Electrochimica Acta 50 (2004) 51–57
Magnetic field influence on mass transport phenomena Sophie Legeaia , Michelle Chateluta , Olivier Vittoria,∗ , Jean-Paul Chopartb , Omar Aaboubib a
b
Laboratoire d’Electrochimie Analytique-USR 059-UFR Chimie-Biochimie, Universit´e Claude Bernard Lyon I, 43, bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France Laboratoire de Dynamique des Transferts aux Interfaces-UMR 6107-UFR Sciences, Universit´e de Reims Champagne-Ardennes, Moulin de la Housse BP 1039-51687 Reims Cedex 2, France Received 19 May 2004; received in revised form 1 July 2004; accepted 12 July 2004
Abstract The oxidation reactions of hexacyanoferrate(II) and hydroquinone in KCl media were studied on disk platinum electrodes using chronoamperometry under a strong magnetic field (1.74 T). The limiting current measured under magnetic field iB can be expressed as a function of parameters that control the mass transfer phenomenon by iB = KCa Db dc νe εf Bg nh . C represents the electroactive species concentration (mol m−3 ), D the diffusion coefficient of the electroactive species (m2 s−1 ), d the working electrode diameter, ν the kinematic viscosity of the electrolyte (cS), ε the dielectric constant of the solution, B the magnetic field strength (T), n the number of electrons involved in the redox process and K is a proportionality constant. Contribution of B to the limiting current is well established (g = 1/3), whereas the contribution of D has to be confirmed (b = 1). The aim of this work was to specify the influence of the other parameters for which various results have been published in recent literature. We concluded that iB = KC4/3 Dd5/3 ν−2/3 ε−7/4 B1/3 n, quantifying for the first time, to our knowledge, the drastic influence of the electrolyte dielectric constant. © 2004 Elsevier Ltd. All rights reserved. Keywords: Magnetoelectrochemistry; Magnetohydrodynamic effect; Magneto-induced convection; Mass transport; Potassium ferro-ferricyanide
1. Introduction It is now well established that the currents observed in electrochemical processes are modified when a magnetic field is coupled to the electric field in a perpendicular direction [1–8]. This effect, called “magnetohydrodynamic effect” (MHD), is generally explained by the appearance of a Lorentz force. It leads to a convective movement of the species to the electrode surface, and for the electrochemical systems limited by the mass transfer, it induces an increase of the electrolytic currents [9–15]. O’Brien and Santhanam observed these convective movements during pulsed electrodeposition, using laser interferometry [16].
∗
Corresponding author. Tel.: +33 4 72 43 14 13; fax: +33 72 44 84 79. E-mail address:
[email protected] (O. Vittori).
0013-4686/$ – see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2004.07.012
However, although basic hydrodynamic equations governing mass transport under the magnetic force are well understood, rigorous analytical solutions are not available because of the nonlinear character of those equations and the fact that neither the velocity nor the concentration profile near the electrode are known a priori. Only semi-empirical or empirical treatments enable to establish expressions governing mass transport phenomena under magnetic field influence [5,17–29]. Moreover, results published on this topic for the last 20 years are varying depending on authors. Chopart explained this diversity by the great difficulty to get results only corresponding to MHD effect. In fact, other side effects like natural and/or thermal convective flows or sometimes effects resulting from the magnetic properties of involved electrochemical species may superimpose to MHD effect [17,18]. The variation law of the limiting current measured under a strong magnetic field as a function of parameters that control
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the mass transfer phenomenon can be expressed by equation 1: iB = KCa Db d c νe εf B1/3 nh
(1)
