Energetics of interacting magnetized domains

Many of the pattern forming features of ferrofluids, lipid monolayers, type-I superconductors, and magnetic bubbles can be understood by treating them as dipolar ~uniformly magnetized or polarized! domains. Here, we investigate the early stages of pattern formation in a system consisting of two quasi-two-dimensional dipolar domains. We calculate the linearized interaction energy for these domains and find that the lowest energy states are those for which each drop has an indentation directed toward the other. These theoretically preferred states compare favorably with results of experiments performed with ferrofluids. Dipolar domain energetics @1# has been used as a model for the pattern formation process in a wide range of physical systems including Langmuir films @2,3#, magnetic bubbles @4#, type-I superconductors @5,6#, and ferrofluids @7#. Using this and other formalisms, a single domain has been exten- sively studied @8-10# and this has led to a good understand- ing of the pattern selection mechanism. A hydrodynamic ap- proach has also proved useful in this context @11# and the resulting patterns are virtually indistinguishable from those obtained using the energetic approach. These studies have focused primarily on the fingering instability for a single domain but have not investigated how the presence of mul- tiple domains might affect the patterns that form. Those that have considered multiple domains @12# have used numerical simulations to focus on the characteristic distance between stripes in a highly disordered array. Here, we seek to find the lowest energy configurations for a system of two domains in the earliest stages of the pattern formation process. These configurations are extremely important because it is from these early stages that the final state patterns emerge. In some sense, the entire structure of the final pattern becomes ''frozen in'' during these early times. To begin, let us describe one particular physical system. Consider a ferrofluid drop of volume V trapped between two closely spaced glass plates separated by a distance h ~a Hele- Shaw cell!. In this geometry, the motion is essentially two dimensional and the boundary of the drop looks like a thin ribbon. We assume throughout this paper that the interface of the fluid is perpendicular to the glass plates and therefore focus our attention on the shape of the cross section. Incom- pressibility of the fluid keeps the cross-sectional area fixed even though the shape can change. When no magnetic field is applied, the shape of the drop is determined solely by surface tension and will take the form of a circle of radius R0. If a magnetic field is applied perpendicular to the glass plates, the ferrofluid becomes magnetized. This produces an outward magnetic force that competes with surface tension and results in the domain evolving into a branched pattern. We assume that the magnetization M is uniform throughout the domain and oriented perpendicular to the Hele-Shaw cell. A point on the interface of the domain can be described by a vector r in the xy plane, and a point z. The vector r gives the location along the length of the ribbon and z gives the height. After integrating over the height of the ribbon, the magnetic energy of an arbitrarily shaped simply connected domain can be written @1# Emag52pM 2 V2M 2 h R ds R ds8 t