1 Introduction

This review summarizes work carried out over the past 20 years to utilize magnetic fields for controlling flow and mass transfer in weakly conducting fluids. It will mainly focus on lab-scale (dimension L < 1 m) applications in aqueous solutions, e.g. electrolytes or sea water, with a typical electrical conductivity \(\sigma\) of about 1–10 S/m. Control can often be achieved by utilizing the Lorentz force density, which is defined as the vector product of current density j in the liquid and the magnetic induction B [1]. As typical flow velocities and magnetic fields will be u < 1 m/s, B < 1 T, the current density induced by flow \(\sigma \left( {{\varvec{u}} \times {\varvec{B}}} \right)\) might be too small to achieve control. Therefore, in the following we consider cases where an external current density j0 is applied, compared to which the induced current density can be neglected. Furthermore, the magnetic Reynolds number \(R_{m} = \mu_{0} \sigma uL\) will be small (\(\mu_{0}\) is the vacuum permeability), and the induced magnetic field can be neglected compared to the applied field B0 [1]. The Lorentz force density therefore reads

$${\varvec{f}}_{{\varvec{L}}} = {\varvec{j}}_{0} \times {\varvec{B}}_{0}$$
(3.1)

Utilizing the Lorentz force mostly relies on adding momentum to the electrolyte to energize wall-parallel flow, which then also impacts mass transfer at electrodes. The second magnetic force considered is the magnetic gradient force or Kelvin force density, which requires the fluid to be paramagnetic or diamagnetic and a spatial gradient of the magnetic field applied. It reads [2]

$${\varvec{f}}_{{\varvec{K}}} = \frac{{\chi_{sol} }}{{\mu_{0} }}B_{0} \nabla B_{0}$$
(3.2)

The magnetic susceptibility \(\chi_{sol}\) of the solution comprises the diamagnetic contribution of the water molecules and the concentration-dependent contribution of further particles or ions. It reads

$$\chi_{sol} = \chi_{H2O} + \mathop \sum \limits_{i} \chi_{i}^{mol} c_{i}$$
(3.3)

where ci denotes the molar concentration of species i in the solution. Parts of the force named concentration gradient force in earlier literature can often be neglected, as clarified in [3]. Although it is not discussed here, it should also be mentioned that effects on particles in liquids (e.g. magnetic torque) can be achieved by magnetic fields, and that the dipole–dipole interaction force can be utilized for magnetic control. For further details and examples see [2, 4]. As chemical reactions taking place at electrodes or an existing spatial variation of the electric current density in the electrolyte may change the density of the electrolyte, effects of solutal and thermal buoyancy may need to be considered as well.

In the following, applications in flow control and electrochemical processes will be reviewed. The latter are of specific interest, as the electric current necessary for Lorentz force control is an intrinsic part of the system and does not to have to be applied externally. What will be considered is water electrolysis to create shear flows by the Lorentz force to enhance bubble departure from the electrode and electrochemical metal deposition where the space–time yield and the uniformity of the deposit can be enhanced. Furthermore, structuring of electrodeposits down to the nanoscale can be improved by applying magnetic fields. For the sake of brevity, the references given mostly focus on own work, in which a broader view on the topics may also be found. Finally, it should be mentioned that apart from the applications given below, the magnetic gradient force can further be used to control interfaces of paramagnetic liquids [5, 6], to enhance the volumetric enrichment of rare earth elements in solution [7, 8] or to manipulate flow driven in microfluidic channels for analysis purposes [9].

