  We study the theoretical phase behavior of an asymmetric binary mixture of
hard spheres, of which the smaller component is monodisperse and the larger
component is polydisperse. The interactions are modeled in terms of the second
virial coefficient and can be additive hard sphere (HS) or non-additive hard
sphere (NAHS) interactions. The polydisperse component is subdivided into two
sub-components and has an average size ten or three times the size of the
monodisperse component. We give the set of equations that defines the phase
diagram for mixtures with more than two components in a solvent. We calculate
the theoretical liquid-liquid phase separation boundary for two phase
separation (the binodal) and three phase separation, the plait point, and the
spinodal. We vary the distribution of the polydisperse component in skewness
and polydispersity, next to that we vary the non-additivity between the
sub-components as well as between the main components. We compare the phase
behavior of the polydisperse mixtures with binary monodisperse mixtures for the
same average size and binary monodisperse mixtures for the same effective
interaction. We find that when the compatibility between the polydisperse
sub-components decreases, three-phase separation becomes possible. The shape
and position of the phase boundary is dependent on the non-additivity between
the subcomponents as well as their size distribution. We conclude that it is
the phase enriched in the polydisperse component that demixes into an
additional phase when the incompatibility between the sub-components increases.
