# Magnetic Microhydrodynamics: An Emerging Research Field. Introduction

Bernard Doudin, Andrejs Cēbers, Michael Coey

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Bernard Doudin, Andrejs Cēbers, Michael Coey. Magnetic Microhydrodynamics: An Emerging Research Field. Introduction. Bernard Doudin; Michael Coey; Andrejs Cēbers. Magnetic Microhydrodynamics. An Emerging Research Field, 120, Springer International Publishing, pp.1-7, 2024, Topics in Applied Physics, 10.1007/978-3-031-58376-6_1. hal-04994711

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# Chapter 1
## Introduction

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Bernard Doudin, Andrejs Cēbers, and Michael Coey

### 1.1 Introduction

This introduction situates the subject matter and scope of the book and provides a brief summary of the contents, identifying the magnetic materials and illustrating the range of magnetic effects that can be observed. The importance of size and scaling is emphasised. Some future challenges and prospects are highlighted.

The topic of this collection of articles on applied physics is 'Magnetics and Micro-hydrodynamics', a domain that is situated close to, but separate from three established areas of research (Fig. 1.1)—*Magnetohydrodynamics* [1], the study of the magnetic properties and dynamics of electrically-conducting fluids such as plasmas, liquid metals and ionic solutions, *Ferrohydrodynamics* [2], the study of the motion of strongly polarizable magnetic liquids (ferrofluids) in a magnetic field and *Microfluidics*, the study of fluid flow confined in sub-millimeter scale structures [3]. As investigations accumulate in this generous interstitial space, we see an interesting subfield with characteristics of its own emerging. This book, a collection of original studies and topical reviews is a first attempt to map out the shape of the new subfield. The following 12 chapters are arranged in four groups of three, each part focussed on a different aspect of the physics, chemistry or applications. First we provide thumbnail outlines of the chapters, emphasizing the role of magnetic field in each case.

B. Doudin (✉)

IPCMS, University of Strasbourg and Centre National de La Recherche, Strasbourg, France
e-mail: bernard.doudin@ipcms.unistra.fr

A. Cēbers

Department of Physics, University of Latvia, Riga 1004, Latvia

M. Coey

School of Physics, Trinity College, Dublin 2, Ireland
<figure><img src="image_3.png" /><figcaption>**Fig. 1.1** Situation of magnetic microhydrodynamics MMH</figcaption></figure>

The first three articles discuss elements of the theory of *magnetic fields and liquids*, especially the magnetic force and energy densities involved. The first of them, by Tim Butcher on ‘Magnetic Action at a Distance’, considers the Lorentz force density $f_L = (j \times B)$ where $j$ is the electric current density and the Kelvin or magnetic field gradient force density $f_k = (\chi/2\mu_0)\nabla B^2$ that depends on the susceptibility $\chi$ of the medium, provided $\chi \ll 1$ ensuring that the response is linear. (An alternate formulation of the same force is the Korteweg-Helmholtz or concentration gradient force density). The magnetic fields $B$ are ~ 1 T and field gradients range from 1 to $10^7$ Tm$^{-1}$. In an incompressible fluid confined by solid walls, vortex flow arises in systems whenever the gradients of field and susceptibility are noncollinear. A susceptibility of the magnetized medium $\chi \ll 1$ is a common situation. It applies to dilute suspensions of polymeric microbeads loaded with superparamagnetic iron oxide nanoparticles ($\chi \approx 10^{-2}-10^{-3}$), as well as paramagnetic solutions of transition metal ions ($\approx 10^{-3}-10^{-5}$), and water itself ($\chi = -9 \times 10^{-6}$) (Table 1.1).

