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Here we extend a phase-field model for epitaxial step-flow growth originally derived by Liu and Metiu to capture the case of different adatom diffusivities at neighboring terraces as well as an arbitrary Ehrlich-Schwoebel (ES) barrier. Our extended model approach bridges the atomic to continuum scale in the sense that it takes into account atomic attachment kinetics in full detail and likewise allows to simulate long range transport processes above the surface efficiently. To verify the model we present a matched asymptotic analysis of the derived model equations, which shows that in a special limit the presented model can be related to the Burton-Cabrera-Frank (BCF) model with different kinds of attachment coefficients at either side of a step edge. We demonstrate the capability of our approach by presenting numerical simulations with an Ehrlich-Schwoebel (ES) barrier, which reproduce the well-known step meandering instability. Thereby we show how mathematical analysis helps to specify and validate a phase-field model and thus contributes to the further development of this modeling approach at the nano- to microscale.