Influence of a Magnetic Field on the Electrochemical Double

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Influence of a Magnetic Field on the Electrochemical Double Layer Peter Dunne, and John Michael David Coey J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b07534 • Publication Date (Web): 06 Sep 2019 Downloaded from pubs.acs.org on September 6, 2019

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The Journal of Physical Chemistry

Influence of a Magnetic Field on the Electrochemical Double Layer Peter Dunne*†, J. M. D. Coey School of Physics and CRANN, Trinity College Dublin, Dublin 2, Ireland Abstract The electrode/electrolyte interface is the seat of electrochemical redox reactions which are controlled by the structure of ions and solvent molecules at an interfacial double layer, 0.5 – 10 nm thick. Here we show that a magnetic field can exert a large and unexpected influence on the double layer – a 0.5 T in-plane field modifies its capacitance and charge transfer resistance by up to 50%. The effects are observed in nitrobenzene, a model electrochemical system, and they indicate a shift of the outer Helmholtz plane of up to 0.25 nm caused by Maxwell stress acting on the paramagnetic NB•– radicals in solution close to the electrode. There are prospects of using magnetic fields to explore the internal terra incognita lying a nanometer or so from the electrode surface, and control critical interface phenomena involving nucleation or surface wetting.

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Introduction In recent years, magnetic field interactions in electrochemical cells have been pursued with great interest with the aim of developing new technologies, and understanding new phenomena1. Magnetic fields often have a surprising influence on physical and chemical processes occurring in electrochemical cells. The rate of electrodeposition of metal2,3, deposit morphology4, hydrogen production5, corrosion6, rest potential7, alloy composition8, and magnetic properties9, are all sensitive to applied magnetic field. The origin of these effects is most commonly magnetohydrodynamic (MHD) in nature, involving the Lorentz force10

FL = j  B

(1)

where j is the current density and B is the magnetic flux density. When the magnetic field is nonuniform, and induces a magnetisation M = χH, a second force, the field gradient, or Kelvin, force operates in the electrochemical cell. For the typical currents and ion concentrations in electrochemical cells the field gradient force can be written as FH = µ0  HH 11 or FB =

1  m cBB μ0

(2)

where µ0 is the permeability of free space, χm the molar magnetic susceptibility, c, the ion concentration, and B the magnetic flux density. The Kelvin force can induce local convection, but only when c ⊥ B

11,12

. In a uniform magnetic field there is no net magnetic force on a

concentration gradient, but there is a stress, described by the Maxwell stress tensor13, which describes the total electromagnetic energy density of a system in terms of a surface stress. This forms part of the electromagnetic stress tensor written as

Tij =  Ei E j +

1 1  Bi B j −   E 2 + B 2   ij  2   1

(3)


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where ε is the material permittivity, µ is the material permeability, and δij is Kronecker's delta. For a paramagnetic liquid phase surrounded by a diamagnetic medium this stress leads to tension and elongation of the paramagnetic phase for c ⊥ B (Fig 1.a), and a compression for c ⊥ B (Fig 1.b). The phenomenon is best seen in ferrofluids, where a spherical drop is visibly distorted by a magnetic field13,14.

Figure 1. A representation of the magneto-wetting response of a circular electro-generated paramagnetic NB•– cloud distorting elliptically due to Maxwell stress for: a) a magnetic field parallel to the electrode surface where wetting of the interface in improved, and b) the opposite case where a perpendicular magnetic field suppresses wetting. c) Schematic of the double layer structure close to an electrode: solvated ions and neutral molecules are arranged in a tightly bound Stern layer comprising of the inner and outer Helmholtz planes (IHP, OHP), and a diffuse layer further from the electrode. Below is the equivalent circuit. m, 1, 2 are the potentials at the metal surface, IHP, and OHP respectively.

The significance of understanding interactions in the electrode/solution interface in electrochemistry cannot be overstated. In the space of a few tens of nanometres electron kinetics, diffusion, and hydrodynamics combine to determine much of the important electrochemistry. There are two distinct interfacial regions, the diffusion layer and the double layer15. The diffusion layer is typically 1 – 100 µm thick, and ion motion is driven by diffusion. In contrast the double


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layer is a highly compact region much closer to the electrode surface (Fig. 1.c) on the order of 0.5 – 10 nm thick, with ion motion dominated by the strong electrostatic fields in the compact double layer, and a thicker diffuse double layer. The diffusion layer can be influenced by convection, forced or natural, while the double layer is sensitive to surface tension, and excess charge distributions. While the majority of reports focus on magnetically induced thinning of the diffusion layer, Aogaki and others have noted that microMHD flows inside the diffusion layer may play an important role in nucleation and growth