where C represents the electroactive species concentration (mol m−3 ), D the diffusion coefficient of the electroactive species (m2 s−1 ), d the working electrode diameter, ν the kinematic viscosity of the electrolyte (cS), ε the dielectric constant of the solution, B the magnetic field strength (T), n the number of electrons involved in the redox process and K is a proportionality constant. Results published in recent literature are often inconsistent with each other. The only contribution to limiting current that is accepted unanimously is the one of the magnetic field strength, B [10–15,17,21,22,24,25]. The aim of this work was to specify the contributions of the other parameters to the limiting current, especially the one of the dielectric constant that has, to our knowledge, never been taken in account. 2. Experimental 2.1. Apparatus The magnetic field was generated by an electromagnet (BRUCKER WP 80, Germany). The pole pieces were of 150 mm diameter and 26 mm apart. The induction was uniform and equal to 1.74 T in the magnet gap. The coils temperature was controlled by a water flow. Chronoamperometric, voltammetric and electrochemical impedance spectroscopy (EIS) studies were performed using a PGZ 301 potentiostat monitored by the Voltamaster 4.0 software (Radiometer analytical S.A., France). 2.2. Cell and electrodes The narrow gap between the pole pieces required the design of a special three electrodes cell as reported in Fig. 1. Three platinum disks whose diameters were respectively 0.2, 0.5, and 1 cm (that is 0.0314, 0.196, and 0.785 cm2 area) were used as working electrodes. The auxiliary electrode was a large square platinum plate (1.28 cm2 area) and the reference was a KCl saturated calomel electrode (SCE).
fluence of the magnetic field, and outside the gap for experiments realized without magnetic field to avoid any remnant field influence. The cell was placed in the field cavity so that the working electrode surface faced downward and was parallel to the lines of the magnetic flux that run horizontally. The magnetic field was then perpendicular to the electric field (Fig. 1). The measurements were carried out after 30 min temperature stabilization at 25 ◦ C in the field cavity. For each experiment three measurements were done and the average value calculated. Dynamic EIS measurements were performed at a potential corresponding to three-fourth of the diffusion-limiting current and were plotted in the Nyquist plane (real part − imaginary part). Chronoamperometry was performed at 350 and 650 mV versus SCE for the oxidation reactions of hexacyanoferrate(II) and hydroquinone, respectively. Experiments were run alternatively in and out the field cavity. Diffusion coefficients of electroactive species were determined versus electrolyte composition using cyclic voltammetry (20 mV/s) and a rotating platinum disk electrode (0.2 cm diameter). The rotating rate was varied between 500 and 1500 rpm. Limiting current is then given by: C δ
(2)
δ = 1.61D1/3 ν1/6 ω−1/2
(3)
il = nFAD where
F is the Faraday constant (96484.6 C), A the electrode area, δ the thickness of the diffusion layer and ω the rotating rate of the electrode. The contributions of all parameters were determined for the hexacyanoferrate(II) oxidation reaction that involves a one-electron transfer. Hydroquinone oxidation, that is a twoelectrons process, was also studied in order to determine the
2.3. Reagents Potassium hexacyanoferrate(II) (≥99%), hydroquinone (≥99%), potassium chloride (≥99.5%), sodium chloride (≥99%) and lithium chloride (≥99%) were purchased from Aldrich (France). Sulphuric acid (≥97%) was purchased from Laurylab (France). All solutions were made with distilled water. 2.4. Procedure The cell containing 25 cm3 of the solution was placed in the field cavity for the experiments performed under the in-
Fig. 1. Cell pattern. ( ) 1 cm.
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53
influence of the number of electrons n involved in the redox process. The classical electrochemical laws were first checked without any magnetic field influence in order to ensure that the species under interest were diffusionally transported.