Gerd Mutschke provides a description of magnetic control of mass transfer in weakly conducting fluids, based on the experience of his group with electrochemical systems, especially for electrolysis of water and metal deposition. The Lorentz force on current flow patterns formed around growing gas bubbles on the electrodes during electrolysis influences hydrogen or oxygen bubble release; oxygen, a weakly paramagnetic gas with $\chi \ll 1$ is subject in addition to a Kelvin force. In metal deposition, local magnetic vortex flow at the electrode stirs and thins the layer of dissolved cations and enhances diffusive mass transport of dissolved metal ions to the cathode where they are reduced to metal. The metal deposits from a mixture of metal ions can be structured by the Kelvin force when the susceptibility and electronegativity of the ions differ.
<table><caption>**Table 1.1** Magnetic materials</caption><thead><tr><th>Material</th><th></th><th>Size</th><th>Susceptibility</th><th>Magnetization<br>kAm<sup>-1</sup></th><th>Coercivity<br>kAm<sup>-1</sup></th></tr></thead><tbody><tr><td><strong>Diamagnets</strong></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Water</td><td>Liquid</td><td>—</td><td>−9 × 10<sup>-6</sup></td><td>—</td><td>—</td></tr><tr><td><strong>Paramagnets</strong></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>3d ion solutions</td><td>Liquid</td><td>—</td><td>10<sup>-5</sup>−10<sup>-4</sup></td><td>—</td><td>—</td></tr><tr><td>4f ion solutions</td><td>Liquid</td><td>—</td><td>10<sup>-4</sup>−10<sup>-3</sup></td><td>—</td><td>—</td></tr><tr><td>4f nanoparticles</td><td>Solid<sup>b</sup></td><td>20 nm</td><td>10<sup>-3</sup></td><td>—</td><td>—</td></tr><tr><td><strong>Superparamagnets</strong></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Iron oxide nanoparticles</td><td>Solid<sup>b</sup></td><td>5−50 nm</td><td>3</td><td>400</td><td>—</td></tr><tr><td>Polymeric microspheres<br>with dispersed iron oxide<br>nanoparticles</td><td>Solid<sup>b</sup></td><td>~ 1−3 µm</td><td>0.1</td><td>30</td><td>—</td></tr><tr><td>Ferrofluids</td><td>Liquid</td><td>—</td><td>2</td><td>30</td><td>—</td></tr><tr><td><strong>Ferromagnets</strong></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Hematite<sup>a</sup> microparticles<br>Cubes and other shapes</td><td>Solid</td><td>~ 1−2 µm</td><td>10<sup>-2</sup></td><td>2</td><td>300</td></tr><tr><td>Iron microspheres<br>Carbonyl iron</td><td>Solid<sup>b</sup></td><td>3−10 µm</td><td>3</td><td>1700</td><td>5</td></tr><tr><td>Nd-Fe-B magnets<br>Various sizes</td><td>Solid</td><td>1 mm−5 cm</td><td>1</td><td>1200</td><td>1000</td></tr></tbody></table>

<sup>a</sup> Hematite is a canted antiferromagnet with a very weak ferromagnetic moment

<sup>b</sup> May be dispersed in liquid or soft matter

The third contribution, by Andrejs Cēbers, on phenonomenological models of magnetizable fluids (including ferrofluids with $\chi \approx 1$, beyond the linear approximation) presents several models, considering conservation laws for the energy of the magnetic field and the fluid medium and the mass and momentum of the fluid medium. A choice of the electromagnetic energy flux and stress due to the field yields a relation for entropy production that includes magnetic relaxation. Unlike the other chapters, which use standard SI units, the reader here will encounter the Gaussian cgs units that are still widely used in theoretical magnetism, and will need to replace factors of $4\pi$ and c, the velocity of light, by $\mu_0$ and $\epsilon_0$, the permeability and polarizability of free space [$c^2 = \frac{1}{\mu_0\epsilon_0}$] to apply them in experimental practice.

Next follows the *Movers and Shakers* section. First is the chapter by Arvind Dev and co-workers, who have looked in detail into the scaling of the liquid-in-liquid flow of water in a moving channel confined by walls of ferrofluid stabilized by a magnetic quadrupole field produced by permanent magnets [4]. Near-ideal plug flow in the 'magnetic antitube' channel is achieved, and the use of surfactant to minimize interfacial tension and reduce the Laplace pressure at the channel wall is explored. Friction is reduced by more than 99% at the moving walls. Shear forces, which are
an impediment to moving fragile objects along microchannels, are minimized. X-ray imaging with synchrotron radiation is needed to view the liquid tube behind the ferrofluid walls. The technology can be scaled in principle down to the 1 µm scale.

The first shakers are micron-scale hematite cubes studied by Mārtiņš Brics and co-workers. Hematite is an antiferromagnet where the sublattice moments are slightly canted due to the Dzyaloshinsky-Moriya interaction, creating a weak resultant ferro-magnetic moment and a susceptibility two orders of magnitude less than that exhibited by the ferrimagnetic iron oxides magnetite and maghemite. The moment direction lies at 12° to a cube diagonal, and it is possible to exert torque on the particles in magnetic field. In a small static field the cubes form straight or kinked chains under the action of dipole-dipole interactions, but in a rotating field the cubes and chains rotate or roll in characteristic ways, including swarming behaviour in two dimensions, that depend on the magnitude and frequency of the rotating magnetic field.

