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Single crystal turbine blades, made from Ni-base superalloys, are commonly manufactured by directional solidification. Due to strong segregation of the alloying elements, inhomogeneous microstructures are observed in the as-cast state, which exhibit microsegregation and interdendritic gamma-prime. Both are removed by a solution heat treatment. In order to simulate the microstructural evolution during solidification and solution heat teatment of a five component model superalloy (Ni-Al-Cr-Ta-W), we use the phase-field method coupled to thermodynamic and kinetic databases. A unit cell approach is employed to describe the microstuctural evolution, in a transverse section through a directionally growing array of dendrites. The size of the unit-cell is chosen according to the primary dendrite spacing, the starting temperature according to the dendrite tip undercooling and the cooling rate according to the product of temperature gradient and solidification front velocity. A multi-component diffusion scheme, including cross-diffusion effects, is applied for solid state diffusion. The simulated as-cast microstructure is used as input for the simulation of the homegenization heat treatment, including dissolution of interdendritic gamma-prime and removal of microsegregation. key experiments were performed by directional solidification and heat treatment. The simulations reproduce the experimentally observed, formation of interdendritic gamma-prime for sufficiently high solidification velocities and gamma-prime dissolution kinetics during the homogenization heat treatment.