16–19

,

with magnetically imposed chirality on electrodeposited surfaces being shown to yield enantiomerselectivity20. The nitrobenze (NB) redox reaction has previously been used as a model system in MHD studies of the diffusion layer21,22. The benefits to using nitrobenzene are that it undergoes an outer sphere reversible one-electron transfer23

NB + e−

NB•–

(4)

which forms a stable paramagnetic radical anion in aprotic media upon reduction24, and remains in a liquid state at the electrode. For neutral species such as nitrobenzene it has been shown that the reduction-diffusion limited current is enhanced by MHD stirring of the reduced species NB•– 25

. Systematic studies of MHD and field gradient effects on a series of organic compounds have

shown very clear current enhancements using electromagnets and permanent magnets25–30. As the double layer is hidden deep inside any diffusion layer or hydrodynamic boundary layer, and the magnetic energy of a paramagnetic ions in 1 T field is five orders of magnitude lower than kBT, there has been a general consensus that both the double layer and the underlying reaction kinetics are insensitive to magnetic fields31–33. Even under extreme conditions, such as Ni electrodeposition across ferromagnetic nanocontacts34, with field gradients B ~ 107 T m-1, the


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field gradient force alters the deposition rate by inducing localized convection, thereby changing the Ni2+ concentration at the electrode, rather than by a shift of the intrinsic kinetics. Furthermore, new spin-orientation-dependent overpotentials predicted35,36 and observed in water splitting37–39, have also shown no change in kinetics. Nevertheless, there is an increasing agreement that long-range ion-ion correlations do play a role in the response of concentrated paramagnetic ion solutions to magnetic fields40–44, such as the dense clouds of NB•– produced at electrodes. Here we show that uniform magnetic fields can in fact play a role in both double layer structuring and reaction kinetics as a form of magnetowetting, distorting the local NB•– cloud (Fig. 1.a,b), and show how it might work.

Methods A standard three-electrode electrochemical cell was employed for all measurements using a CHI 660C potentiostat, consisting of Pt disc working electrode of diameter 0.5 mm, unless otherwise stated, a Pt mesh counter electrode and an Ag/Ag+ reference electrode containing 0.01 M AgNO3 and 0.2 M tetrabutylammonium perchlorate (TBAP) in acetonitrile (ACN). 0.1 M nitrobenzene (NB) or 0.1 M anthracene was chosen as the active species with 0.2 M (TBAP) as the supporting electrolyte in ACN. Potentials are quoted versus the reference electrode unless otherwise stated. Ar gas was bubbled for 30 minutes prior to each experiment, and between each measurement the Pt working electrode was polished in successive steps down to 0.5 µm diamond lapping paper resulting in a mirror-like finish. A uniform magnetic field (0.1% in a 10 mm diameter spherical volume) of up to 1.5 T was applied parallel or perpendicular to the working electrode surface using a large bore electromagnet, or up to 2.5 T using a large air bore superconducting magnet. The electromagnet was water cooled, and to work at constant temperature, the magnet was energized to 0.5 T, held at this field for more than two hours, and then energized/deenergized to the desired


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field. The electromagnet was compared to a ‘Multimag’ with two rotatable concentric permanent magnet Halbach arrays45, which provide variable intense and uniform magnetic fields without electrical currents, and the above-mentioned open bore superconducting magnet. No difference was found. Impedance spectroscopy was carried out between 100 kHz and 10 mHz at a series of potentials on either side of the NB redox potential. The high-frequency response on a Nyquist plot was the typical depressed semicircle expected for a parallel R-CPE (constant phase element) circuit. The impedance of the cell and capacitance – voltage and resistance – voltage curves as a function of applied magnetic field were measured at 10 kHz, where only the double layer and charge transfer resistance contribute to the AC response. The solution resistance was measured before and after each measurement using the positive feedback method and iR compensation was used throughout each scan, allowing a simple two-circuit model to be used to calculate the charge transfer resistance and double layer capacitance. For cyclic voltammetry, measurements were acquired without active compensation, and the iR distortion was post-corrected using the convolution method of Bond46. Furthermore, the fits to the impedance spectra (Nyquist plots) gave iR values in agreement with both methods. Additional iR compensation checks were made using a Luggin capillary reference electrode, and lower concentrations of NB.


