# Influence of Large Magnetic Field Gradients at the Electrochemical Interface

Jinu Kurian, Peter Dunne, Vincent Vivier, Gwenael Atcheson, Ruslan Salikhov, Ciaran Fowley, Munuswamy Venkatesan, Olav Hellwig, Michael Coey, Bernard Doudin

► To cite this version:

Jinu Kurian, Peter Dunne, Vincent Vivier, Gwenael Atcheson, Ruslan Salikhov, et al. Influence of Large Magnetic Field Gradients at the Electrochemical Interface. Bernard Doudin; Michael Coey; Andrejs Cēbers. Magnetic Microhydrodynamics: An Emerging Research Field, 120, Springer, pp.111–130, 2024, Topics in Applied Physics, 978-3-031-58375-9. 10.1007/978-3-031-58376-6_9. hal-04956800

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Chapter 9

Influence of Large Magnetic Field
Gradients at the Electrochemical
Interface

Jinu Kurian, Peter Dunne, Vincent Vivier, Gwenaël Atcheson,
Ruslan Salikhov, Ciaran Fowley, Munuswamy Venkatesan, Olav Hellwig,
Michael Coey, and Bernard Doudin

9.1 Introduction

There is renewed interest in the importance of magnetic fields and magnetic mate-
rials on (electro)chemical reactions [1, 2], driven by possible energy applications,
ranging from charge storage, to solar cells and hydrogen production [3]. The situation
is complex, with multiple processes simultaneously at play, ranging from microhy-
drodynamics and magneto-chemistry to spin-dependent surface chemistry [4, 5]. As
a result, experiments are mostly performed by trial-and-error. Nearly all experiments
make use of large permanent magnets or are conducted within the bore of large-scale
solenoid flux sources. The region between electrode and electrolyte can be divided
into two; a few nm thin region of electrolyte at the electrode known as double layer,

J. Kurian (✉) · P. Dunne · B. Doudin

CNRS, IPCMS UMR 7504, Université de Strasbourg, 23 Rue du Loess, 67034 Strasbourg, France

e-mail: jinukurian9048@gmail.com

P. Dunne

e-mail: peter.dunne@ipcms.unistra.fr

B. Doudin

e-mail: bernard.doudin@ipcms.unistra.fr

V. Vivier

CNRS, Laboratoire de Réactivité de Surface, UMR 7197, Sorbonne Université, 4 Place Jussieu,

75005 Paris, France

e-mail: vincent.vivier@sorbonne-universite.fr

G. Atcheson · M. Venkatesan · M. Coey

AMBER and School of Physics, Trinity College, Dublin, Ireland

e-mail: atchesog@tcd.ie

M. Venkatesan

e-mail: venkatem@tcd.ie
or Stern layer, which then extends as the microns to mm thick diffusion layer. Most of the magnetic field effects influence the diffusion layer, which controls the mass transport in the electrochemical cell [2, 3] and only indirectly impact the diffuse part of the Stern layer [6]. However, key electrochemical reactions occur at short distance from the electrode surface (Fig. 9.1), and specifically within the first few nanometers of solvents and reactants, in the double layer that develops at the liquid (electrolyte)-solid (electrode) interface [6, 7]. Ideally, one would therefore like to probe magnetic effects by applying intense magnetic fields confined to only a short distance from the electrode surface.

A magnetic field can influence an electrochemical reaction mainly by means of the Lorentz force [8] and the Kelvin force [9]. The Lorentz force, which can be expressed as

$$F_L = jXB \qquad (9.1)$$

will induce convection when the current density $\vec{j}$ and magnetic field $\vec{B}$ are non-collinear. In a typical electrochemical system with $B = 1$ T and $j = 10^3$ A/m$^2$, the Lorentz force will be of the order of $10^3$ N/m$^3$, comparable to the buoyancy-driven convective force. This magnetohydrodynamic (MHD) force modifies the convection, resulting in compression of the diffusion layer, which is similar to the effect of mechanical stirring. However, due to its dependence on the local current density, MHD effect can generate edge-effect induced vorticity [10], flow patterns in electrodeposits [11], and can even influence bubble formation due to micro-MHD flows [12].