3. Results 3.1. Evidence for convective movements The MHD effect is generally explained by the appearance of a Lorentz force. It leads to a convective movement of the species to the electrode surface, and for the electrochemical systems limited by the mass transfer, it induces an increase of the electrolytic currents [9–15]. In order to confirm the existence of this convective movement, dynamic EIS measurements were realized with and without magnetic field influence and plotted in the Nyquist plane (Fig. 2). Impedance diagrams present two characteristic parts. The high-frequency part characterizes the charge transfer process in parallel with double-layer charging (enlarged in the right part of Fig. 2) and the low-frequency part characterizes the mass transport process. It can be noticed that the magnetic field does not seem to exert any influence on the charge transfer process, whereas a huge difference is observed in the low-frequency part. In the absence of any magnetic field influence, a Warburg straight line characteristic of a diffusional mass transport process is observed. Conversely, applying a magnetic field to the electrochemical cell leads to a mixed diffusion–convection mass transport process, characterized by a loop with a 45◦ start.
Fig. 3. Variation of the limiting current iB under magnetic field as a function of the hexacyanoferrate(II) concentration C. One centimeter diameter working electrode. T = 25 ◦ C. B = 1.74 T. [KCl] = 0.4 M.
To quantify the dependence of iB on C and d, the concentration of Fe(CN)6 4− was varied between 1 and
100 mM using KCl 0.4 M as supporting electrolyte. Limiting currents were measured using successively the three platinum disks (0.2, 0.5, and 1 cm diameter) as working electrode. The dependence of iB on C, represented in Fig. 3 for the 1 cm diameter working electrode, is not linear. It curves upward until C = 40 mM, meaning that higher bulk concentrations cause more efficient convective stirring. This could be due to the fact that higher bulk concentrations cause steeper concentration gradients and therefore higher currents [24,25]. In more concentrated solutions, iB curves slightly downward probably because of adsorption or migration phenomena. The plot of log (iB ) versus log (C) is linear for hexacyanoferrate(II) concentrations up to 40 mM, the square of the regression coefficient R2 being 0.9999. From the slope it was determined that the limiting current is proportional to C4/3 in this concentration range. The plots of log (iB ) versus log (d) is linear and from the slope it was determined that the limiting current is proportional to d5/3 (R2 = 0.9991). The plot of all iB values measured in this study versus (C4/3 d5/3 ) validates the above proportionalities (Fig. 4).
Fig. 2. Evidence for convective movements. [Fe(CN)6 4− ] = 10 mM. [KCl] = 0.4 M. Two millimetres diameter working electrode. Potential step: 5 mV, step duration: 5 s, ac sine wave amplitude: 10 mV. T = 25 ◦ C. B = 1.74 T.
Fig. 4. Variation of the limiting current iB under magnetic field as a function of C4/3 d5/3 . Hexacyanoferrate(II) oxidation. [KCl] = 0.4 M. T = 25 ◦ C. B = 1.74 T.
3.2. iB dependence on the working electrode diameter and the bulk concentration of the electroactive species
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S. Legeai et al. / Electrochimica Acta 50 (2004) 51–57 Table 2 Electrolyte kinematic viscosity ν and hexacyanoferrate(II) diffusion coefficient D as a function of LiCl concentration
Fig. 5. Variation of the limiting current iB under magnetic field as a function of Dν−2/3 . [Fe(CN)6 4− ] = 10 mM. Two millimeters diameter working electrode. T = 25 ◦ C. B = 1.74 T.