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Results 1. High Frequency Electrochemical Response

Figure 2. a) Nyquist plot showing the magnetic field effect on the double layer capacitance and charge transfer resistance during the reduction of NB at a potential of -1.2 V. Inset shows the orientation of the magnetic field b) Cyclic voltammogram at 100 mV s-1 with no external magnetic field. Magnetic field evolution at 10 kHz Vac for: c) capacitance – voltage and d) resistance – voltage curves. A reduction of both the double layer capacitance and charge transfer resistance is observed at potentials < - 1V. The arrow in c) indicates the trend for increasing magnetic fields. A fit to data in c) using the three element Grahame model, Eq. is shown in e). Epzc denotes the point of zero charge, marked by arrows at 0 T and 0.5 T.

Before performing impedance spectroscopy, and the reversible, one electron reduction of NB was confirmed by cyclic voltammetry at zero field (Fig. 2b). During active reduction of NB, such as at -1.2 V, both the double layer capacitance and charge transfer resistance are suppressed for


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increasing magnetic fields applied parallel to the electrode surface. This is shown by the decreasing semi-circles at high frequencies (1 – 100 kHz) in a Nyquist plot in Fig. 2.a. To gain insight into this response of the double layer, we turn to the voltage-dependent impedance responses shown in Fig. 2.c for the double layer capacitance, and Fig. 2.d for the charge transfer resistance measured at 10 kHz, a series of magnetic fields parallel to the electrode surface. We measured up to 46% enhancement of the kinetic rate constant at 0.5 T and -1.2 V from the magnetic field variation of R in Fig. 2.d. At first glance, the change is astonishing; the energy of a spin-½ ion in a magnetic field is about five orders of magnitude lower than kBT, the thermal energy driving diffusion. However, the potential 2 at the Outer Helmholtz Plane, OHP (Fig. 1.c) where NB is reduced is not the same as the potential at the electrode surface, m, because of the potential drop across the diffuse layer. Frumkin corrections must be applied to the apparent kinetic rate constant to compensate for this potential difference and obtain the true rate constant15. In the case of nitrobenzene, the uncorrected values can underestimate the true rate constant by a factor three61. The implications for this will be dealt with in section 4 below. At potentials < -1.0 V we observe a decrease in both the double layer capacitance C and charge transfer resistance R. At these potentials, NB is actively reduced to paramagnetic NB•–. The pseudocapacitance has a minimum, the point of zero charge Epzc, which shifts to more negative potential with magnetic field, indicative of anion adsorption15. The capacitance in a potential window -1.5 V to -1.0 V is lowered by up to 50% in a magnetic field. In contrast, the diffuse double layer contribution of the supporting electrolyte, observable for potentials > -1.0 V is not significantly altered. Similarly, the plateau observed far from the experimental Epzc due to the compact Helmholtz layer is insensitive to magnetic field. In contrast, when the magnetic field is


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applied perpendicular to the electrode surface the capacitance behavior is reversed (Fig. 3). Under reduction (E < -1 V) the capacitance increases rather than decreases with stronger magnetic fields.

Figure 3. The magnetic field evolution of the voltage-dependent double layer capacitance for B ⊥ electrode surface. 2. Three Element Grahame Double Layer To account for the behaviour of the double layer capacitance, we employed a three-element Grahame model, with example fitted results for B// shown in Fig. 2.e. The model consists of the compact layer capacitance in series with the diffuse layer capacitance of the supporting electrolyte and solvent, and a pseudo-capacitance due to the adsorption of NB•–, which can only occur for potentials more negative than Ered, i.e. at potentials where NB•– is produced. The total capacitance is described thus:  2eε ε c N 1/2  d  ze(0 − 1 )   r 0 1 A C= +   cosh  N1  kbT 2k BT     ε r ε 0 A     2eε ε c N   ze(0 − 2 )   r 0 2 A +   cosh  N 2  k BT 2k T    B   1/2

−1

−1

  + kad  

−1

(5)

where d is the compact layer thickness, εr is the relative permittivity, ε0 the permittivity of free space, A is the electrode area, e the elementary charge, c1 is the concentration of TBA+ in the diffuse layer, c2 is the concentration of NB•– in the diffuse layer, NA Avogadro's number, kB is Boltzmann's constant, T is the temperature, z is the ion charge, 0 is the electrode potential, 1 and


