Amplitude Equation for the Rosensweig Instability

Since its discovery in 1967 [1], the normal field or Rosensweig instability attracted the attention of experimentalists and theoreticians, alike. The phenomenon describes the transition of an initially flat ferrofluid surface to hexagonally ordered surface spikes as soon as an applied magnetic field exceeds a certain critical value. Ferrofluids are suspensions of magnetic nanoparticles in a suitable carrier liquid. One of the most prominent property is the superparamagnetic behavior in external magnetic fields, which accounts for the large magnetic susceptibility and the high saturation magnetization in rather low magnetic fields. If one starts to crosslink a mixture of a ferrofluid and a polymer solution with cross-linking agents, a superparamagnetic elastic medium, called ferrogel, is obtained [2]. As in usual ferrofluids, the initially flat surface of ferrogels becomes linearly unstable beyond a critical magnetic field [3]. A nonlinear analysis of the Rosensweig instability, however, turned out to be very complicated mainly due to the fact that the instability necessarily involves a deformable surface. In addition, the driving force is provided by the boundary conditions at the deformed surface and not by the bulk equations. A second problem arises from the fact that this instability is intrinsically dynamic in nature, although the linear threshold and the critical wavelength can be obtained in a static description (as an energetic minimum neglecting viscous effects). A nonlinear description, however, has to treat this instability as a breakdown of traveling surface waves.