A refined theory of magnetoelastic buckling matches experiments with ferromagnetic and superparamagnetic rods

Abstract

In its simplest form the magnetoelastic buckling instability refers to the sudden bending transition of an elastic rod experiencing a uniform induction field applied at a normal angle with respect to its long axis. This fundamental physics phenomenon was initially documented in 1968, and, surprisingly, despite many refinements, a gap has always remained between the observations and the theoretical expectations. Here, we first renew the theory with a simple model based on the assumption that the magnetization follows the rod axis as soon as it bends. We demonstrate that the magnetoelastic buckling corresponds to a classical Landau secondorder transition. Our model yields a solution for the critical field as well as the shape of the deformed rods which we compare with experiments on flexible ferromagnetic nickel rods at the centimeter scale. We also report this instability at the micrometer scale with specially designed rods made of nanoparticles. We characterized our samples by determining all of the relevant parameters (radius, length, Young modulus, magnetic susceptibility) and, using these values, we found that the theory fits extremely well the experimental results for both systems without any adjustable parameter. The superparamagnetic feature of the microrods also highlights the fact that ferromagnetic systems break the symmetry before the buckling. We propose a magnetic "stick-slip" model to explain this peculiar feature, which was visible in past reports but never detailed. magnetoelasticity | buckling | nanoparticles | instability | magnetic I n the past decade there has been an emerging field of research on new magnetic and elastic soft materials whose shape can be remotely controlled by application of an external magnetic field (1-3). Indeed, at many scales and in various domains, magnetic filaments (4, 5), gels (6), and so on (7) show great promise in numerous domains of application (8). With the progress in the design of these materials, their magnetic susceptibility increases, and brings them closer to the behavior of the more conventional magnetic alloys. Thus, they can benefit a more ancient domain of research that described their magnetoelastic properties. Magnetoelasticity generally describes various phenomena that couple magnetization and mechanical deformation of solid-state objects (9). Excluding magnetostriction, the domain splits into two categories depending on whether the system is driven by free macroscopic currents or only by bound currents. In this latter case, the system may be studied only for its equilibrium configuration, or for its dynamic behavior (10). The complexity of the solution of the magnetic field in the general case justifies that a comprehensive description of the phenomenon is restricted to some trivial geometries such as rods of large aspect ratios (11). The main contribution to the field was made in 1968 with an elegant work (12) that paved the way both for experiments and theory but whose measures showed a critical field twice lower than expected. Further studies (1...

Optical detection of nanometric thermal fluctuations to measure the stiffness of rigid superparamagnetic microrods

“…In future works, rods synthesized with more monodisperse magnetic properties would advantageously permit an in-depth comparison between both methods. Alternatively, its combination with the thermal measures may provide a direct measure of χ on individual rods (32).…”

“…Microrods were prepared by slow completion of oppositecharged polymers with iron-oxyde nanoparticles during a dialysis performed under a permanent magnetic field of ∼200 mT (Fig. 1) according to a protocol already described (30,32). Here, we used two stock solutions of citrated particles (1% vol fraction 52 g/L) of which the diameter follows a log-normal distribution around 8 ± 2 nm and 13 ± 3 nm, respectively.…”

“…The system was piloted by ImageJ and Micro-Manager softwares. The magnetic setup, described previously (32), is shown in Fig. S4.…”

“…Our study was performed on synthetic microrods made of iron-oxide nanoparticles (with a mean diameter of 8 nm or 13 nm) interconnected by a polymer (29,30). These superparamagnetic rods with a radius r ∼ 350 nm and a length L ∼ 20−70 µm were recently used to illustrate the magnetoelastic instability (31,32) and served as a passive (3) or magnetically driven (33)(34)(35) rheological probe to study Newtonian (33) and complex fluids (3,34,35), including the cytoplasm of living cells (36). We reached the required precision to detect fluctuations as small as 4 nm by using a standard superresolutive method to locate the rod centerline (11,13) and by developing an algorithm to track noisy motions of the microscope stage (25).…”

Flexible Magnetic Filaments and their Applications

“…[ 26 ] when studying bending of 20 80 m − μ long Ni rods (radius 46 m μ ). Buckling was observed by clamping one end at an angle 0 ϑ with respect to the perpendicular to the fi eld direction.…”