  Soft-core attractive potentials can give rise to a phase diagram with three
fluid phases at different densities (gas, low-density liquid and high-density
liquid), separated by first order phase transition lines ending in critical
points. Experiments show a phase diagram with these features for phosphorous
and triphenyl-phosphite. Liquid-liquid phase transition could be relevant for
water, silica, liquid metals, colloids and protein solutions, among others.
Here we compare two potentials with short-range soft-core repulsion and narrow
attraction. One of them is a squared potential that is known to have
liquid-liquid phase transition, ending in a critical point, and no anomaly in
density. The normal, monotonic, behavior of density for isobaric cooling is
surprising if compared with molecular liquids, such as water, where a
hypothetical critical point is proposed as rationale for the anomalous behavior
of density. The second potential is a continuous version of the first. We show
that the phase diagram associated to this new potential has, not only the
liquid-liquid phase transition, but also the density anomaly. Our result,
therefore, shows that the behavior in density is strongly dependent on the
derivative of the potential.
