- Nichtlineare Dynamik (1) (remove)
- Surface Deformations of Magnetic Continua in Homogeneous Fields (2009)
- In this thesis, experiments with magnetic liquids and gels are presented. Ferrofluids are synthetically created suspensions of magnetic nanoparticles in a carrier liquid. By adding a gelator, such a ferrofluid can be turned into a ferrogel. The magnetic properties of these substances are similar to a usual paramagnet with the important difference, that the susceptibility of the former is higher by a factor of 10^3 to 10^6. By the application of a homogeneous field, a transformation of the shape of a magnetic sample can be induced. In this thesis, four experiments on the surface deformation in homogeneous magnetic fields are presented. Two geometric configurations are considered: a horizontally extended flat layer with a free surface as well as a spherical sample. In both cases, the application of a homogeneous magnetic field leads to changes of the shape of the free boundary. In the case of the spherical geometry, the sample is deformed into a prolate ellipsoid under the action of the field, the so called magnetodeformational effect. In case of the extended flat layer, an abrupt shape transition into a patterned state takes place, the normal field or Rosensweig instability. In contrast to the smooth deformation of the sphere, this is an instability, which breaks the translational symmetry, and the transition occurs at a certain threshold value of the magnetic induction. Each of the four experiments in this thesis is briefly summarized in the following paragraphs. Part I of the thesis considers ferrofluids. In chapter 2, the ideal geometry of an infinitely extended flat layer is intentionally reduced to a cylinder such that only a single spike in the centre exists, and the solution space becomes rotationally symmetric. This makes the problem very feasible for experimental methods and numerical simulations. Two measurement techniques are applied and compared to each other, namely an X-ray technique, where the surface deformation is extracted from radioscopic images, and a laser technique, which focuses a laser spot onto the surface. The experiments and the simulations, the latter performed in close cooperation with a group in Athens, show a convincing agreement within a few percent. It remains an open question, whether the result can be deduced in analytic form, however. In chapter 3, a highly viscous ferrofluid is utilized to study the nonlinear dynamics of the normal field instability at very low Reynolds numbers. The linear growth rate for the growth and decay of the pattern at small amplitudes is extracted from the measurements and compared with an existing theoretical model. In addition, the measurement technique provides the reconstruction of a fully nonlinear amplitude equation, which is qualitatively compared to model equations. These nonlinear amplitude equations can only describe the dynamics of the growth in the immediate vicinity of the critical point so far. For a quantitative comparison, there is a need for a model with an extended range of validity. Additionally, localized patterns are observed which arise spontaneously in the neighbourhood of the unstable solution branch, which have previously been observed with the help of an external disturbance Part II of the thesis deals with thermoreversible ferrogels. Chapter 4 studies the magnetodeformational effect. A ferrogel sphere is exposed to homogeneous magnetic field. When the field is applied suddenly, the sphere not only elongates in the direction of the field, but also vibrates about the new equilibrium. On a longer time scale, the deformation continuously increases due to the viscoelastic properties of the gel. Both phenomena can well be described by a harmonic oscillator model, where the spring constant changes with time. From the deformation parallel and perpendicular to the applied field, Poisson´s ratio can be calculated, which turns out to be close to the limit of incompressibility. The absolute values of the deformation are compared to recent theoretical models. The resulting deviation of about 10% is attributed to the viscoelastic properties of the ferrogel, which are not taken into account in the static models. In chapter 5, the normal field instability is realized for the first time with a ferrogel. A flat layer of a thermoreversible ferrogel is exposed to a homogeneous magnetic field at different temperatures, where the gel is viscoelastic. This is a consequence of the need for a very soft material, such that the growth of the pattern is not completely suppressed by the elastic forces. The magnetic field is periodically modulated in time, and the amplitude of the instability is measured, which is modulated with the same frequency. The comparison with rheological measurements reveals a scaling of the modulated amplitude with the complex viscosity of the ferrogel. A comparison with the theoretical model for a ferrogel is difficult due to the viscoelasticity of the gel.