3.3. iB dependence on the kinematic viscosity of the electrolyte and the diffusion coefficient of the electroactive species
0.01
0.05
0.1
0.5
1
2
5
8
ν (cS) D × 106 (cm2 /s)
1.00 5.73
1.01 5.67
1.02 5.66
1.07 5.64
1.14 5.57
1.31 4.79
2.04 3.33
3.53 2.14
The corresponding values of diffusion coefficient and kinematic viscosity are given in Tables 1 and 2 for NaCl and LiCl electrolytes, respectively. The kinematic viscosities were taken from the Handbook of Chemistry and Physics [30]. Diffusion coefficient values were determined with Eqs. (2) and (3) using cyclic voltammetry (20 mV/s) and a platinum rotating disk electrode (0.2 cm diameter) in 0.1 mM K4 Fe(CN)6 solutions. 3.4. iB dependence on the dielectric constant of the electrolyte
The influence of the kinematic viscosity of the electrolyte ν, was determined in 10 mM K4 Fe(CN)6 solutions using NaCl and LiCl as supporting electrolytes. The electrolyte concentration was varied from 10 mM to saturated solutions. Limiting currents were measured using the 1 cm diameter platinum disk as working electrode. The variation of the electrolyte kinematic viscosity leads to a variation of the electroactive species diffusion coefficient. In order to determine the influence of ν, the contribution of the hexacyanoferrate(II) diffusion coefficient should then be taken in account. As the proportionality between the limiting current and the diffusion coefficient has been determined in previous literature [17,18,24,25], Eq. (1) could be written as follows: log(iB ) = log(D) + e log(ν) + k
[LiCl] (M)
(4)
where k represents the other parameters remaining constant. The plots of [log (iB ) − log (D)] versus log (ν) for the two supporting electrolytes allowed then the determination of coefficient e. From the slopes it was determined that e is equal to −2/3 (R2 = 0.9806). It seems so that the decrease of the mass-transport-limited current is due to the friction forces becoming more effective as the electrolyte viscosity increases, preventing magnetohydrodynamic convection of the solution. The plot of iB versus (Dν−2/3 ) validates the above result and confirms the proportionality between iB and D established in previous literature (Fig. 5).
The influence of the dielectric constant was determined in K4 Fe(CN)6 10 mM solutions by adding different amounts of ethanol, from 0 to 11.7 % mol/mol. Limiting currents were measured using the 1 cm diameter platinum disk as working electrode and KCl 0.4 M as supporting electrolyte. The change in the electrolyte composition leads to the variation of the diffusion coefficient and of the electrolyte kinematic viscosity. As previously, the contribution of these parameters should be taken in account in order to determine the dependence of the current on the only dielectric constant. Eq. (1) could be written as follows: log(iB ) = log(D) − 23 log(n) + f log(e) + k
(5)
where k’ represents the other parameters remaining constant. The plot of [log (iB ) − log (D) + 2/3 log ()] versus log (ε) allowed then the determination of coefficient f. From the slope it is determined that f is equal to −7/4 (R2 = 0.9512). The proportionality between iB and ε−7/4 shows the drastic influence of the dielectric constant on the current observed
Table 1 Electrolyte kinematic viscosity ν and hexacyanoferrate(II) diffusion coefficient D as a function of NaCl concentration [NaCl] (M)
0.01
0.05
0.5
1
2
4
5
ν (cS) D × 106 (cm2 /s)
1.00 5.73
1.01 5.68
1.03 5.30
1.06 5.07
1.13 4.68
1.37 3.68
1.58 3.43
Fig. 6. Variation of the limiting current iB under magnetic field as a function of Dν−2/3 ε−7/4 . [Fe(CN)6 4− ] = 10 mM. Two millimetres diameter working electrode. T = 25 ◦ C. B = 1.74 T.
