Charged lattice gas with a neutralizing background

We study a model that was first introduced to describe the ordering of two different types of positive ions in the metal planes of layered hydroxides ${\mathrm{Ni}}_{1-x}{\mathrm{Al}}_x\left(\mathrm{OH}{)}_2\right({\mathrm{CO}}_3{)}_{x/2\mathrm{}}\cdot y{\mathrm{H}}_2\mathrm{O}.$ The ordering is assumed to occur due to long-range Coulomb interactions, and overall charge neutrality is provided by a negative background representing the hydroxide planes and ${\mathrm{CO}}_3^{2-}$ anions. The previous study was restricted to the ground-state properties. Here we use a Monte Carlo technique to extend the study to finite temperatures. The model predicts that, at some values of the concentration x, the system can exhibit an instability and phase separate. In order to evaluate the precision of these Monte Carlo procedures, we first study a linear chain with finite-range interactions where exact solutions can be obtained using a transfer-matrix method. For a linear chain with infinite-range interactions, we use a devil’s staircase formalism to obtain the dependence of the energy of the equilibrium configurations on x. Finally we study the two-dimensional triangular lattice using the same Monte Carlo techniques. In spite of its simplicity, the model predicts multiple first-order phase transitions. The model can be useful in applications such as modeling of the ordering of intercalated metal ions in positive electrodes of lithium batteries or in graphite.