On the other hand, the magnetic field gradient force can be expressed as

$$F_{\nabla B} = \frac{1}{2\mu_0} c \chi_m (\mathbf{B} \cdot \nabla) \mathbf{B}, \qquad (9.2)$$

M. Coey

e-mail: jcoey@tcd.ie

R. Salikhov · C. Fowley · O. Hellwig

Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany

e-mail: r.salikhov@hzdr.de

C. Fowley

e-mail: c.fowley@hzdr.de

O. Hellwig

e-mail: o.hellwig@hzdr.de

O. Hellwig

Institute of Physics, Chemnitz University of Technology, Reichenhainer Strasse 70, 09107

Chemnitz, Germany
<figure><img src="image_3.png" /><figcaption>**Fig. 9.1** Schematic of the double-layer structure close to an electrode surface; the Stern layer consists of compactly arranged solvated ions and neutral molecules, which forms the inner and outer Helmholtz planes (IHP, OHP) and a diffuse layer due to the thermal agitation in the solution.
Reprinted with permission from [6]. Copyright 2019 American Chemical Society</figcaption></figure>

where $\mu_0 = 4\pi 10^{-7} H/m$ and $c$ is the concentration of paramagnetic species having a small enough molar magnetic susceptibility $\chi_m$ (as always encountered in electrochemical solutions). This field gradient force, proportional to the product of the scale-independent magnetic field and its gradient, will be active whenever there is a concentration gradient that is not parallel to the non-homogenous field [8, 13, 14]. In an electrolyte containing paramagnetic ions with $c = 1 \text{ M}$, $\chi_m = 10^{-8} \text{ m}^3/\text{mol}$, $\mathbf{B} = 1 \text{ T}$ and $\nabla B = 100 \text{ T/m}$, the magnetic field gradient force is of the order of $10^3 \text{ N/m}^3$, comparable to natural convective forces. It has been shown that the magnetic field gradient force can levitate diamagnetic materials in paramagnetic media [15], influence the efficiency of the hydrogen reduction [16], electrodeposit direct and inverse patterns [17–19], and can even stabilize liquid based frictionless microfluidic channels [20, 21].

Reduction of the size of magnetic sources to the nanoscale opens new possibilities, with enhanced field gradient forces and broadband control that is orders of magnitudes larger than currently reported in the literature. We therefore focus on increasing one of the two magnetic forces at play, namely the magnetic field gradient force originating from the spatial derivative of the magnetic field, which increases as we decrease the size of the system. This gain in force magnitude is obtained by generating a field and field gradient acting on a very small volume about 10 nm region
at the interface between a metallic film and a solution, exactly where it will have the greatest impact on electrochemical reactions. Our approach relies on the use of magnetic thin films stacks, made of alternating Co and Pt layers, optimized to keep their magnetization normal to the plane of the films. We therefore take advantage of technological advances developed for perpendicular magnetic recording applications. We investigate how these planar electrodes compare to pure Pt films for a benchmark single $e^-$ exchange reaction. Our strategy is therefore to simplify to the maximum the type and location of the magnetic force at play on a benchmark electrochemical reaction process.

## 9.2 Experimental Methods

Magnetic thin film stacks of Ta(5.0)/Pt(2.0)/[Co(0.8)/Pt(0.8)]<sub>N</sub>/Pt(3.0) [numbers in nm] with N = 10, 20 and 50 multilayers were DC magnetron sputtered onto thermally oxidized Si substrates. The Ta/Pt in the stack served as a buffer layer for the growth of Co/Pt repeats while the outer Pt functioned as a capping layer to prevent oxidation and maintain a stable surface. Films consisting just of Ta(2.0)/Pt(20.0) are also sputter deposited as a non-magnetic film benchmark. All multi-repeat growth thicknesses were confirmed using X-ray reflectivity measurements (not shown here). All Co/Pt based samples are magnetically characterized using superconducting quantum interference device (SQUID) magnetometry and magnetic force microscopy (MFM).

A three-electrode system was used to characterize the electrochemical ferricyanide/ferrocyanide redox reaction. This is a well-documented single $e^-$ redox reaction with an appropriate energy and kinetic window for experiments that is widely used as a standard for testing electrochemical systems [22, 23]. Pt or Co/Pt films were used as the working electrode (WE), Pt mesh as the counter electrode (CE) and an Ag/AgCl electrode in 3 M KCl as the reference electrode (RE). The WE area was typically 1–10 mm², defined using electro-inactive epoxy masking the sides of the thin film sample. This protected the thin film electrode against oxidation and ensured the absence of magnetic fringe field effects (where we do not control the magnitude and direction of the stray field) that could perturb the interpretation of the results if the sides of the magnetic electrode are exposed to the solution. The electrolyte, made of 0.2 M potassium ferricyanide ($K_3[Fe(CN)_6]$, > 98% purity) as electroactive species and 1 M potassium chloride (KCl, 99% purity) as supporting electrolyte, was freshly prepared prior to measurements.

