Separation of Wigner structures for 2D equimolar binary mixtures of Coulomb particles

We study the lowest energy configurations of an equimolar binary mixture of classical pointlike particles with charges $Q_1$ and $Q_2$, such that $q=Q_2/Q_1\in [0,1]$. The particles interact pairwisely via 3D Coulomb potential and are confined to a 2D plane with a homogeneous neutralizing background charge density. In a recent paper by M. Antlanger and G. Kahl [Cond. Mat. Phys. {\bf 16}, 43501 (2013)], using numerical computations based on evolutionary algorithm, six fully mixed structures were identified for $0\le q\lesssim 0.59$, while the separation of $Q_1$ and $Q_2$ pure hexagonal phases minimizes the energy for $0.59\lesssim q<1$. Here, we introduce a novel structure which consists in the separation of two phases, the pure hexagonal one formed by a fraction of particles with the larger charge $Q_1$ and the other fixed one containing different numbers of $Q_1$ and $Q_2$ charges. Using an analytic method based on an expansion of the interaction energy in Misra functions we show that this novel structure provides the lowest energy in two intervals of $q$ values, $0<q\lesssim 0.04707$ and $0.58895\lesssim q\lesssim 0.61367$. This fact might inspire numerical methods, for both Coulomb and Yukawa interactions, to test more general separations which go beyond the separation of two pure phases.