Modelling of magnetic vortex microdisc dynamics under varying magnetic field in biological viscoelastic environments

TL;DR

This work develops a 2D macrospin-based model to describe magneto-mechanical actuation of vortex microdiscs embedded in viscoelastic media under oscillating and rotating magnetic fields. By coupling a Maxwell-type viscoelastic description with elastic resistance, the authors derive analytical expressions and validate them with simulations, revealing how viscosity, stiffness, and field frequency govern transitions between synchronous and asynchronous disc motion and the role of magnetisation flips. The study shows energy transfer to the surrounding medium can be significant and potentially perturb cellular processes while heating remains negligible in the explored frequency range, offering a practical framework to predict rheology-driven effects in cells or extracellular matrices. Despite the simplifications (notably the 2D approximation and Maxwell rheology), the model provides actionable guidance for designing magneto-mechanical stimulation experiments and understanding how viscoelastic properties influence vortex microdisc actuation in biological environments.

Abstract

Magnetically driven microparticles provide a versatile platform for probing and manipulating biological systems, yet the physical framework governing their actuation in complex environments remains only partially explored. Within the field of cellular magneto-mechanical stimulation, vortex microdiscs have emerged as particularly promising candidates for developing novel therapeutic approaches. Here, we introduce a simplified two-dimensional model describing the magneto-mechanical response of such particles embedded in viscoelastic media under varying magnetic fields. Using a Maxwell description of the medium combined with simplified elasticity assumptions, we derive analytical expressions and support them with numerical simulations of particle motion under both oscillating and rotating magnetic fields. Our results show that rotating fields typically induce oscillatory dynamics and that the transition to asynchronous motion occurs at a critical frequency determined by viscosity and stiffness. The amplitude and phase of this motion is governed by the competition between magnetic and viscoelastic contributions, with particle motion being strongly impaired when the latter dominate. Energy-based considerations further demonstrate that, within the frequency range explored of few tens of Hertz, no heat is generated -- distinguishing this approach from magnetic hyperthermia -- while the elastic energy transferred to the surrounding medium is, in principle, sufficient to perturb major cellular processes. This work provides a simple framework to anticipate the first-order influence of rheological properties on magnetically driven microdisc dynamics, thereby enabling a better understanding of their impact in cells or extracellular materials and bridging the gap between experimental observations and theoretical modelling.

Figures (10)

• Figure 1: a) Illustration of the vortex micromagnetic behaviour in a disc-shaped particle: from vortex state (absence of external field) to saturation. b) Schematic representation of superparamagnetic-like behaviour of the vortex microdiscs. There is no remanent magnetisation in the absence of magnetic field. Particles made of permalloy with radius of the order of 1 $\mu$m and thickness below 100 nm saturate with field amplitude in the range 40-100 mT depending on their thickness leulmi_comparison_2013.
• Figure 2: a) Microdisc embedded into a 3D viscoelastic material. b) 2D representation of an external field, rotating or oscillating in the ($x,z$) plane. The field orientation at time $t$ is given by $\theta(t)$. The field is invariant in the $y$ direction. c) The particle is trapped into a material, either extracellular or intracellular, approximated by a Maxwell viscoelastic material. Its rheological properties are equivalent to a spring and a dashpot in series.
• Figure 3: a) The Stoner-Wohlfarth model determines the orientation of the magnetisation $\bf{M}$ relative to the plane of the particle (labelled by the angles $\delta$ or $\delta' = \pi - \delta$) as a function of the orientation of the field $\bf{B}$ (angle $\theta$). b) In our model, the amplitude of the field is assumed constant while its orientation varies in time. We approximate the magnetic hysteresis curve to first order: the magnetisation of the disc flips from one saturated state to the other when the field is perpendicular to the plane of the disc.
• Figure 4: Geometrical parameters of the model when the particle can rotate.
• Figure 5: Evolution of the angle $\delta$ between the anisotropy plane and the field as function of the field angle, $\theta$, for an immobilised particle (Stoner-Wohlfarth's model).