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2 are the potentials of zero charge, kad is a specific adsorption coupling constant, and N1 and N2 are ideality factors. The first term is the geometric, or compact Helmholtz capacitance, the second is the diffuse double layer capacitance of the supporting electrolyte and solvent, the third term is the pseudocapacitance of the adsorbed reduced species NB•–, and the fourth term considers the interaction of non-specific adsorption to electrode potential. Several simplifications and assumptions were made in the application of this model, firstly in the diffuse double layer, the peak width includes ideality factors N1, N2. This is because two important interactions cannot be modelled or measured in this system. In the Stern model (the diffuse layer), the number density of ions near the electrode was modelled by a simple Boltzmann distribution15. This ignores any recombination currents in the diffuse layer, which can arise for NB either from the back reaction (oxidation) step, or protonation, which can be written as i = zFAkr [ni − ni0 ] where kr is the recombination current coefficient. A recombination current will

cause the linewidth of the capacitance peak to narrow, signified by an ideality factor less than 1, at maximum recombination the ideality factor becomes 0.5. Another factor not modelled directly is the surface roughness. If there are surface roughness features on the same scale as the diffuse layer thickness, then there will be a distribution of potential. This is a function of position at the electrode surface, thereby leading to a broadening of the diffuse layer capacitance as a function of voltage47. For a polycrystalline metal surface, the measured potential at the point of zero charge (Epzc) is a global average, individual faces can have an excess charge, and this too broadens the diffuse layer capacitance peak. In summary, recombination currents will lead to an ideality factor Ni < 1, while surface roughness will broaden it (Ni > 1).


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Another important simplification employed is a discrete pseudo-capacitance onset potential. This is not strictly correct as this diffuse capacitance gradually forms as the voltage sweeps negative, until the reduction potential is reached. This leads to an overestimation of the diffuse layer capacitance close to Epzc as shown in Fig. 2.d. Finally, the coupling of adsorption to the electrode potential was assumed to be linear in the measured region, as has been seen in other systems15.

Figure 4. Model results fitting equation (5) to the capacitance data in Fig 2.c, with B // electrode surface. The left column corresponds to the diffuse double layer of the supporting electrolyte TBA+ the right to the pseudocapacitive diffuse double layer of NB•–. The magnetic field dependencies are plotted as top row (a,b): diffuse double layer ion concentration; middle row (c,d): point of zero charge for the ion diffuse layer compared to the global Epzc from measurements; and bottom row (e,f): ideality factor N. When B is parallel to the surface the concentration of NB•– in the diffuse double layer increases (Fig. 4.b). This improved wetting between NB•– and the electrode occurs in association with a negative shift of the local Epzc, a characteristic of anionic adsorption (Fig. 4.d). The chemical protonation reaction is highly concentration dependent and at larger field values the increased NB•–


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concentration drives the ideality factor < 1 (Fig. 4.f). Similar features are observed for the supporting electrolyte ion TBA+. Its concentration likewise increases with B (Fig. 4.a), and along with an increase of its Epzc (Fig. 4.c), as expected for cationic adsorption. However, since TBA+ is stable in solution, no recombination current is observed (Fig. 4.e), and the concentration remains steady over time. Thus, the ideality factor for the supporting electrolyte diffuse double layer is sensitive only to the surface roughness and therefore is independent B.

Figure 5. Model results fitting equation (5) to the capacitance data in Fig 3.f, with B ⊥ electrode surface. The left column corresponds to the diffuse double layer of the supporting electrolyte TBA+ the right to the pseudocapacitive diffuse double layer of NB•–. The magnetic field dependencies are plotted as top row (a,b): diffuse double layer ion concentration; middle row (c,d): point of zero charge for the ion diffuse layer compared to the global Epzc from measurements; and bottom row (e,f): ideality factor N. Quite different results are observed when B is applied perpendicular to the surface. Here the concentration of NB•– remains relatively unchanged with magnetic field (Fig. 5.b), yet there is evidence of adsorption from the negative Epzc shift in Fig. 5.d. Fig. 5.f shows that the


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pseudocapacitive diffuse double layer becomes insensitive to recombination currents with increasing magnetic fields, with the surface roughness dominating. The magnetic field effects on the TBA+ in the diffuse double layer are less obvious. While its concentration decreases with B (Fig. 5.a), its Epzc shows no clear trend (Fig. 5.c), and its ideality factor remains independent of B, with one outlier at 0 T (Fig. 5.e). 3. Excluding Systematic Errors and Mass Transport Effects In order to exclude any systematic errors in the high frequency impedance response of the cell, many steps were taken to validate the experimental setup, the appropriate working conditions, and the models applied to the resultant data. 3.1 iR Compensation The iR compensation method of Bond46, requires the use of semi-integral analysis of cyclic voltammograms. Both semi-derivation and semi-integration have been used on electrochemical measurements to extract diffusional and kinetic properties of species of interest48–51. The difference between the true potential, and applied potential is the well-known equation46 Eapplied (t ) = Etrue (t ) + Ru i(t )

