Towards efficient methods for the study of pattern formation in ferrofluid films

Hexagonal and labyrinthine patterns appear in thin ferrofluid films after application of a magnetic field perpendicular to the film. The pattern size and the stability of the hexagonal and labyrinthine structures can be predicted by free energy approaches. Several approximations are used in the literature to accelerate the calculation of the magnetic energy. They are usually based on the use of a uniform, average or constant magnetization. In the uniform approximation the magnetization at all points in the pattern is assumed to be equal to its value at the center of the stripes or cylinders in the labyrinthine or hexagonal patterns. Recent papers indicate that this approximation gives qualitatively wrong results. This is corroborated here by a comparison with accurate results. When a volume- averaged magnetization is used during the calculation of the demagnetization field, from which the magnetic energy is evaluated, the theoretical results are only slightly modified with respect to the accurate results. Thus, we can propose a new method which gives results in good agreement with the accurate values and accelerates the calculations by a factor of 1000. The influence of the approximations is explained by a study of the evolution of the demagnetization field in the patterns. This study indicates that the volume-averaged approximation might only be reliable for patterns with a homogeneously distributed magnetic fluid. Another approximation of a constant magnetization, which is widely used in the literature, assumes that the magnetization does not change during the pattern formation in contrast to the uniform and average approximations. A different way of computing the constant magnetization than that usually employed markedly improves the agreement with the accurate results. This is explained by the derivation of a direct relationship between the approximations of a constant and an average magnetization.