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We present a phase-field model for particle capture during directional solidification. Its predictions for critical growth velocities for particles of different sizes are compared with experimental results for capture of silicon carbide (SiC) particles during directional solidification of silicon. The phase-field model allows us to systematically test the influence of different assumptions about attractive and repulsive forces and the capture mechanisms, including the effects of particle shape and of partial engulfment of the particle by the interface. We identify common properties of models that show agreement with experiments, trying to determine the underlying physical effects by abductive inference. We find that predictions vary only slightly between models with different repulsive forces and that the shape of the particle can have a larger effect on the critical growth velocity than the exact nature of the repulsive force or the capture process. We assess to what extent a good description of experimental critical growth velocities implies that the model accurately describes the actual physical processes and propose additional ways to test the validity of models.