Bending and Position Hysteresis of Magnetic Microfibers in Nonuniform Magnetic Fields

Magnetic microfibers are fibers that behave as a flexible paramagnetic body, for example, polymer fibers filled with superparamagnetic particles. A cantilevered magnetic microfiber will bend in response to an applied magnetic field. In a nonuniform field, generated for example by a single electromagnet or by a magnetic dipole, a magnetic microfiber displays position hysteresis as the field strength increases and decreases. This paper presents a model for determining stable shapes of a cantilevered magnetic microfiber in a nonuniform magnetic field. The model determines stable shapes by finding local minima of the potential energy using a Rayleigh-Ritz method. The model predicts the position hysteresis behavior observed in magnetic microfibers. Experimental data ware collected using two electromagnets with different geometries. The model simulation and experimental data compare well both qualitatively and quantitatively. The model will be useful for designing actuators based on magnetic microfibers and for characterizing the magnetic properties of fabricated fibers. A rigid bar model is also introduced, which captures the qualitative behavior of the fiber and illustrates the source of the position hysteresis behavior. INTRODUCTION A slender, flexible, unconstrained paramagnetic body in a uniform magnetic field will be acted on by torques that will cause the body to align itself with the direction of the field. In a nonuniform magnetic field, the body will additionally be acted on by forces that cause the body to move toward regions where the field has higher magnitude. If such a paramagnetic body is cantilevered, it will bend in the direction of an applied magnetic field, produced for example by an electromagnet. Magnetic microfibers are fibers that act as a flexible paramagnetic body. The fibers used in the experimental section of this paper are hollow fibers filled with superparamagnetic nanoparticles. The model presented in the paper applies more broadly to magnetic microfibers, which includes the variety of paramagnetic composite fibers, such as found in [1-4]. More generally, magnetic microfibers are being investigated for potential use as electromagnetically actuated, fiber-based manipulators for microfluidic applications. The manipulators consist of a cantilevered fiber, typically 1-5cm in length, positioned between electromagnets with axes orthogonal to the fiber. Magnetic microfibers offer an additional tool for mesoscale fluid manipulation, for example picking and placing drops of liquid in lab-on-a-chip applications. The present work is specifically motivated by the use of magnetic microfibers as the basis for an artificial butterfly proboscis [4]. For a typical configuration of electromagnet and fiber, there will be a single stable shape when the magnitude of the magnetic field is small or large. When the magnitude of the magnetic field is in an intermediate range, there are two stable shapes that the fiber can take. This leads to hysteresis in the position of the fiber as the field is changed from low to high and back again. This hysteresis was first observed experimentally. One important objective of this paper is to provide a model that accurately captures this behavior. Recent interest in magnetic microfibers has been driven by new methods of fabricating paramagnetic composite fibers and new applications for the fibers [1-9]. Much of the recent magnetic microfiber modeling effort builds directly on the analysis of paramagnetic beads in magnetostatic fields [10], which has a long history in the literature, motivated initially by the use of ligand-decorated paramagnetic beads for separating compounds out of solution. Paramagnetic particles in colloids form into chains in the presence of external fields, the dynamics of which are analyzed in [5,6]. Micrometer-sized