(6)

where t is time, Ru is the uncompensated solution resistance, and i is the current passed. The procedure to measure Ru is shown diagrammatically in Fig. Figure 6, or using the following equation

Ru = [ E(t1 ) − E(t2 )] / [i(t1 ) − i(t2 )]

(7)

Ru for this configuration is found to be 366 , and after the compensation, the current, i, and semiintegrals, e, recover the ideal shape expected for a one-electron transfer process. The separation between the anodic and cathodic peaks decreases from 220 mV to 67 mV, and the semi-integral gap closes for intermediate potentials. The semi-integral trace does not overlap at extreme potentials due to convolution of spherical diffusion52, and irreversible protonation of the radical.


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Figure 6. a) typical cyclic voltammogram and semi-integral of 0.1 M NB used to calculate Ru using Eq.(7); b) cyclic voltammogram and semi-integral after compensation. Additionally, the forward and reverse peaks in the semi-derivative (Fig. 7) become symmetric, with both peak maxima at the redox potential, E0, confirming the effective compensation of Ru.

Figure 7. Cyclic voltammogram of 0.1 M NB with a 0.85 mm diameter Pt working electrode before (light blue) and after correction (dark blue), and semi-derivative before (light green) and after (dark green).


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3.2. Grahame Model Validity and Multi-Frequency Impedance Spectroscopy To ensure the reliability of the fixed-frequency method we performed numerous impedance spectroscopy measurements from 100 kHz down to 10 mHz. The capacitance – voltage curves in Fig. 8.a, were constructed from the impedance response at a fixed frequency of 10 kHz; any contributions from the Warburg (diffusion layer) impedance are at frequencies below 200 Hz, and iR compensation was used for the solution resistance. This reduces the measured cell impedance to that of a resistor R(E) and capacitor C(E) in parallel. Table 1 shows the fitted parameters to a capacitance-voltage scan at 10 kHz under no applied magnetic field. A Helmholtz capacitance of 11 nF implies that the relative permittivity in this compact layer is ~3.5, given that the Stokes radius for acetonitrile and nitrobenzene are 0.36 nm and 0.55 nm respectively53. This is a common result due to the electrostriction breaking the structure of the solution. The effective pressure on the molecules in the compact layer is in the region of gigapascals54, aligning all the molecular dipoles and thus lowering the relative permittivity15,47,55. The concentration of NB•– in the diffuse layer was found to be much less than the concentration of the neutral species in bulk; 135 µmol L-1 compared to 0.1 µmol L-1, and the point of zero charge 2 is slightly more negative than Ered, supporting the conclusion that it is due to a pseudo-capacitance that only occurs when NB is reduced at the electrode surface. Water contamination leads to an irreversible chemical reaction (protonation) in solution24 decreasing the concentration of NB•– at the electrode surface; manifesting itself as an ideality factor < 1 agreeing with the fitted value N2 = 0.8. It is not possible to distinguish any surface roughness broadening as it is hidden by the recombination current narrowing. The opposite is true for the supporting electrolye. An ideality factor >1 is observed, suggesting that the surface roughness dominates the ideality factor, and little or no recombination current can be expected as TBA+ is not reactive in this potential window24.


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Figure 8. Double layer capacitance as a function of electrode potential a) using the simple RC model at 10 kHz for NB (black) without magnetic field and a model fit using equation (5) (red); and b) calculated from full fits to impedance spectra between 100 kHz to 10 mHz

Table 1. Fitting results from three capacitor model for 0.1 M NB at 10kHz CH

c1

N1

nF µmol L−1 11

197

1

c2

V

µmol L−1

1.4 -0.93

135

N2

2

kad

V

nF V

0.8 -1.06

0.56

The main peak is due to the pseudo-capacitance of the reduced species NB•– which can only appear on the negative side of Ered; the neutral species will make no measurable contribution to the double layer capacitance, which is dependent on excess charges. This peak can be seen more clearly when full impedance spectroscopy is carried out (Fig. 8.b). Here the impedance is measured in the frequency range 100 kHz to 10 mHz, and the resultant data fitted assuming a standard Randles circuit. This validates the approach of modelling the impedance response at 10 kHz with a two-element circuit. 3.3. Luggin Capillary Reference Electrode To ensure that the high-frequency resistance measured is indeed the charge transfer resistance, particularly when a magnetic field is applied, two further approaches were adopted a) using a