2 Flow Control

First attempts to utilize the Lorentz force for propelling ships date back to the 1960s in the US for experimental submarines. The basic principle is that electrical current from wall-mounted electrodes at opposite sides and a perpendicularly oriented magnetic field generate thrust in a channel [10]. Apart from problems related to gas evolution and corrosion issues, the efficiency of those MHD thrusters is typically low in electrolytes, as the power loss per volume scales as j2/σ. Therefore, applying strong magnetic fields is of advantage. In 1994, the MHD ship “YAMATO-1” tested in Japan was the first to use superconducting magnets for propulsion [11]. Recent developments in designs for propelling cruise ships are reported in [12]. As the total losses remain considerable, a way out is to reduce the actuation volume. In that sense, boundary layer control, well studied in aerodynamics for flaps and airfoils, by electromagnetic means appears to be more attractive. Based on a setup of surface flush-mounted alternating stripes of electrodes and magnets [13] the boundary layer can be energized by adding streamwise momentum, whereby the depth of forcing can be easily controlled by the width of the stripes. The Lorentz force created by the actuator exponentially decays in wall-normal direction, which was found to be advantageous for stabilizing the boundary layer and delaying transition to turbulence [14, 15]. Several efforts have been undertaken to control the separation of flow at inclined hydrofoils, useful for stabilizing large cruise ships. The area of control can thereby be restricted to a small front part at the suction side of the foil. In order to further improve the efficiency of control, proper time-oscillating instead of static forcing can be applied to reduce the separation and thus to enhance the rudder efficiency [16, 17]. Although the method works well also in turbulent flow at high Reynolds numbers, its application remains limited due to the strong increase of energy consumption with rising flow velocity [18].

3 Gas Evolution During Water Electrolysis

Electrochemical splitting of water into hydrogen and oxygen today appears attractive for storing the fluctuating renewable energy supply in chemical form. The efficiency of alkaline water electrolysis at high current densities suffers from intense gas evolution, thereby increasing ohmic and kinetic losses [19]. At vertical electrodes, beside buoyancy and pumping the electrolyte, magnetic fields allow generation of a Lorentz force that drives additional shear to enhance upward bubble transport [20]. Apart from the behavior at large planar electrodes, the phenomena at microelectrodes are also of interest as they could serve as a generic model for electrodes with small islands of catalytic materials or surface elevations at the micro- or nano-scale. Here it was found that surface-parallel magnetic fields generate a shear flow similar to the behavior at large electrodes that strongly favours small-size bubble departure [21]. Surface-normal magnetic fields can be expected to generate counter-rotating azimuthal flows of different strength around the bubble in the upper and lower hemisphere due to tangential deflection of the current density vectors near the bubble surface. At large electrodes, a faster bubble departure is observed, whereas at microelectrodes their departure is retarded. Here, the azimuthal flow driven in the lower hemisphere was found to dominate. Then, due to centrifugal acceleration, a secondary flow is driven towards the bubble/electrode, which retards the bubble departure [22, 23]. Finally, it should be mentioned that inhomogeneous magnetic fields may also appear from magnetization of the electrodes and influence the evolution of paramagnetic oxygen bubbles on the anode side by the magnetic gradient force.

4 Electrodeposition of Metal

4.1 Magnetic Stirring by the Lorentz Force

Applying magnetic fields to electrochemical reactions has been intensively studied in the past [24, 25], largely encouraged by enhanced mass transfer found for electrode-parallel magnetic fields. According to Eq. (3.1), electrode-normal oriented current then causes a Lorentz force that drives an electrode-parallel flow, normal to the direction of the magnetic field. The effect on mass transfer can easily be understood from the corresponding enrichment of the boundary layer with bulk electrolyte, thus increasing the diffusional mass flux towards the electrode. Recent progress is based on an improved understanding of the magnetic forcing. Applying homogeneous magnetic fields to cuboid cells with vertical wall electrodes is typically not very successful, as only the small rotational part of the Lorentz force, regardless of the field orientation, creates weak horizontal stirring of the electrolyte [26,27,28]. Instead, tailored inhomogeneous magnetic fields can be used to create strong vertical stirring in the cells. The forced flow counteracts the stratification of the electrolyte caused by solutal buoyancy and is able to increase the homogeneity of the deposit thickness considerably [29, 30]. However, as small electrodes are frequently submerged in larger cells, the flow structure generated in the cell may be quite complex, and it is difficult to draw conclusions based only on the direction of the magnetic field with respect to the electrode, as will be shown below.

