Magnetic dimer at a surface: Influence of gravity and external magnetic fields
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THE EUROPEAN PHYSICAL JOURNAL E
Regular Article
Magnetic dimer at a surface: Influence of gravity and external magnetic fields Ebenezer Kemgang, Herv´e Mohrbach, and Ren´e Messinaa Universit´e de Lorraine, Laboratoire de Physique et Chimie Th´eoriques, LPCT - UMR CNRS 7019, 1 Boulevard Arago, 57070 Metz, France Received 14 January 2020 and Received in final form 2 June 2020 Published online: 9 July 2020 c EDP Sciences / Societ` a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2020 Abstract. The interaction of two dipolar hard spheres near a surface and under the influence of gravity and external perpendicular magnetic fields is investigated theoretically. The full ground-state phase diagram as a function of gravity and magnetic field strengths is established. A dimer (i.e., two touching beads) can only exist when the gravity and magnetic field strengths are simultaneously not too large. Thereby, upon increasing the magnetic field strength, three dimeric states emerge: a lying state (dimer axis parallel to the substrate), an inclined state (intermediate state between the lying and standing ones) and a standing state (dimer axis normal to the substrate). It is found that the orientation angles of the dimer axis and the dipole moment in the newly discovered inclined phase are related by a strikingly simple Snell-Descartes-like law. We argue that our findings can be experimentally verified in colloidal and granular systems.
1 Introduction Magnetic self-assembly is of both fundamental and practical importance. A quite wide spectrum of related key applications can be found ranging from water treatment industry [1,2], through micro-swimming [3,4] to data storage [5,6] and medical technologies [7,8]. The self-assembly behavior of magnetic particles is highly sensitive to confinement (e.g., at a surface [9,10] or within a narrow channel [11, 12]) and exposure to external fields (e.g., gravity and/or magnetic field [13–19]). On a more conceptual side, the bulk crystallization of dipolar hard spheres is by now well understood [20] as well as the influence of the density on the crystal phases [21, 22]. Two-dimensional crystallization can typically be reached i) by imposing a slit-like geometry with an aperture comparable to the size of the particles and/or ii) by strong gravity near a surface. When the constitutive particles do not possess a permanent magnetic moment, then a triangular lattice emerges under the influence of a strong normal magnetic field generating repulsive induced dipoles [13, 23–25]. Without any external imposed magnetic field, dipolar particles (i.e., owing permanent moments) self-assemble into a close-packed triangular lattice too but with a continuous in-plane degeneracy of the dipole orientation [26, 27]. Magnetic clusters, i.e. assembly of a few (say N ) a
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magnetic particles (here with permanent moments), although thoroughly studied and increasingly more during these two-three last decades [4, 10, 15, 28–34], remain a formidable challenge concern
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