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Luggin capillary reference electrode, where the iR drops are significantly smaller, and b) using 10 mM NB, a concentration that is an order of magnitude lower than the supporting electrolyte. The same magnetic field effect is found either with the Luggin capillary, Fig. 9, or with 10 mM nitrobenzene, Fig. 11. In former case, the iR drop is 40  at 0 T (Fig. 9.a), with little to no distortion of the voltammograms, and in the latter case the iR drop is 3.0 k. (Fig. 11.a)

Figure 9. a) Cyclic voltammogram of 0.1 M NB under 0 to 2.5 T external fields using a Luggin capillary and 0.85 mm diameter Pt working electrode; and Nyquist plots for impedance spectroscopy at three different voltages versus E0: b) 7 mV, c) -43 mV, and d) -113 mV. Note that the low frequency Warburg impedance above 10 Hz is insensitive to fields up to 0.5 T.

3.4. Diffusion Layer Thinning and Electromigration Minimization of electro-migration effects were important in selection of the active species; the reactant NB is neutral in solution. This allows larger concentrations of NB to be used without seeing migratory mass transport during reduction. Furthermore, if the valence of the ions and the


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concentration of the ions were higher, then direct molecule-molecule interactions would become more important, especially in the double layer. The cyclic voltammograms in Fig. 9.a show the typical reversible reaction expected for NB reduction at zero field. Unlike a rotating disc electrode, the magnetic field acts only on moving charge species, or species with unpaired spins (NB is a neutral species with no unpaired spin). This leads to the asymmetry in current enhancement as the Lorentz force induced convection acts only on the cathodic currents during production of NB•–, as has been similarly observed elsewhere28. This Lorenz force convection induces significant thinning of the diffusion layer in concentrated solutions (0.1 M NB) for fields B  1 T at negative potentials, as shown in Fig. 9.a. Hence, we limited the impedance measurements to B  0.5 T where the Warburg impedance does not deviate from 45° unless f < 10 Hz. This is shown in the Nyquist plots of Figs. 9.b-d, which were performed at potentials +7, -43, and -113 mV vs. E0, respectively. Furthermore, the fitted Warburg resistances show little variation below 0.5 T (Fig. 10.a). In this regime, we observe an undisturbed large cloud of NB•– at the working electrode, while still retaining a decrease in the double layer capacitance (Fig. 10.b).

Figure 10. a) fitted Warburg resistances of 0.1 M NB, and b) double layer capacitances from the Nyquist plots in Fig 9.b-d.


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At lower concentrations such as 10 mM presented in Fig. 11, the Warburg resistance is less sensitive to applied magnetic fields, varying between 27 k at 0 T to 26 k at 0.5 T (Fig. 11.c), while the double layer capacitance drops from 14.8 nF to 11.8 nF (Fig. 11.b).

Figure 11. a) Cyclic voltammogram and its semi-derivative for 10 mM NB at 100 mVs-1, Ep = 64 mV; b) capacitance versus voltage curves at 10 kHz for increasing magnetic fields; c) impedance spectra at -1.1 V from 100 kHz to 1 Hz as a function of applied perpendicular magnetic fields between 0 and 1.5 T. 3.5. Diamagnetic Systems We observe these double layer effects only for paramagnetic solutions, as the susceptibility of diamagnetic solutions is significantly smaller in magnitude. This was confirmed in this case by a series of control experiments where we replaced nitrobenzene with 0.1 M anthracene56, a neutral diamagnetic molecule57. When electrochemically oxidized, anthracene undergoes a series of irreversible electrochemical and electrochemical reactions (ECEC)58–60, resulting in a mixture of bianthrone and anthraquinone as products. The paramagnetic anthracene radical cation is extremely unstable24, immediately reacting with residual water, and thus has far too short a lifetime to contribute to the magnetics of the electrode/solution interface. Within the reproducibility of the measurements, no change was observed in either the diffuse double layer capacitance (Fig. 12.a) or charge transfer resistance close to Epzc (Fig. 12.b).


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Figure 12. Magnetic field evolution at 10 kHz Vac for: a) capacitance – voltage and b) resistance – voltage curves of 0.1 M anthracene in 0.2 M TBAP/ACN.

4. Accounting for the Apparent Kinetic Effects As mentioned previously, we measured a striking enhancement of the kinetic rate constant by up to 46% at 0.5 T and -1.2 V in Fig. 2.d, which needs to be adjusted using Frumkin corrections. In fact, a shift of the distance from the OHP to the electrode, x and a relative change in OHP potential, 2 due to magnetic fields can be derived from the Frumkin-corrected15 R, and integrated C (see appendices 2 and 3 for derivations). The magnetically induced change OHP potential, 2, is given by

2 =

 R0  RT ln  ctB  ( − n)  Rct 

(8)

where α is the transfer coefficient ~0.5 for NB, R is the gas constant, kB, kinetic rate constant measured under field B, and k0 is the rate constant measured under zero magnetic field. While the shift in OHP is given by  tanh(ne20 / 4k BT )  x = − xDL exp   B  tanh(ne2 / 4k BT ) 

(9)


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where 20 is the OHP potential in zero field, and 2B is the OHP potential under a magnetic field B, and xDL is the Debye length, see appendix 3 Eq. (A15).

