  We consider a simple model consisting of particles with four bonding sites
("patches"), two of type A and two of type B, on the square lattice, and
investigate its global phase behavior by simulations and theory. We set the
interaction between B patches to zero and calculate the phase diagram as the
ratio between the AB and the AA interactions, {\epsilon}^\star_AB, varies. In
line with previous work, on three-dimensional off-lattice models, we show that
the liquid-vapor phase diagram exhibits a re-entrant or "pinched" shape for the
same range of \epsilon^\star_AB, suggesting that the ratio of the energy scales
- and the corresponding empty fluid regime - is independent of the
dimensionality of the system and of the lattice structure. In addition, the
model exhibits an order-disorder transition that is ferromagnetic in the
re-entrant regime. The use of low-dimensional lattice models allows the
simulation of sufficiently large systems to establish the nature of the
liquid-vapor critical points and to describe the structure of the liquid phase
in the empty fluid regime, where the size of the "voids" increases as the
temperature decreases. We have found that the liquid-vapor critical point is in
the 2D Ising universality class, with a scaling region that decreases rapidly
as the temperature decreases. The results of simulations and theoretical
analysis suggest that the line of order-disorder transitions intersects the
condensation line at a multicritical point at zero temperature and density, for
patchy particle models with a re-entrant, empty fluid, regime.
