  The thermodynamic instabilities of a binary mixture of sticky hard spheres
(SHS) in the modified Mean Spherical Approximation (mMSA) and the Percus-Yevick
(PY) approximation are investigated using an approach devised by X. S. Chen and
F. Forstmann [J. Chem. Phys. 97, 3696 (1992)]. This scheme hinges on a
diagonalization of the matrix of second functional derivatives of the grand
canonical potential with respect to the particle density fluctuations. The
zeroes of the smallest eigenvalue and the direction of the relative eigenvector
characterize the instability uniquely. We explicitly compute three different
classes of examples. For a symmetrical binary mixture, analytical calculations,
both for mMSA and for PY, predict that when the strength of adhesiveness
between like particles is smaller than the one between unlike particles, only a
pure condensation spinodal exists; in the opposite regime, a pure demixing
spinodal appears at high densities. We then compare the mMSA and PY results for
a mixture where like particles interact as hard spheres (HS) and unlike
particles as SHS, and for a mixture of HS in a SHS fluid. In these cases, even
though the mMSA and PY spinodals are quantitatively and qualitatively very
different from each other, we prove that they have the same kind of
instabilities. Finally, we study the mMSA solution for five different mixtures
obtained by setting the stickiness parameters equal to five different functions
of the hard sphere diameters. We find that four of the five mixtures exhibit
very different type of instabilities. Our results are expected to provide a
further step toward a more thoughtful application of SHS models to colloidal
fluids.
