Abstract
We derive the dispersion relation of surface waves for magnetic gel membranes or thin films at the interface between two fluids in the presence of an external magnetic field normal to the free surface. Above a critical field strength surface waves become linearly unstable with respect to a stationary pattern of surface protuberances. This linear stability criterion generalizes that of the Rosensweig instability for ferrofluid and ferrogel free surfaces to take into account bending elasticity and intrinsic elastic and magnetic surface properties of the film or membrane, additionally. The latter is of interest for uniaxial ferrogel film or membranes, which show a locked-in permanent magnetization.
Highlights
Free surfaces of ferrofluids, stable colloidal solutions of single-domain magnetic nanoparticles in a carrier fluid, are known to undergo a transition from a flat state to a stationary pattern of surface spikes above a certain threshold of an applied normal magnetic field [1]
A polymer network crosslinked in the presence of a ferrofluid [2], elasticity of the network constitutes an additional stabilizing mechanism increasing the threshold value of the external magnetic field without, changing the characteristic wavelength of the most unstable linear mode [3]
We are interested in the situation of a thin layer of a magnetic gel sandwiched between two fluids in the presence of a normal field
Summary
Stable colloidal solutions of single-domain magnetic nanoparticles in a carrier fluid, are known to undergo a transition from a flat state to a stationary pattern of surface spikes above a certain threshold of an applied normal magnetic field [1]. In contrast to the case of a free surface of a bulk ferrogel, a film or membrane can buckle as a whole bringing bending elasticity into the picture Since the latter has a different wave vector dependence than ordinary elasticity, one can expect that in this case the critical wave vector at the onset of instability does depend on the elastic properties of the gel. Besides isotropic ferrogels there are uniaxial ones, where the latter are obtained by performing the crosslinking process in the presence of a magnetic field [5, 6] In some cases their elastic anisotropy can be very small (if present at all) [5] and can safely be neglected. There are various reversible and irreversible dynamic crosscouplings between flow, elasticity and magnetization, of which we only keep those that are presumably the relevant ones for the present problem
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