  The mass flux of a low-density granular binary mixture obtained previously by
solving the Boltzmann equation by means of the Chapman-Enskog method is
considered further. As in the elastic case, the associated transport
coefficients $D$, $D_p$ and $D'$ are given in terms of the solutions of a set
of coupled linear integral equations which are approximately solved by
considering the first and second Sonine approximations. The diffusion
coefficients are explicitly obtained as functions of the coefficients of
restitution and the parameters of the mixture (masses, diameters and
concentration) and their expressions hold for an arbitrary number of
dimensions. In order to check the accuracy of the second Sonine correction for
highly inelastic collisions, the Boltzmann equation is also numerically solved
by means of the direct simulation Monte Carlo (DSMC) method to determine the
mutual diffusion coefficient $D$ in some special situations (self-diffusion
problem and tracer limit). The comparison with DSMC results reveals that the
second Sonine approximation to $D$ improves the predictions made from the first
Sonine approximation. We also study the granular segregation driven by a
uni-directional thermal gradient. The segregation criterion is obtained from
the so-called thermal diffusion factor $\Lambda$, which measures the amount of
segregation parallel to the temperature gradient. The factor $\Lambda$ is
determined here by considering the second-order Sonine forms of the diffusion
coefficients and its dependence on the coefficients of restitution is widely
analyzed across the parameter space of the system. The results obtained in this
paper extend previous works carried out in the tracer limit (vanishing mole
fraction of one of the species) by some of the authors of the present paper.