Impedance spectra were measured under potentiostatic conditions, applying a 10 mV perturbation signal having frequency in the range from 1 Hz to 100 kHz. Linearity of the data obtained was confirmed using Kramers–Kronig transformations. A Pt wire coupled in parallel with the reference electrode through a 0.1 µF capacitor was used to avoid high frequency artifacts [24]. Impedance data were fitted with a Randles circuit composed of a charge transfer resistance ($R_{ct}$) in series with a semi-infinite Warburg diffusion resistance ($Z_w$), both in parallel with the double layer capacitance expressed as a constant phase element (CPE), and all in series with the
solution resistance which includes uncompensated as well as cable resistances. The effective double layer capacitance was estimated from the CPE element using the relation [25, 26],

$$C_{eff} = P^{1/\alpha} \left[ \frac{1}{R_s} + \frac{1}{R_{ct}} \right]^{(\alpha-1)/\alpha}, \quad (9.3)$$

where P is related to the CPE impedance as $Z(f) = 1/[P(j2\pi f)^\alpha]$ where f and $\alpha$ are frequency and distribution factors respectively. The solution resistance was obtained before each measurement by a positive feedback method and agrees with the resistance value obtained from impedance spectroscopy. Cyclic voltammetry measurements were obtained after correcting for the ohmic drop, $iR_s$. Chronoamperograms were measured at potential of 0.25 V after waiting 10 min to reach the steady state.

## 9.3 Magnetic Source Design

Co/Pt films are best known for their perpendicular magnetic anisotropy; their magnetic properties and magnetization switching are well documented in the literature [27–29]. In the present study, magnetic force microscopy (MFM) is used to image the magnetic domains of the thin films in their remanent state. Figure 9.2 shows the multi-state maze-like domain patterns of Co/Pt films with different number of Co repeats. To estimate the domain width, the contrast difference across stripes at different positions is fitted using a Gaussian function and the FWHM of the fit is taken as the average domain size. A minimum domain width D of 130 ± 25 nm is found when N = 20 (Table 9.1). The change of domain width with number of Co/Pt repeats can be well understood within the model of periodic stripe domains with uniaxial magnetic anisotropy developed by Kittel [30, 31] and further extended by Kooy and Enz [32]. According to this model, the total magnetic energy at remanence can be expressed in terms of magnetostatic energy and a domain wall energy, the balance of which determines the domain width. Similar observation of variation of domain width with number of repeats has been reported by Diao et al. [33] and Hellwig et al. [28].

Table 9.1 summarizes the relevant parameters characterizing our Co/Pt thin films. The anisotropy field, $H_k$, is defined as the field required to saturate the sample along its hard axis. Considering the perpendicular magnetic anisotropy of the Co/Pt films, $H_k$ can be approximated as the in-plane field required to saturate the magnetization parallel to the film surface. The effective anisotropy can be found using the relation,

$$K_{eff} = \frac{M_s \cdot H_k}{2}. \quad (9.4)$$

In terms of volume anisotropy $K_v$ and surface anisotropy $K_s$,
<figure><img src="image_4.png" /><figcaption>**Fig. 9.2** MFM images (5.0 µm × 5.0 µm) of [Co(0.8)/Pt(0.8)]<sub>N</sub> multilayer samples at tip lift height 50 nm with **a** N = 10, **b** N = 20 and **c** N = 50 Co/Pt repeats in the remanent state after out-of-plane saturation</figcaption></figure>

$$K_{eff} = K_v + \frac{2K_s}{t} \qquad (9.5)$$

where *t* is the magnetic film thickness. The multiplier 2 is because there are two interfaces on either side of the thin film. Using Kittel's magnetic domain model, the Bloch wall width can be estimated as

$$\lambda_w = \pi \sqrt{A/K_u} \qquad (9.6)$$

where *A* is the exchange stiffness, taken to be about 10 pJ/m [34] and *K*<sub>u</sub> is related to surface anisotropy by *K*<sub>u</sub> = *K*<sub>s</sub>/<i>t</i>. Thus, the domain wall thickness is found to be of the order of 5–10% of the domain width, depending on the number *N* of Co/Pt repeats. These results are in line with more detailed studies on similar multilayers in the literature [35, 36].

