Theoretical study of the field-induced pattern formation in magnetic liquids.

When a thin layer of magnetic fluid confined with an immiscible nonmagnetic liquid is subjected to a perpendicular field, the formation of hexagonal and labyrinthine patterns is observed experimentally. To develop a coherent theoretical description of this phenomenon, the free energy functionals of both types of magnetic structures are derived. Both energy functionals have the same form, which explains that the theoretical results found in this paper for hexagonal and labyrinthlike striped patterns are analogous. The size of the patterns is determined by minimizing the free energy. The influence of the method for computing the magnetic energy on the theoretical results is studied. An accurate computation of the magnetic energy proves important in predicting the experimental pattern size as a function of external field and of layer height. How the results change, when a constant magnetization is assumed during the pattern formation is also investigated. The transition between hexagonal and striped structures is studied by a comparison of their free energies. The ratio of the magnetic to the nonmagnetic liquid is found to be an important factor for the relative stability of the patterns. In agreement with experiments, striped structures are observed at large phase ratios, whereas at small phase ratios hexagonal patterns predominate.