Stable crystalline lattices in two-dimensional binary mixtures of dipolar particles

The phase diagram of binary mixtures of particles interacting via a pair potential of parallel dipoles is computed at zero temperature as a function of composition and the ratio of their magnetic susceptibilities. Using lattice sums, a rich variety of different stable crystalline structures is identified including AmBn structures. [A (B) particles correspond to large (small) dipolar moments.] Their elementary cells consist of triangular, square, rectangular or rhombic lattices of the A particles with a basis comprising various structures of A and B particles. For small (dipolar) asymmetry there are intermediate AB2 and A2B crystals besides the pure A and B triangular crystals. These structures are detectable in experiments on granular and colloidal matter. While the freezing transition and the corresponding crystal lattice in one-component systems is wellunderstood by now [1,2], binary mixtures of two different particle species exhibits a much richer possibility of different solid phases For example, while a one-component hard sphere system freezes into the close-packed face-centeredcubic lattice [3], binary hard sphere mixtures exhibit a huge variety of close-packed structures depending on their diameter ratio. These structures include ABn superlattices, where A are the large and B the small spheres, with n = 1, 2, 5, 6, 13. These structures were found in theoretical calculations [4], computer simulations [5,6] and in realspace experiments on sterically stabilized colloidal suspensions [7,8]. Much less is known for soft repulsive interparticle interactions; most recent studies on crystallization include attractions and consider Lennard-Jones mixtures [9, 10] or oppositely charged colloidal particles [11–13]. In this letter we explore the phase diagram of a binary mixture interacting via a soft repulsive pair potential proportional to the inverse cube of the particle separation. Using lattice sums, we obtain the zero-temperature phase diagram as a function of composition and asymmetry, i.e. the ratio of the corresponding prefactors in the particleparticle interaction. Our motivation to do so is threefold: i) First there is an urgent need to understand the effect of softness in general and in particular in two spatial dimensions. The case of hard interactions in two spatial dimensions, namely binary hard disks, has been obtained by Likos and Henley [14] for a large range of diameter ratios. A complex phase behavior is encountered and it is unknown how the phase behavior is affected and controlled by soft interactions. ii) The model of dipolar particles considered in this letter is realized in quite different fields of physics. Dipolar colloidal particles can be realized by imposing a magnetic field [15]. In particular, our model is realized by micron-sized superparamagnetic colloidal particles which are confined to a planar water-air interface and exposed to an external magnetic field parallel to the surface normal [15–19]. The magnetic field induces a magnetic dipole moment on the particles whose magnitude is governed by the magnetic susceptibility. Hence their interaction potential scales like that between two parallel dipoles with the inverse cube of the particle distance. Binary mixtures of colloidal particles with different susceptibilities have been studied for colloidal dynamics [20], fluid clustering [21,22], and the glass transition [23]. A complementary way to obtain dipolar colloidal particles is a fast alternating electric field which generates effective dipole moments in the colloidal particles [24]. This set-up has been applied for two-dimensional binary mixtures in Ref. [25]. In granular matter, the model has been realized by mixing millimeter-