Estimates of the magnitude of the stray magnetic field and the related field gradient as a function of distance from the film surface are key for understanding the field effects at the electrochemical interface. To calculate the field magnitude, thin films are considered as 2D sheets magnetized normal to surface, defined along the *z*-axis. Furthermore, individual domains are assumed to be infinitely long and rectangular

<table><caption>**Table 9.1** Summary of relevant parameters of Co/Pt films. The saturation magnetization of the Co layer *M*<sub>s</sub> and the anisotropy *H*<sub>k</sub> of the Co/Pt stack are obtained using SQUID magnetometry</caption><thead><tr><th>N</th><th>Total tCo (nm)</th><th>M<sub>s</sub> ± 0.03 (MA/m)</th><th>H<sub>k</sub> ± 0.03 (T)</th><th>K<sub>eff</sub> (kJ/m<sup>3</sup>)</th><th>D (nm)</th><th>λ<sub>w</sub> (nm)</th></tr></thead><tbody><tr><td>10</td><td>8</td><td>1.53</td><td>0.71</td><td>535 ± 14</td><td>250 ± 40</td><td>12.5</td></tr><tr><td>20</td><td>16</td><td>1.52</td><td>0.79</td><td>600 ± 17</td><td>130 ± 25</td><td>12.1</td></tr><tr><td>50</td><td>40</td><td>1.54</td><td>0.97</td><td>746 ± 23</td><td>200 ± 25</td><td>11.5</td></tr></tbody></table>

The domain size *D* is obtained from MFM images analysis and the domain wall width λ<sub>w</sub> is calculated using Eq. (9.6)
and the field components can be analytically derived using the Amperian current model as [37],

$$B = B_x \hat{x} + B_z \hat{z} \qquad (9.7)$$

$$B_x = \frac{\mu_0 M_r}{4\pi} \left[ \ln\left(\frac{(x+a)^2 + (z-b)^2}{(x+a)^2 + (z+b)^2}\right) - \ln\left(\frac{(x-a)^2 + (z-b)^2}{(x-a)^2 + (z+b)^2}\right) \right] \qquad (9.8)$$

$$B_z = \frac{\mu_0 M_r}{2\pi} \left[ \arctan\left(\frac{2b(x+a)}{z^2 - b^2 + (x+a)^2}\right) - \arctan\left(\frac{2b(x-a)}{z^2 - b^2 + (x-a)^2}\right) \right] \qquad (9.9)$$

where 2a is the width of domain and 2b is the thickness of the magnetic layer. A python code for such calculation is available online [38]. In a magnetic film, the sources of the magnetic stray field are the edges of the magnetic domains. Hence, a monodomain film with minimal edges would generate a smaller field gradient (near zero) compared to the large near-surface field gradient generated by the multi-domain magnetic films. In such a state, the field gradient is maximum near the domain wall where the magnetization direction changes rapidly. Figure 9.3a shows a contour plot of the field gradient generated by a [Co/Pt]20 film with a large resulting field gradient of the order of $10^6 - 10^7$ T/m generated by the multi-domain magnetic structure near its surface. Figure 9.3b shows the average field gradient values for three different repeats. It is found that N = 20 multilayer generates a field gradient of magnitude comparable to that of the N = 50 multilayer and it is expected that the gradient field effects will be more localized near the surface of N=20 multilayer as the values drop faster compared to that of the N = 50 sample. N = 20 multilayers are therefore chosen as the magnetic working electrode to investigate the field induced effects presented in the electrochemistry experiments. It is worthwhile to note that these field gradient values can have local maxima much larger than the plotted average values of the gradient magnitude along the x-axis as a function of the distance from sample surface shown in Figure 9.3b. Furthermore, this field calculation is based on the assumptions such as (i) domain wall regions with tilted magnetization are considered to be magnetically dead, (ii) equal-width domains; and (iii) a perfect interface and a flat surface.

## 9.4 The Electrochemical System

The one e⁻ transfer outer sphere ferricyanide/ferrocyanide redox couple is chosen as the model electrochemical system. It is commonly used as a standard redox probe due to its electrochemical reversibility [39] and stability [40]. The reaction is:

$$[Fe(CN)_6]^{4-} \leftrightarrow [Fe(CN)_6]^{3-} + e^{-}, \qquad (9.10)$$
a)

<figure><img src="image_5.png" /></figure>

b)

