Magnetic-field-induced nonequilibrium structures in a ferrofluid emulsion

Using optical microscopy, we studied magnetic-field-induced structures in a confined ferrofluid emulsion where the magnetic field is applied quickly as a step function. Columnar, bent-wall-like, and labyrinthine structures in three dimensions are observed, corresponding to disks, “worms,” and branchlike patterns in cross-sectional area normal to the magnetic-field direction. These two-dimensional structures are characterized by both the ratio of worms to total aggregates and the average complexity 〈C〉 of the aggregates in a given image. “Phase” diagrams are obtained to characterize disk (columnar) to worm (bent-wall) structural transitions as a function of the thickness of the cell used to confine the sample along the field direction, the particle volume fraction, and the rate of the magnetic-field application. The distribution of aggregate complexity for a given image is characterized by the skewness and quality factor to describe the symmetry and width of complexity distribution. The results show that increasing either cell thickness L, particle volume fraction Φ, or magnetic field ramping rate R increases the average complexity of the formed patterns as $〈C〉=1.8\mathrm{}{\Phi }^{3.11}L+0.141\mathrm{}{\log }_{10}\mathrm{}\left(R\right)+0.83,$ as well as the symmetry and the range of the complexity. This relation can be understood qualitatively. For fast ramping rate R or increasing Φ (decreasing the interparticle distance and thus increasing the particle interaction), the strong magnetic interaction between particles does not allow particles enough time to explore the lowest-energy state (columnar structures) before being locked into local energy minima (labyrinthine structures). The L dependence of the complexity supports molecular dynamics simulation results: Chains form first and then aggregate to form complex structures; longer chains have a larger range of attraction.