4.2 Structuring Deposits with the Magnetic Gradient Force

The magnetic gradient force comes into play from the magnetic property of constituents of the electrolyte and the exposure to an inhomogeneous magnetic field. Experiments of Tschulik et al. [31,32,33] and Dunne et al. [34,35,36] on the deposition of different metals at electrodes that are magnetically structured on a milli- and micrometer scale delivered results that were surprising at first sight. The deposition rate of paramagnetic ions (e.g. Cu2+) in electrolytes of moderate concentration was found to increase in the vicinity of small permanent magnets or magnetized ferromagnetic elements arranged underneath the cathode. This was surprising, as the magnetic susceptibility of the electrolyte was negative because it was dominated by the diamagnetic water molecules (Eq. 3.3). Adding electrochemically inert and strongly paramagnetic ions (e.g. Mn2+) to the electrolyte turned the former elevation of the deposit into a valley (see Fig. 3.1a, b). Depositing instead diamagnetic ions such as Bi3+ did not lead to any local alteration of the deposit thickness in the vicinity of the magnetic elements, which could be understood from the very small magnitude of the magnetic susceptibility of diamagnetic ions in general when compared to paramagnetic ions. However, adding electrochemically inert Mn2+ ions locally reduced the deposition rate, similar to the combined copper case already mentioned (see Fig. 3.1c, d).

Fig. 3.1
4 schematics compare the effects of electrochemically inert strongly paramagnetic and diamagnetic ions on the magnetic fields.

Sketch of the local flow forced near the cathode (black, top position) where a metal (a, b: copper, c, d: bismuth; all in light blue) is deposited. The electrolyte contains copper or bismuth ions with (b, d) or without (a, c) manganese ions in excess. The magnetic element above the cathode is drawn in red

Full size image

Combined fluid-mechanical and electrochemical reasoning is the key to understand these results. As only the rotational part of the magnetic gradient force is able to drive a flow, only the gradient of the magnetic susceptibility of the solution determines the direction of the flow forced [37]. When considering the case of plain copper ions, despite the electrolyte has a negative magnetic susceptibility, its gradient is determined by the concentration gradient of the copper ions only. Thus, a flow towards the electrode is driven by the magnetic gradient force that enriches the boundary layer and therefore enhances the deposition at the magnetic element. Adding electrochemical reasoning, the inverse copper patterning in case of an electrolyte with Mn2+ ions can also easily be understood. The key point is that even though the Mn2+ ions do not take part in the electrochemical reaction, their concentration near the cathode increases, as electrical neutrality must hold true outside the double layer [38]. As the gradient of the Mn2+ concentration dominates the gradient of the solution susceptibility, its sign is opposite compared to the pure copper case, and the direction of flow driven by the magnetic gradient force is reversed. Thus, a wall-parallel flow of depleted electrolyte is approaching the magnetic element where it leaves the electrode, thereby yielding locally a lower deposition rate. The same argument seamlessly explains the case of Bi3+ deposition with inert Mn2+ ions, as again the inert but strongly paramagnetic ions dominate the susceptibility gradient [39,40,41]. A broader summary of the reasoning can be found in [42, 43].

The question how tall the structures may become during deposition was addressed only recently. During growth, the surface departs from the region of largest magnetic gradients underneath the cathode, and the structuring effect is reduced. Furthermore, at larger times, solutal buoyancy starts to disturb the structuring effect. It is therefore advantageous to perform the deposition in a pulse-reversed mode to frequently rebuild the concentration boundary layer and retard the action of buoyancy at horizontal electrodes [44].

4.3 Nano-Structuring of Metal Layers

The last question addressed here is whether magnetic fields can be beneficial for the manufacturing of nanostructured layers of metals. In electrodeposition, nano-structuring is conventionally achieved by adding capping agents (e.g. Cl− ions) to the electrolyte to damp or enhance growth in different crystallographic directions [45]. Recently it was shown that electrode-normal magnetic fields force a circumferential flow near conical elevations of metal deposits by action of the Lorentz force. The resulting centrifugal acceleration then gives rise to a secondary downward flow that enhances conical growth [46]. The magnetic gradient force at ferromagnetic cones was found to drive a downward flow as well, thus supporting the conical growth [47]. However, electrodeposition of nano-structured nickel layers in a magnetic field has not yet shown that capping agents can completely be omitted [48]. A scaling analysis performed recently found that the supporting flow driven by the magnetic forces gets weaker as the size of the cone reduces. At the nanoscale, the support of the magnetic gradient force remains substantial, whereas the support of the Lorentz force has decayed [49]. However, experiments to deposit nickel nanostructures did not yet give strong support, as global cell flow driven by the Lorentz force supersedes the beneficial local flow near the cones. Therefore, improved cell and electrode designs are currently under discussion to reduce the global cell flow and thus enable magnetic support for depositing nanostructures [50].