<figure><img src="image_6.png" /><figcaption>**Fig. 9.3** **a** Near-surface magnetic field gradient contour plot calculated for the [Co/Pt]₂₀ multilayer. The schematics represents the cross-sectional view of Co/Pt repeats with gap between the alternative domains corresponding to the dead region. Relevant parameters: magnet thickness = 16.0 nm, domain size = 0.13 µm, magnetization = 1.54 MA/m, magnetically dead region = 9%. **b** line plot of the average field gradient magnitude as a function of distance from the film surface for the three stacks described in Table 9.1</figcaption></figure>

where the ferrocyanide anion, [Fe(CN)₆]⁴⁻, is reversibly oxidized to form the ferri-
cyanide anion, [Fe(CN)₆]³⁻. Ferrocyanide is diamagnetic with a low spin ferrous
Fe²⁺ while ferricyanide is paramagnetic with iron in the high spin ferric Fe³⁺ state.
In this redox reaction, both species are soluble in water and remain as solvated ions,
minimizing the changes to the electrode surface during electrochemical measure-
ments, unlike electrodeposition or corrosion. The reaction's redox potential ensures
that gas generation due to water splitting (hydrogen or oxygen evolution) is negligible
during electrochemical measurements, thus minimizing micro-convective effects due
to bubble formation [41].

Prior to the effects induced by magnetic field, it is important to characterize the
properties of the reaction in detail. Cyclic voltammetry was used to estimate the
kinetics as well as the diffusion parameters. It is known that the kinetics of the
[Fe(CN)₆]³⁻/[Fe(CN)₆]⁴⁻ electrode process is highly dependent on the cleanliness of
the electrode [42]. In the present study, the redox peaks in the cyclic voltammogram
are found to be absent when either Pt or Co/Pt films are used as WE without any
prior cleaning. Various cleaning procedures have been reported in the literature to
improve the electrode activity [43]. We followed the procedure described in [42],
which starts with sonication of the thin film in acetone for 2 min followed by a rinse
using ethanol and isopropanol. For the electrochemical cleaning, the potential was
swept between -1.0 V and +1.3 V at a scan rate of 100 mV/s in 0.1 M KCl solution
until a steady state response was observed.

After WE pretreatment, the ferricyanide/ferrocyanide redox reaction is character-
ized by sweeping the potential and measuring the current response under a uniform
surface normal field of 400 mT so that the Pt electrode properties can be compared to
those of the Co/Pt electrodes in their saturated mono-domain state (Figure 9.4a). The
<figure><img src="image_7.png" /><figcaption>Fig. 9.4 a Cyclic voltammogram at a scan rate of 50 mV/s and b dimensionless parameter Ψ of Pt and Co/Pt electrode in solution containing 0.2 M K₃[Fe(CN)₆] and 1 M KCl</figcaption></figure>

solution resistance, R_s, is found to be 21 Ω using a positive feedback technique. The open circuit potential (OCP) of the system is measured as 0.41 V. When the potential is swept from 0.6 to 0.0 V, reduction of ferricyanide occurs while the oxidation of reduced species occurs in the reverse sweep. According to the Randles–Sevcik equation for a reversible system of the type O + ne⁻ → R, the peak current at room temperature follows the relation, which holds for low scan rate only,

$$i_{p,c} = -2.69 - 2.6910^5 n^{\frac{3}{2}} A D_0^{\frac{1}{2}} [O]_{\infty} v^{1/2}, \quad (9.11)$$

where v is the scan rate in V/s, A is electrode area in cm², D is analyte diffusion coefficient in cm²/s, [O]_∞ is the bulk analyte concentration in mol/cm³, and n is the number of electrons transferred in the redox reaction. In order to estimate the diffusion coefficient, i_p is plotted against v¹/² and fitted assuming a linear relation (Eq. 9.11). Irrespective of the electrode, the diffusion coefficient of the ferricyanide reduction reaction is found to be ~6 × 10⁻⁶ cm²/s and that of ferrocyanide oxidation is ~5 × 10⁻⁶ cm²/s. As the diffusion coefficient indicates the quantity of diffused particles per unit time, the difference is ascribed to the lower density of the ferricyanide compared to that of the ferrocyanide [44]. The obtained values are comparable to the literature values 5–7.2 × 10⁻⁶ cm²/s for ferricyanide and 4.5–6.4 × 10⁻⁶ cm²/s for ferrocyanide [45, 46].

