Two-dimensional patterns of magnetic particles at air-water or glass-water interfaces induced by an external magnetic field: Theory and simulation of the formation process

We describe theoretically the formation process of two-dimensional patterns of magnetic particles at an interface under the action of an external magnetic field. First, we developed a theoretical model for the forces acting on the particles, which are the magnetic attraction toward the axis of symmetry, magnetic repulsion due to the induced dipoles, electrostatic repulsion due to the surface net charge and electric dipoles of the particles, and lateral capillary forces in the case of a liquid interface. Then, we developed a Brownian dynamics algorithm and simulated the pattern formation. Our simulations reproduced the hexagonally aligned two-dimensional patterns, which were experimentally observed and reported in the complementary experimental work, that is, they reproduced the distance between the particles and the time of pattern formation. Irregular patterns such as square and pentagonal ones that were experimentally observed were also simulated, We calculated that when the distance between particles is larger than 20 mu m (i.e., in case or a weak magnetic field), the magnetic force is sufficient to explain the pattern formation. But, when the distance is smaller than 20 mu m, the electrostatic repulsive force may play an essential role in determining the interparticle distance. Our calculations showed that the lateral capillary forces and the electrostatic dipolar repulsive forces were negligible for the pattern formation process. Finally, our method for simulating the pattern formation may be applied for understanding these processes and predicting the patterns that might be formed at liquid or solid interfaces.