principle of the MUSCA technique: The investigated potential range (from Ei to Ef ) is discretized in potential steps; at each potential En ¼ Ei þ nΔE (n > 0) a potential step ΔE is applied and the current in ðtÞ is recorded as function of time until it ideally drops to zero. An averaged current is then calculated Z Δt in ðΔtÞ ¼ Δt1 in ðtÞdt where Δt is chosen so that the corresponding value 0
vap ¼ ΔE=Δt is comparable to the scan rate v used in a LSV experiment.
2. RC circuit It includes a constant capacitance Cd in series with an uncompensated resistance Ru . The LSV trace for a scan in the anodic direction where the electrode potential, E, is scanned according to E ¼ Ei þ vt is given by (v: scan rate, Ei : initial potential): E Ei i ¼ Cd v 1 exp vRu Cd In the MUSCA technique [4], where at each potential En ¼ Ei þ nΔE (n > 0), a potential step ΔE is applied and the current is recorded as function of time until it drops to zero. The ensuing current is: in ðtÞ ¼
ΔE t exp Ru Ru C d
which is then averaged as: in ðΔtÞ ¼
1 Δt
Z
Δt
in ðtÞdt ¼ Cd
0
ΔE Δt 1 exp Δt Ru Cd
Defining vap ¼ ΔE=Δt,
DOI of original article: https://doi.org/10.1016/j.ensm.2019.07.038. https://doi.org/10.1016/j.ensm.2019.07.029 Received 8 July 2019; Received in revised form 21 July 2019; Accepted 22 July 2019 Available online xxxx 2405-8297/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: C. Costentin, J.-M. Saveant, Ohmic drop correction in electrochemical techniques. Multiple potential step chronoamperometry at the test bench, Energy Storage Materials, https://doi.org/10.1016/j.ensm.2019.07.029
C. Costentin, J.-M. Saveant
in ¼ Cd vap 1 exp
Energy Storage Materials xxx (xxxx) xxx
ΔE vap Ru Cd
;
Hence: login ¼ log Cd vap þ log 1 exp
ΔE vap Ru Cd
as represented in Fig. 1 for various values of ΔE and fixed values of Ru and Cd . These variations are exactly the same as the ones observed experimentally in Fig. 3 of reference [4]. Two limiting behaviors are observed: when vap >>
ΔE ΔE ði:e: Δt << Ru Cd Þ : login log Ru C d Ru
when vap <<
ΔE ði:e: Δt >> Ru Cd Þ : login ¼ logCd þ logvap Ru C d
Fig. 2. Plot of LSV charging of a Ru Cd ¼ 2 s circuit at 25 (red), 50 (green) and 100 (blue) mV/s. Horizontal lines correspond to MUSCA in =Cd v with ΔE ¼ 25 mV and v ¼ ΔE=Δt ¼ 25 (red), 50 (green) and 100 (blue) mV/s. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Hence, for in not to be affected by “ohmic drop”, it is required that vap < (i.e. Δt >> Ru Cd ). < RΔE u Cd Comparing now the LSV trace with the “reconstructed” trace via MUSCA with a potential step and a measurement time such as vap ¼ v amounts to compare: EEi in ¼ Cd v 1 exp vRΔE with i ¼ Cd v 1 exp vR u Cd u Cd
0
the electrode and the reaction site is: E ¼ E Ru i We consider a fast electron transfer, obeying the Nernst law: ΓB FðE E0 Þ : ¼ exp RT ΓA 0 Since, Γ A þ Γ B ¼ Γ :
If E Ei >> vRu Cd , the LSV trace is obviously unaffected by ohmic drop. However, the putative interest of “MUSCA” would be to decrease the ohmic drop on the rising parts of the charging curves in Fig. 2. The results of the application of MUSCA are represented by the horizontal lines in Fig. 2 for values of the parameters similar to those presented in Ref. [4]. It clearly appears that MUSCA is much worse than the corresponding LSV experiments in which no ohmic drop correction has been attempted.