For high scan rates, the electrochemical system behaves as a quasi-reversible one, and the separation between redox peaks in CV curves is measured to get insight into the kinetic rate constant. At a scan rate of 50 mV/s, the peak separation ΔE_p for a Pt electrode is found to be 77 mV while that of a Co/Pt electrode is 98 mV. A higher ΔE_p value compared to the 57 mV separation for a reversible reaction can be related to the slow kinetics [47] or electrode surface contamination [48]. The heterogeneous standard kinetic rate constant k₀ can be extracted from the ΔE_p dependence on the scan rate v using the empirical relation [49],
$$ \psi = (-0.6288 + 0.0021X)/(1 - 0.017X), \qquad (9.12) $$

where $X = \Delta E_p \times n$, the rate constant $k^0$ can be obtained using the Nicholson approach [47] where $\psi$ is given by

$$ \psi = k^0/[\pi DnF_v/RT]^{1/2}, \qquad (9.13) $$

To estimate the standard rate constant, $\psi$ is calculated using Eq. (9.12) and is plotted against $v^{-1/2}$ (Fig. 9.4b). Fitting the plot with a linear function and using Eq. (9.13), k0 of Co/Pt and Pt electrodes are estimated as 3.1 × 10⁻³ and 13.0 × 10⁻³ cm/s, respectively. The $k_o$-value obtained for the Pt film is within the wide range 0.01–0.4 cm/s reported for Pt electrode based ferricyanide/ferrocyanide systems [48, 50]. The lower rate constant for Co/Pt could be attributed to a non-perfect coverage of Pt as top layer of the magnetic stack. The dependence of $k_o$ on the magnetic field will be discussed in the next section.

## 9.5 Magnetic Field Effects on the Electrochemical Reaction

The impact of a magnetic field on Pt and Co/Pt film electrodes is studied using cyclic voltammetry in two ways (Fig. 9.5a, b) the external field is applied out of the plane of the film (OP) or parallel to the current density, and (c) (d) the field is applied in the plane of the film (IP) or perpendicular to the current density. Figure 9.5a confirms that there are no field-induced effects in the OP field geometry on a Pt benchmark electrode. Thus, it is safe to assume that micro-MHD and edge-effect, induced field driven convective effects are minimized in the system in this magnetic configuration. In the case of the Co/Pt electrode (Fig. 9.5b), a small change in reduction current can be observed with the perpendicular field which might be due to localized field gradient effects (see the discussion in the next section). However, no appreciable shift in peak separation is observed. It confirms that the observed difference in $k_o$ between Pt and Co/Pt electrodes is likely to be a chemical surface effect. We also checked that upon covering the Co/Pt with a thicker 10 nm Pt cap layer, the $k_o$ increased to 10 × 10⁻³ cm/s, becoming comparable to that of a pure Pt film. However, the thick Pt overlayer will severely diminish the nanoscale field gradient effects on the electrochemical interface (see Fig. 9.3b). Hence, the field effect studies were performed with 3 nm capped Co/Pt films, keeping in mind the difference in kinetic activity, but still pertinent to check if kinetics can change when modifying the magnetic state of the electrode.

Lorentz force induced effects can be probed when a uniform magnetic field is applied parallel to the electrode surface. Figure 9.5c, d illustrate the MHD effect on both Pt and Co/Pt with an IP magnetic field configuration. During reduction of ferricyanide, Lorentz force effects are negligible in a potential range 0.6–0.25 V/Ag/AgCl which corresponds to the kinetics limited region whereas the diffusion limited region (0.275–0.6 V/Ag/AgCl) is considerably influenced by the in-plane magnetic field. We can also observe from the cyclic voltammograms that the Lorentz force effects on
<figure><img src="image_8.png" /><figcaption>**Fig. 9.5** Cyclic voltammograms of a Pt and b Co/Pt with 2.8 nm cap layer under different out-of-plane magnetic field. CV curves of c Pt and d Co/Pt film under different in-plane magnetic fields. Scan rate of all CV is 50 mV/s</figcaption></figure>

the two reaction peaks are asymmetrical, which can be explained by the difference in absolute current density values. As the electrolyte consists of ferricyanide solution, the current density is higher in the reduction region (left) compared to the oxidation region (right) in Fig. 9.5c, d, resulting in a difference in the Lorentz force magnitude. Similar in-plane field behavior is observed for both Pt and Co/Pt film electrodes and therefore indicates that the magnetic domain configuration of the Co/Pt samples does not play a significant role in this field configuration.

Cyclic voltammograms give qualitative information on the mass transport response to the magnetic field. In order to better understand the Lorentz force effects on the diffusion layer, electrochemical impedance spectroscopy (EIS) is used. Figure 9.6a shows the evolution of the impedance spectra when varying the IP magnetic field magnitude at a bias voltage of 0.15 V/Ag/AgCl, where active reduction of ferricyanide is mass transport limited. The Nyquist plot consists of a semi-circular curve in the high frequency region (left), which is related to the interfacial properties and a slanted straight line in the low frequency region (right side)
<figure><img src="image_9.png" /><figcaption>**Fig. 9.6** a Evolution of impedance spectra of Co/Pt electrode at a bias voltage 0.15 V/Ag/AgCl under different magnetic field applied along the plane of film. Randles circuit fit parameters b charge transfer resistance c effective double layer capacitance and d Warburg element as a function of in-plane field</figcaption></figure>

