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We examine the influence of free non-adsorbing polymer on the phase behavior of rodlike colloids. We model the rodlike particles by spherocylinders, as we have essentially complete information on the phase diagram of such objects, for arbitrary aspect ratio (see chapter 5). In the simulations, we model the polymer-induced attraction by a suitable generalization of the square-well model for spherical particles. We find evidence for demixing transitions in the isotropic, nematic and solid phases. We compare our simulation results with predictions based on the perturbation theory of Lekkerkerker et al. [113]. A crucial input in this theory is the so-called free-volume fraction of the hard spherocylinder reference system. In the work of Lekkerkerker and Stroobants, this quantity is estimated using scaled-particle-theory. Here, we determine the free-volume-fraction using numerical simulations.