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Collapsar disks have been proposed to be rich factories of heavy elements, but the major question of whether their outflows are neutron-rich, and could therefore represent significant sites of the rapid neutron-capture (r-) process, or dominated by iron-group elements remains unresolved. We present the first global models of collapsars that start from a stellar progenitor and self-consistently describe the evolution of the disk, its composition, and its outflows in response to the imploding stellar mantle, using energy-dependent M1 neutrino transport and an alpha-viscosity to approximate turbulent angular-momentum transport. We find that a neutron-rich, neutrino-dominated accretion flow (NDAF) is established only marginally--either for short times or relatively low viscosities--because the disk tends to disintegrate into an advective disk (ADAF) already at relatively high mass-accretion rates, launching powerful outflows but preventing it from developing a hot, dense, and therefore neutron-rich core. Viscous outflows disrupt the star within ~100s with explosion energies close to that of hypernovae. If viscosity is neglected, a stable NDAF with disk mass of about 1Msun is formed but is unable to release neutron-rich ejecta, while it produces a relatively mild explosion powered by a neutrino-driven wind blown off its surface. With ejecta electron fractions close to 0.5, all models presumably produce large amounts of Ni56. Our results suggest that collapsar models based on the alpha-viscosity are inefficient r-process sites and that genuinely magnetohydrodynamic effects may be required to generate neutron-rich outflows. A relatively weak effective viscosity generated by magnetohydrodynamic turbulence would improve the prospects for obtaining neutron-rich ejecta.