The last article in this section by Mattia Ostinato et al. describes what happens in simulations of two layers of polymeric microparticles loaded with superparamagnetic iron oxide nanoparticles that are confined between glass slides separated by less than two particle diameters. In a low frequency (1 Hz) rotating field, the spheres form dimers that rotate coherently. Then at a critical frequency there is a transition from the coherent state to an ‘exchange’ state where the dimers begin to break up and reconnect with other partners. The fraction of the particles that are active in this sense is taken as the order parameter of the ‘synchronous-exchange’ transition, an out-of-equilibrium phase transition belonging to a special universality class that includes other examples such as forest fires and financial crises.

The third section deals with magnetic field effects on water and ionic solutions. A wide range of effects on water are reviewed by in the chapter by Sruthy Poulose and co-workers. Effects on the shapes of pendant or sessile droplets are small, because sub-tesla fields have little or no effect on the static or dynamic surface tension of water or ionic solutions. Kelvin forces will modify the droplet shapes, and zero-susceptibility solutions of paramagnetic ions in diamagnetic water are useful to eliminate them. But dramatic field effects of up to 100% or more on the evaporation rate of water in confined spaces cannot be explained by any thermodynamic effect of the field on water chemistry—the magnetic energy is eight orders of magnitude less than the hydrogen bond energies. The explanation is based on field-induced transitions between nuclear *ortho* (triplet) and *para* (singlet) isomers of water with parallel or antiparallel proton spins, which behave as quasi-independent gasses in the vapour phase.

Jinu Kurian et al. use the very large magnetic field gradients created at a multido main thin film Co–Pt multilayer cathode with perpendicular magnetic anisotropy to influence the model one-electron electrochemical ferricyanide redox reaction at the cathode. The field gradient of order $10^7$ Tm$^{-1}$ is localized in the double layer, within about 20 nm of the electrode. The system is investigated by electrochemical impedance spectroscopy. Effects of the magnetic field gradient on the double layer capacitance and the charge transfer resistance are found, but limited to typically 7% when we switch on the Kelvin force by the applied field. Although the field in the
double layer is 0.2 T and the average field gradient force is as high as $10^7$ Nm$^{-3}$, they are insufficient to modify greatly diffusion near the electrode. No appreciable influence of the field gradient force on the reaction kinetics was detected.

Fenshe Sun and co-workers describe a new method to measure the rate constant of a rare earth solvent extraction system using an immiscible water-oil system in a thin horizontal Hele-Shaw cell with a 1 mm gap and an interface between Dy(III) in aqueous solution and an organic oil phase. The concentration of dysprosium in the boundary layer is measured interferometrically with a Mach-Zehnder interferometer and the interface mass flux at the initial time is determined for varying Dy(III) and oil concentrations. The system is found to be quasi first-order for Dy and quasi second-order for the oil. The method, which involves no magnetic field, can be used for a wide range of reactions of transparent liquids, and unlike conventional stirring methods, requires only about 1 mL of liquid.

The final section is concerned with biomedical applications, which is where the potential impact of bringing magnetism to life on a sub-micron scale is greatens. The first two chapters are concerned with the mechanical properties of biomolecules. Peter Galie's chapter on magnetic elastomers and hydrogels for studies of mechanobiology uses synthetic magnetoelastic materials produced by loading PDMS or fibrous polymeric hydrogels with 0.1 wt% of spheres of ferromagnetic carbonyl iron a few microns in diameter. The susceptibility is $\approx 4 \times 10^{-4}$, and the hydrogel is anisotropically stiffened by the magnetic dipole-dipole interaction between the spheres, with a shear modulus of $10^2-10^4$ Pa that matches the intercellular medium of soft tissue. Cells are very sensitive to the substrate on which they are grown, and respond in seconds to a change in environmental stiffness by means of intracellular Ca flows. The study compares linear elastomeric and nonlinear fibrous hydrogel systems.