Figure 13. a) Relative distance change from the OHP to the electrode surface as a function of potential and parallel magnetic field using Eq. (9). b) Relative change in the OHP potential for three electrode potentials, indicated by the vertical lines in a), as a function of magnetic field using Eq. (8). The change in the OHP position x saturates for fields around 0.3 T close to Epzc, but it is compressed by up to 0.1 nm at more positive potentials and by 0.25 nm at more negative potentials. The compression is reversible, as illustrated by Fig. 13.b, where the OHP potential change 2 is shown at three potentials -1.2 V, -1.0 V, and -0.9 V. The field sweeps are symmetric about B = 0. Consistent with the other data, the largest enhancement of 2 is for electrode potentials yielding high concentrations of NB•–, shadowing the OHP compression highlighted by the vertical lines in


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Fig. 13.a. Therefore, the apparent change in kinetics comes from a shift of the OHP towards the electrode surface, not from any change in the underlying kinetics of the reaction.

This raises the key question, why does the OHP shift in a magnetic field?

Discussion To answer this, we consider the surface tension at the metal-electrolyte interface. The Lipmann equation relates the surface tension γ at the electrode to the excess charge m at the electrode interface15

  =− m E A

(10)

where E is the potential drop across the metal-solution interface and A is the electrode area; ultimately the double layer capacitance is related to surface tension via C =  m / E . During NB reduction, NB•– is continually generated close to the interface where it forms a concentrated paramagnetic space-charge cloud. In the absence of magnetic or electric fields, the shape of the NB•– cloud would be determined solely by the competing interface tensions between itself, the bulk solution and the electrode surface at the three-phase contact line. Electric fields are known to lower the surface tension and distort the shape of a droplet –a phenomenon known as electrowetting62. Lipmann's thermodynamic approach identifies surface tension as the Gibbs free energy required by the droplet to cover the surface; an electric field can change the surface tension by an amount15

 = −

2 C E − Epzc ) ( 2A

(11)


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By formal analogy with electric charges, we can consider that magnetic ‘charges’ qm are induced at the interface of the droplet by a magnetic field B, and they are subject to a force f = qmB which redistributes them to minimize energy†. Although there is no net force on the droplet in a uniform field, because there are no uncompensated magnetic charges, there is a stress, described by the Maxwell stress tensor in Eq. (3). We assume a simple axially symmetric 2D model, with the electric field –to first approximation– perpendicular to the electrode, and the magnetic field parallel or perpendicular to the electrode. For E, the dipoles of the nitrobenzene molecules will be polarized parallel to the electric field generated by the electrode, and similarly the magnetic field induces a partial moment in the paramagnetic cloud parallel to itself. Thus, there are no significant off-diagonal terms for either the electric or magnetic stresses so long as the magnetic field is perpendicular or parallel to the surface. The stress tensor then takes the two following forms:

 B 2 / 2 0  TB =   2  E / 2 0

(12)

0 0  TB ⊥ =   2 2  0  E / 2 + B / 2 

(13)

and

If, however, the external magnetic field B is at some angle θ to the electrode surface, off-diagonal shear terms appear in the tensor:

 ( B 2 / 2 ) cos 2  TB , =  2  ( B /  ) cos  sin 

( B 2 /  ) cos  sin 

   E 2 / 2 + ( B 2 / 2 )sin 2  

(14)

†

The magnetic charges are a formal convenience for magnetic calculations. There are no isolated magnetic poles in nature, according to Maxwell's equation   B = 0 .


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We use this equation below to estimate the effect of a magnetic field on the deformation of the paramagnetic cloud and its influence on surface tension.