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Table 3 Electrolyte dielectric constant ε, electrolyte kinematic viscosity ν and hexacyanoferrate(II) diffusion coefficient D as a function of ethanol amount Ethanol amount (%mol/mol)
0
1.27
2.62
4.04
5.56
7.17
11.7
ε (cS) D × 106 (cm2 /s)
78.3 76.1 74.0 71.8 69.7 67.5 62.1 1 1.16 1.32 1.51 1.73 1.96 2.50 5.74 5.20 4.55 4.21 3.77 3.46 2.49
under magnetic field. To our knowledge, this parameter has never been taken in account in previous literature. The plot of iB versus (Dν−2/3 ε−7/4 ) validates the above result and confirms the proportionality between iB and Dν−2/3 established previously (Fig. 6). The corresponding data are given in Table 3. As previously, hexacyanoferrate(II) diffusion coefficients were determined with Eqs. (2) and (3), using cyclic voltammetry and a platinum rotating disk electrode in 0.1 mM K4 Fe(CN)6 solutions. The values of the dielectric constant and the kinematic viscosity of each solution were taken from the Handbook of Chemistry and Physics [30]. It was finally checked that the dielectric constant does not exert any influence on the limiting current measured without magnetic field influence, showing that the limiting current dependence on the dielectric constant is only due to the magnetohydrodynamic phenomenon. 3.5. Empirical expression of mass-transport limited current iB under magnetic field
mination of the proportionality constant K (Fig. 7). From the slope it was determined that K = 1.2336 × 1011 A mol−4/3 m1/3 s cS2/3 T−1/3 . In order to determine the dependence of the limiting current on the number of electrons involved in the redox process, the hydroquinone oxidation involving a two-electrons transfer was studied. Limiting currents were measured using the 1 cm diameter platinum disk as working electrode and H2 SO4 0.1 M as supporting electrolyte. Eq. (1) could then be written as follows: iB = K C4/3 Dd 5/3 ν−2/3 ε−7/4 B1/3 K
These above results and those reported in recent literature [17–25] allowed us to establish an empirical expression of iB for the hexacyanoferrate(II) oxidation. As this reaction involves a one-electron transfer, Eq. (1) could be written as follows: iB = KC4/3 Dd 5/3 ν−2/3 ε−7/4 B1/3
Fig. 8. Variation of the limiting current iB under magnetic field as a function of C4/3 Dd5/3 ν−2/3 ε−7/4 B1/3 . Hydroquinone oxidation. T = 25 ◦ C. B = 1.74 T.
(7)
Knh .
where = versus As previously, the plot of iB (C4/3 Dd5/3 ν−2/3 ε−7/4 B1/3 ) allowed the determination of the proportionality constant K (Fig. 8). From the slope it was determined that K = 2.3190 × 1011 A mol−4/3 m1/3 s cS2/3 T−1/3 . It could then be determined from K and K that f is equal to 1, showing that iB is proportional to n.
(6)
The plot of all iB values measured in this study versus (C4/3 Dd5/3 ν−2/3 ε−7/4 B1/3 ) allowed then the deter-
4. Discussion The empirical expression of iB was established for our experimental setup: iB = KC4/3 Dd 5/3 ν−2/3 ε−7/4 B1/3 n
Fig. 7. Variation of the limiting current iB under magnetic field as a function of C4/3 Dd5/3 ν−2/3 ε−7/4 B1/3 . Hexacyanoferrate(II) oxidation. T = 25 ◦ C. B = 1.74 T.
(8)
where K is the (1.2 ± 0.1) ×109 A mol−4/3 m1/3 s cS2/3 T−1/3 . In agreement to all literature reports, the limiting current measured under magnetic field influence using a disk electrode was found to be proportional to C4/3 [10–15,17,18,20–25]. The proportionality between iB and d5/3 established here is in agreement with the result published by Aaboubi and co-workers [10–15,17,18,22], but different from the one reported by Leventis and co-workers who determined that iB is proportional to d3/2 [24,25]. It is important to note that log–log plots do not actually allow to distinguish close slope values, like 3/2 and 5/3 [31]. This consideration could explain the various results reported in literature.
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The proportionality between iB and ν−2/3 determined here is in agreement with the result predicted by Chopart and coworkers [17,18] and determined by Fricoteaux et al. [22], but different from the one reported by Leventis and co-workers [24,25]. This difference could be explained by the fact that Leventis et al. used organic solvents in order to vary the kinematic viscosity of the electrolyte. Indeed, adding organic solvents leads to a variation of the dielectric constant, parameter that strongly influence the current measured in presence of a magnetic field: we established above that iB is proportional to ε−7/4 . The values of the dielectric constant and the kinematic viscosity taken from the Handbook of Chemistry and Physics [30] allows to establish the following expression: ε = k1 ν−1/4
(9)
where k1 is a proportionality constant. Replacing ε in the limiting current expression, Eq. (8), it can be written: −7/4
iB = k2 ν−2/3 (ν−1/4 )
Fig. 9. Variation of the limiting current iB under magnetic field as a function of Dν−2/3 ε−7/4 . Hexacyanoferrate(II) oxidation. T = 25 ◦ C. B = 1.74 T. [Fe(CN)6 4− ] = 10 mM. Two millimeters diameter working electrode.
