  The exact solution, via transfer matrix, of the simple one dimensional
lattice Coulomb gas (1d LCG) model can reproduce peculiar features of ionic
liquid capacitors, such as overscreening, layering, and camel- and bell-shaped
capacitance curves. Using the same transfer matrix method, we now compute the
bulk properties of the 1d LCG in the constant voltage ensemble. We unveil a
phase diagram with rich structure exhibiting a low density disordered and high
density ordered phases, separated by a first order phase transition at low
temperature; the solid state at full packing can be ordered or not, depending
on the temperature. This phase diagram, which is strikingly similar to its
three dimensional counterpart, also sheds light on the behaviour of the
confined system.