Figure 14. Voltage-dependence of a) Derived surface energy and b) effective magnetic moment of NB•– for B parallel to the electrode surface; c) derived surface energy and d) effective magnetic moment of NB•– for B perpendicular to the electrode surface. The insets show the orientation of the magnetic field and the deformation of the paramagnetic cloud due to the Maxwell stress. The improved wetting increases the NB•– concentration in the diffuse double layer and compresses the distance of closest approach of the NB•– ions to the electrode, fixed by the OHP (Fig. 1.c). Paradoxically, this improved wetting increases the surface energy (Fig. 14.b) because an excess of negative ions weakens the diffuse double layer capacitance, as seen earlier in Fig.2.c. The increase in surface energy is clearly strongest for E < -1.0 V, and the magnetic energy enhancement is linear in B, with the slope increasing sharply for more negative electrode potentials (Fig. 14.b). The opposite trend is observed in Fig. 14.c, where the perpendicular magnetic field lowers the surface energy. This energy/magnetic-field coupling can be related to a magnetic moment, mexp, with units of nJ T-1 (Fig. 14.b,d). The thermal average moment for a spin–½, , is given by the Curie Law  m = μ 2B B / k BT

(15)


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where µB is the Bohr magneton. The concentration of NB•– at the electrode surface, cNB•− can be estimated from mexp, as mexp = cNB•−  m ,

cNB•− =

mexp  m N aVc

(16)

where Na is Avogadro's number, and Vc is the cloud volume, (assuming full coverage of the electrode and a 200 µm thick diffusion layer of NB•– with a concentration of up to 2 mol L-1 according to the right-hand scale of Fig. 14.b and 14.d. NB•– forms a dense red-orange cloud close to the electrode. Local concentrations of up to 4 mol L-1 have been reported for similar cells elsewhere63. The variation in NB•– concentration with potential agrees with the potential window observed in the cyclic voltammetry (Fig. 2.b) and shows that the presence of paramagnetic NB•– is essential for a significant magnetic field effect on the double layer. From equation (14), if the magnetic field is applied parallel to the electrode surface, the Maxwell stress causes an oblate distortion of the droplet, and the surface tension γ is lowered by about 1 mJ m-2 in a field of 0.5 T (Fig 14.a). The distortion of a spherical paramagnetic cloud of radius r by a magnetic field B into an ellipsoid x2 / a2 + ( y 2 + z 2 ) / b = 1 of equal volume can be described by64

1  0 M = ( 2υ−2/3 − υ−1/6 − υ−5/6 ) 2 r

(17)

where M is the magnetization of the cloud, υ is related to the eccentricity of the ellipsoid ec as υ = 1 − ec2 = b2 / a 2 , and noting that a = r υ-1/3, and b = r υ1/6.

Although equation (17) was originally derived for ferrofluids in air, the boundary conditions remain valid, and only the magnitude of the effects are smaller, indeed magnetic surface tension effects have been observed for paramagnetic CuSO4 – hexane interfaces65, and magnetic-


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surfactant-laden water droplets in air66. Using an initial radius r equal to the working electrode (0.25 mm), a surface tension at -1.4 V and 0 T (Fig 14.a) of 1.5 mJ m-2, and an estimate of the magnetic moment mexp = 0.4 nJ T-1 (Fig. 14.b), equation (17) was solved for υ resulting in an eccentricity ec = 710-3. This is equivalent to an elongation of 4 nm parallel, and a contraction of 2 nm perpendicular to the electrode, with a decrease of curvature at the point of closest contact of about 0.1 m-1. Impedance measurements are macroscopic in nature, averaging over the whole electrode, hence the decrease of curvature is quite sufficient to account for a 0.25 nm average displacement of the OHP towards the electrode in a magnetic field parallel to the surface. Though the model is simple, this result illustrates the effects clearly. The cloud produced at the surface will not be perfectly spherical, but the change in curvature due to the Maxwell stress will be of the same order of magnitude. The oblate-like deformation is responsible for the decrease in surface tension of the paramagnetic liquid, thereby decreasing the total surface charge and capacitance. The deformation will also increase the adsorption of NB•– at the electrode surface as the NB•– volume is drawn closer to the electrode. This will appear as a shifting of the pzc with applied magnetic field, reflected by the Esin-Markov effect, which we indeed observe in Fig. 4.c. The charge transfer resistance will also decrease with applied magnetic field through the Frumkin effect –the potential at the OHP depends on the distance to the electrode– which will increase for decreasing distances from the electrode. Applying the magnetic field perpendicular to the electrode surface results in the opposite effect; there is then a prolate distortion (Fig. 14.c inset) which increases the curvature at the cloudelectrode interface, resulting in a lower surface tension (Fig. 14.c). NB•– concentrations inferred


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from the surface energy were like those for the parallel alignment, and the energy change only becomes negligible at more positive potentials (Fig. 14.d). Furthermore, the change in shape of the cloud neatly explains the observed recombination current effects in the double layer response, i.e. N

Influence of a Magnetic Field on the Electrochemical Double

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