Rafael Tapia-Rojo reports a new design of magnetic tweezers for manipulating biomolecules such as DNA or proteins. Instead of pulling the biomolecule by applying a variable magnetic field gradient from moveable permanent magnets to a micron-scale superparamagnetic polymeric microbead tethered to the molecule, improved control is achieved with a miniature electromagnet in an old-style write head from a tape recorder, which allows fine control of both the magnitude and frequency of the magnetic field gradient force. The method can be used to uncoil proteins and measure their dynamics on a kHz timescale. Two examples are studied. One is protein L where a transient molten globule state appears when the protein is sampled on a millisecond timescale. It is the precursor of the folded states. The other is the Talin mechanosensor, which is exquisitely sensitive to tiny force changes of a few pN, and is shown to exhibit stochastic resonance in the presence of noise.

The last chapter, by Thomas Gevaert and colleagues is an informative review of the dual-purpose magnetic microparticles that can serve both for diagnostics and treatment of disease—an area known as theranostics, which involves pairing of diagnostic biomarkers with therapeutic agents that share a specific target in diseased cells or tissues—cancer in this case. Polymeric microbeads loaded with superparamagnetic iron oxide or gadolinium-based nanoparticles 10–15 nm in diameter can serve as contrast agents in MRI to help locate and identify diseased tissue, and then treat it by heating the local area by means of a kilohertz electromagnetic field in a
process known as hypothermia. Drugs attached to the microbead surface may also be released by heating, or used in assay or diagnosis.

Although the applications of magnetics in these chapters are quite diverse, common themes of Lorentz and magnetic field gradient forces recur, both in static and dynamic contexts in low-susceptibility systems. In addition there are dipole-dipole interactions and torques on dilute dispersions of ferromagnetically-ordered inclusions or induced moments in dilute arrays of superparamagnetic nanoparticles in polymeric microbeads. There are all semi-classical effects. To account for effects in magnetochemistry, quantum mechanics and the influence of weak magnetic fields on the symmetry of the electronic or nuclear wave-functions may be critical. Table 1.1 summarizes the properties of various relevant magnetic materials, which are classified by their susceptibility and magnetic order. Size also matters, as the effect of magnetic field and forces will also increase with the size of the object. However, scaling down can be beneficial: the magnetic dipole field is scale independent, hence the field produced by an array of permanent magnets does not depend on the length scale—the secret to the success of magnetic recording. However the gradient field forces are enhanced on decreasing the scale, which allows the creation of intense force fields at the nanoscale, with experimental designs and improved sensitivity illustrated in several chapters of this book.

We hope that these ideas will encourage people with different scientific interests to appreciate the uses and potential of magnetic field and magnetic materials and use them effectively to build a new magnetoscience on sound foundations. A key first step should be to establish experimentally the magnetic principle behind any proposed effect. The experiments can be quite simple at first, but the vector nature of the magnetic field and the combination of magnetic and other forces can complicate the interpretation and modelling of the systems under investigation, with a benefit of a remarkable richness of the observed phenomena. There is therefore a real need to properly identify the forces at play, and search for the best experimental conditions to avoid misinterpretation of the results.

Some areas of study we think will be important for future development of Magnetic Microhydrodynamics (MMH) are the following.

* A focus on the often-elusive influence of magnetic fields and forces on the properties of water, the ubiquitous component of living matter.

* An effort in experimental design that minimizes ambiguity and ‘artefacts’ in the results.

* Development of models that reveal new behaviour in magnetic microhydrodynamics beyond the linear approximation, analogous to coercivity.

* A truly interdisciplinary approach, where teams are built associating members having an understanding of magnetism with experts in the fields of application such as mechanics, materials chemistry or life science.

One example at least serves to emphasis that a fundamental research program in MMH has potential to achieve significant societal relevance: Several authors evoke the idea of efficient magneto-electrochemical separation of the rare-earth elements, the first claims for which date back to the work of Walter and Ida Noddack in
the 1950s [5]. It remains a challenge to demonstrate an effective process that can be implemented at scale, which would be a step towards a more sustainable energy landscape where rare-earth permanent magnets will be meeting many of our requirements for magnetic field at no continuous expenditure of energy.

## References

1. P.A. Davidson, An Introduction to Magnetohydrodynamics (Cambridge University Press 2010)

2. R.E. Rosensweig, Ferrohydrodynamics (Dover Publications, 2014)

3. P. Tabeling, Introduction to Microfluidics (Oxford University Press, 2010)

4. P. Dunne et al., Nature **581**, 58–62 (2020)

5. W. Noddack, I. Noddack, E. Wicht, Ber. Bunsenges. Phys. Chem. Bunsenges. Phys. Chem. **62**, 77 (1958)

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