and the current is obtained from:
3. Surface wave
i ¼ FS
ΓB ¼
Γ0 0 F ðE Ru iE 0 Þ 1 þ exp RT
dΓ B : dt leading to:
The system consists now of a simple surface redox couple A ¼ B þ e (of standard potential E 0 ) and an uncompensated resistance, Ru . In LSV with no ohmic drop correction. At t ¼ 0, Γ A ¼ Γ 0 and Γ B ¼ 0 (Γ A and Γ B are the surface concentrations of the subscript species and Γ 0 the total surface concentration of reactants). 0 At t > 0, the applied potential is E ¼ Ei þ vt and the potential between
F ðEi þvtRu iE 0 Þ exp RT i F di v Ru ¼ 2 dt FSΓ 0 RT F ðEi þvtRu iE 0 Þ 1 þ exp RT and therefore to:
Fig. 3. LSV of a surface redox wave without (blue trace) and with (green trace) uncompensated resistance Ru ¼ 10 Ω at 0.1 V/s. E 0 ¼ 0 V, Γ 0 ¼ 107 mol/cm2, S ¼ 1 cm2, T ¼ 298.15 K. Red dots: MUSCA data points in ðΔtÞ obtained with ΔE ¼ 25 mV and Δt ¼ ΔE=v ¼ 0.25 s. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 1. login as function of logvap (mV/s) for ΔE ¼ 25 (red), 50 (green), 100 (blue) mV with Ru Cd ¼ 2 s and Cd ¼ 102 F. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) 2
C. Costentin, J.-M. Saveant
di þ0 dt B B F BFSΓ 0 RT @
Energy Storage Materials xxx (xxxx) xxx
i
1
¼
F ðEi þvtRu iE0 Þ RT
C C 2 CRu A F ðE þvtRu iE0 Þ
exp
1þexp
i
Z
v Ru
0
0
4. Conclusion It must therefore be concluded that the MUSCA technique is far to “allow for reconstruction of cyclic voltammograms with considerably lower ohmic drop contribution.” In fact, the cure is worse than the disease: MUSCA severely distorts the CV responses, opening the way to dangerous artefacts in the interpretation of experimental data.
Shown as the blue curve in Fig. 3. Application of MUSCA: At each potential En ¼ Ei þ nΔE (n > 0) a potential step ΔE is applied and the current is recorded as function of time until it drops to zero. For a given n value: Γ
At t ¼ 0, En ¼ Ei þ nΔE, Γ A ¼ Γ 0 Γ B and Γ B ¼
0
References
FðEn E 0 Þ RT
1þexp 0
At t 0, E ¼ Ei þ ðn þ 1ÞHðtÞΔE ¼ En þ ΔE HðtÞ; HðtÞ is the Heaviside step function. dΓ B and Γ B ¼ dt
[1] A.J. Bard, L.R. Faulkner, Electrochemical. Methods: Fundamentals and Applications, second ed., John Wiley & Sons, New York, NY, 2001. [2] J.-M. Saveant, Elements of Molecular and Biomolecular Electrochemistry: an Electrochemical Approach to Electron Transfer Chemistry, John Wiley & Sons, Hoboken, NJ, 2006. [3] C. Costentin, J.-M. Saveant, Elements of Molecular and Biomolecular Electrochemistry: an Electrochemical Approach to Electron Transfer Chemistry, second ed., John Wiley & Sons, Hoboken, NJ, 2019. [4] H. Shao, Z. Lin, K. Xu, P.-L. Taberna, P. Simon, Electrochemical study of pseudocapacitive behavior of Ti3C2Tx MXene material in aqueous electrolytes, Energy Storage Material (2019), https://doi.org/10.1016/j.ensm.2018.12.017.
Γ0 0 F ðE Ru iE 0 Þ 1 þ exp RT
leading to: din þ dt
in
0 B B F Ru BFSΓ 0 RT @
exp
F ðEn þΔERu in E0 Þ RT
1¼
C C 2 C A F ðEn þΔERu in E0 Þ
1þexp
Γ0 Γ0 0 F ðEn þΔEE Þ F ðEn E 0 Þ 1 þ exp 1 þ exp RT RT
The resulting MUSCA data points are shown in Fig. 3 (red dots) for the same typical condition. It is clearly seen that MUSCA data do not provide the announced reconstruction of an “ohmic drop free” CV response.
F ðEi þvtE 0 Þ exp RT Fv i ¼ FSΓ 0 2 RT F ðEi þvtE 0 Þ 1 þ exp RT
0
in ðtÞ dt ¼ FS
Numerical computation of the above equation leads to in ðtÞ and then Z Δt in ðtÞdt. to in ðΔtÞ ¼ Δt1
RT
The current – potential trace is then obtained by numerical calculation. A typical example is given in Fig. 3 (green curve). In the absence of uncompensated resistance, the trace is given by:
EðtÞ ¼ E Ru in ; in ¼ FS
∞
δðtÞ Ru
Cyril Costentin*, Jean-Michel Saveant* Universite Paris Diderot, Sorbonne Paris Cite, Laboratoire d’Electrochimie Moleculaire, Unite Mixte de Recherche Universite - CNRS N 7591, B^ atiment Lavoisier, 15 rue Jean de Baïf, 75205 Paris Cedex 13, France
RT
(δðtÞ is the Dirac function).
Z
∞
At t ¼ ∞, in ¼ 0 and Γ B ðEn þ ΔEÞ Γ B ðEn Þ ¼ 0
in ðtÞ dt FS
*
Corresponding authors. E-mail addresses: [email protected] (C. Costentin), [email protected] (J.-M. Saveant).
3