related to the diffusion layer. The Warburg element, related to the diffusion layer, is found to be very sensitive to the applied field with a power law dependence of $B^{1/3}$ (Fig. 9.6d), as predicted by Aogaki [51] for magneto-convection induced by the Lorentz force. The change of $R_{ct}$ and $C_{dl}$ with the IP field (Fig. 9.6b, c) suggests that both interfacial and diffusion regions are sensitive to the bulk convection. We further confirmed the convective nature of the Lorentz force by studying the reaction under forced convection induced by mechanical stirring and found a similar behavior that we do not show here.

In order to study how a large field gradient at the electrochemical interface can impact the reaction, impedance spectra of the bath using the magnetic Co/Pt WE are recorded when imposing several different OP field conditions (Fig. 9.7a), under 0.25 V/Ag/AgCl potentiostatic conditions. The data is fitted using the Randles circuit like the one for IP field measurements. The bath resistance ($R_s$) $19.7 \pm 0.2$ Ω is found to be almost independent of the applied magnetic field.
<figure><img src="image_10.png" /><figcaption>**Fig. 9.7** **a** Evolution of the impedance spectra of Co/Pt electrode at a bias voltage 0.25 V/Ag/AgCl under different magnetic fields applied normal to the film surface. Randles circuit fit parameters **b** charge transfer resistance, **c** double layer capacitance and **d** Warburg diffusion impedance as a function of magnetic field. The magnetization evolution of [Co(0.8)/Pt(0.8)]₂₀ with OP field is shown in the background of figure **b** and **c** (grey)</figcaption></figure>

However, a significant difference occurs in the high frequency region, where the charge transfer resistance ($R_{ct}$) and the double layer capacitance ($C_{dl}$) both show a clear dependence on the applied magnetic field (Fig. 9.7b, c). Both quantities are increased by 5–6% when the applied field changes from 0 to ±400 mT. These values correspond to the magnetic film in the in multi-domain state (large field gradient) at 0 mT and mono-domain state (minimum field gradient) when in a ±400 mT applied field. As the applied DC bias voltage is close to the half potential $E_{1/2}$, the effective double layer capacitance can be treated as a Gouy-Chapman capacitor corresponding to the diffusive double layer region. A change in $C_{dl}$ with applied field would then imply a change in the concentration of ions in the diffusive double layer region. Thus, a field-gradient driven micro-near-electrode convection is assumed to be the origin of the observed changes of impedance spectroscopy.

To check if the field dependence of the EIS with a magnetic Co/Pt WE was intrinsic to the magnetic properties of the WE, measurements under identical conditions were performed using a non-magnetic Pt film as WE. Figure 9.8 shows the corresponding
<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

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**Fig. 9.8** a Evolution of the impedance spectra of Pt electrode at a bias voltage 0.275 V/Ag/AgCl under different magnetic field applied parallel to the current density. Randles circuit fitted parameters b charge transfer resistance, c double layer capacitance and d Warburg diffusion impedance as a function of magnetic field

Nyquist diagrams as well as the evolution of the Randles circuit parameters under different magnetic fields. It reveals that neither interfacial nor bulk properties of the reaction are sensitive to the applied magnetic field. Hence, the perpendicular field has almost no effect for the nonmagnetic WE. Irrespective of the field value, $R_{ct}$ and $C_{dl}$ remain unchanged within about 1–2%, which shows the reproducibility of the measurements. It confirms the CV data of Fig. 9.5a, with a picture of reproducible and stable reaction along with the minimum external-field induced effects in this particular configuration.

Insight into the possible magnetic effect on the charge transfer coefficient of the reaction can be gained by testing how the product of current $i$ and the charge transfer resistance $R_{ct}$ evolves with the change of magnetic force field amplitude [52]. The steady state current response to the OP field is therefore measured for Pt and Co/Pt films under the same potentiostatic condition (0.25 V/Ag/AgCl) as the one used for EIS measurements (Fig. 9.9). The applied OP field is swept between ± 400 mT in steps of 10 mT. Once the applied field is stabilized (in < 5 s), the current is measured after a 10 s waiting time, implying a delay time of 15 s in total between two
<figure><img src="image_15.png" /><figcaption>**Fig. 9.9** Steady state current as a function of OP applied magnetic field, increasing when the magnetic force field generated by the Co/Pt multilayer electrode increases, and nearly invariant for a non-magnetic benchmark Pt electrode</figcaption></figure>

