Influence of magnetic field on the orientation of anisotropic magnetic particles at liquid interfaces

Abstract

We study theoretically the influence of an external magnetic field on the orientation of an ellipsoidal magnetic particle adsorbed at a liquid interface. Using the finite element program Surface Evolver, we calculate the equilibrium meniscus shape around the ellipsoidal particle and its equilibrium tilt angle with respect to the undeformed interface θ_t when a magnetic field B is applied perpendicular to the interface. We find that as we increase field strength, θ_t increases and at a critical magnetic field B_c1 and tilt angle θ_c1, the particle undergoes a discontinuous transition to the ‘perpendicular’ orientation (θ_t = 90°). Our results agree qualitatively with the simplified theory of Bresme and Faraudo [F. Bresme and J. Faraudo, J. Phys.: Condens. Matter, 2007, 19, 375110] which assumes that the liquid interface is flat, while they agree quantitatively with recent lattice-Boltzmann simulations of Davies et al. [G. Davies et al., Soft Matter, 2014, 10, 6742] which account for the deformation of the liquid meniscus. We also show for the first time that upon reducing the external magnetic field, at a critical magnetic field B_c2 < B_c1, the particle undergoes a second discontinuous transition from the perpendicular orientation to a finite tilt angle θ_c2 < θ_c1. In other words, for micron-sized particles where the thermal energy k_BT is negligible compared to the interfacial energy, the tilt angle vs. magnetic field curve exhibits hysteresis behaviour. Due to the higher degree of accuracy of the Surface Evolver method, we are able to analyse the behaviour of the particles near these orientational transitions accurately and study how the critical quantities B_c1, B_c2, θ_c1 and θ_c2 vary with particle aspect ratio and contact angle.