D = k2 ν−11/48
D∼ = k2 ν−1/4 D
(10)
where k2 represents the other parameters remaining constant. The Eq. (10) is very close to the one established by Leventis and co-workers [24,25]. Finally, the proportionality established here between iB and n is in agreement with the result established by Chopart and co-workers [17,18] but different from the proportionality between iB and n3/2 or n4/3 predicted by Aogaki and coworkers, respectively [21,22]. It should be noticed that the dependence of the limiting current measured under magnetic field on the number of electrons involved in the redox process is difficult to determine experimentally. Indeed, it is not easy to find a statistically significant number of redox-active substances involving a number of electrons different from 1 or 2. To sum up, major discordances between the present work and expressions established in previous literature concern the influences of the kinematic viscosity, ν, and of the dielectric constant, ε, parameters that cannot be dissociated as soon as organic solvents are present in the electrolytic solution. In order to display that the empirical equation established here better fits experimental data obtained in this work than expressions previously published in literature, limiting currents measured for hexacyanoferrat(II) oxidation reaction are represented in Figs. 9–11 as a function of D, ν and ε, whereas d and C are remaining constant. D, ν and ε were affected with the appropriate exponents, according to Eqs. (8), (11) and (12), respectively. () refers to experiments realized in the presence of ethanol whereas () refers to limiting currents measured without any organic solvent. iB = 3.5 × 108 C4/3 Dd 3/2 ν−1/4 B1/3 nf +1
(11)
iB = 3.6 × 104 C4/3 Dd 5/3 ν−2/3 B1/3 n4/3
(12)
Fig. 10. Variation of the limiting current iB under magnetic field as a function of Dν−1/4 . Hexacyanoferrate(II) oxidation. T = 25 ◦ C. B = 1.74 T. [Fe(CN)6 4− ] = 10 mM. Two millimeters diameter working electrode.
Expressions (11) and (12) were established by Leventis and co-workers [22,24,25], respectively. It can be noticed on Figs. 9–11 that the expression established in the present work fits better experimental values than equations established by Leventis et al. and Fricoteaux
Fig. 11. Variation of the limiting current iB under magnetic field as a function of Dν−2/3 . Hexacyanoferrate(II) oxidation. T = 25 ◦ C. B = 1.74 T. [Fe(CN)6 4− ] = 10 mM. Two millimeters diameter working electrode.
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et al. In fact, Eq. (11) fits well experimental current values measured in the presence of ethanol (Fig. 10). This could be explained by the fact that Leventis et al. used only organic solvents in order to vary the kinematic viscosity of the electrolyte, varying simultaneously the dielectric constant of the solution. As previously mentioned, the dependence of iB on this latter parameter was consequently included in the −1/4 exponent attributed by Leventis et al. to the kinematic viscosity. Conversely, Fricoteaux et al. who varied the kinematic viscosity of the electrolyte for constant values of ε established an expression that fits well experimental data obtained here without ethanol (Fig. 11). These latter considerations confirm the fact that the dielectric constant is a parameter that strongly influence the current measured in the presence of a magnetic field.
5. Conclusions This parametric study of the magnetic field influence on the mass transport of electroactive species allowed us to establish an empirical expression of the limiting current, in good agreement with previous literature reports. The limiting current dependences on electrode diameter and kinematic viscosity have been clarified. Moreover, the influence of the dielectric constant has been, to our knowledge, quantified for the first time.
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