consecutive current measurements. While the steady-state current of a Pt electrode is almost invariant with applied field, the Co/Pt electrode shows a clear change in current density with applied OP field, having a maximum magnitude at 0 mT where the film is in multi-domain state, and a minimum when the field is > |200| mT where the film magnetization becomes saturated (Fig. 9.9). Upon changing the field from ±400 to 0 mT, the current density of the Co/Pt WE increases by around 7%, which implies that the product $i*R_{ct}$ is roughly invariant under applied OP field conditions. This support the claim that the charge transfer process, or the reaction kinetics, under these specific potentiostatic conditions are not influenced by the large magnetic force at the interface.

Returning to the OP field dependence on the reaction, the micro-MHD and edge effect driven convective effects can be assumed to be negligible in the system as no field-induced effects are observed for the Pt-film WE. Hence, the external field cannot be considered as a primary source for these changes. We have therefore strong experimental evidence that the localized stray field gradient generated by the Co/Pt stack in the multi-domain state at small applied external field is responsible for the observed changes in the Randles cell components and the steady state current. This force field is expected to be acting near the immediate vicinity of the electrode, within the few tens of nm of the working electrode surface. Aogaki et al. reported that the field-induced forces acting near the electrode can in fact induce convection inside the diffusive layer [53] and influence the morphology of electrodeposits. For the Co/Pt film electrode in a multi-domain state, regarding it as a magnetized 2D sheet (Sect. 3), an average field of 0.2 T is expected at the film surface. As the current density involved is small, the average Lorentz force density at the surface is rather small, of the order of 10 N/m³ ($j = 5$ mA/cm², $B_{av} = 0.2$ T), and cannot compete with the force driving natural convection (~10³ N/m³). On the other hand, the magnetic field gradient force can act as a driving force. In our case, with the molar magnetic susceptibility $\chi_{mol} = 28.8 \times 10^{-9}$ m³/mol for potassium ferricyanide of
molar concentration of $c \approx 0.1$ M, we estimate the gradient force density to be of the order of $10^7$ N/m$^3$ near the electrode surface. This force can inhibit the convection of paramagnetic species and can locally alter the concentration gradient. A study using CoPt nanowires embedded in an alumina membrane has shown that the oxygen reduction current can be enhanced by the magnetic field gradient driven convective inhibition of paramagnetic radicals near the electrode surface [54, 55].

## 9.6 Conclusions

Co/Pt multilayers with perpendicular magnetization are ideally suited for generating large localized magnetic field gradient forces while limiting the amplitude of the Lorentz force. Our multilayers produce values of $\nabla B^2$ [equal to $(B \cdot \nabla)B$] in excess of $10^6$ T$^2$/m at the electrode surface. Use of a single Pt top layer as nonmagnetic cap layer allows us to convincingly compare the differences of behavior between magnetic and non-magnetic electrodes. For a reversible single $e^-$ electrochemical reaction, we show how the large magnetic field gradient force (Eq. 9.2) acting on the paramagnetic species impacts the diffusive double layer at the interface, decreasing its capacitance and its charge transfer resistance, indicating quantitatively how it perturbs the concentration profile near the interface. However, we find no indication of changes of kinetics of the reaction under the extreme local magnetic field gradient forces generated by our optimized nanoscale magnetic source.

We showed that our strategy for building high magnetic field gradient electrodes impacts the electrochemical process precisely where the forces act, specifically within the first tens of nm of electrolyte near the electrode surface. This strategy could be used beyond the presented benchmark case, for chemical processes highly sensitive to the electrode surface processes, such as the oxygen evolution reaction. Dynamical studies could possibly be extended down to very short time scales, limited only by the intrinsic properties of magnetic switching as the one could turn the magnetic field gradient forces on and off by magnetization reversal due to domain wall motion or spin-orbit torque [56, 57], possibly going down to the picosecond scale [58]. This should open new perspectives for impedance spectroscopy, where we can define the timescale of a system's response as well as limiting the spatial location, with an excitation designed to occur only in the immediate vicinity of the working electrode.

**Acknowledgements** This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 766007. We also acknowledge the support of the University of Strasbourg, Institute for Advanced Studies (USIAS) and the Fondation Jean-Marie Lehn, as well as the support from IdEx Unistra (ANR 10 IDEX 0002), SFRI STRAT'US project (ANR 20 SFRI 0012) and EUR QMAT ANR-17-EURE-0024 under the framework of the French Investments for the Future Program.
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