Mullins and Sekerka showed for fixed temperature gradient that the planar interface is linearly stable for all pulling speeds V above some critical value, the absolute stability limit. Near this limit, where solidification rates are rapid, the assumption of local equilibrium at the interface may be violated. We incorporate non-equilibrium effects into a linear stability analysis of the planar front by allowing the segregation coefficient and interface temperature to depend on V in a thermodynamically-consistent way. The absolute stability limit of the cellular mode is modified. A new oscillatory state is formed which, in the absence of latent heat, has a critical wavenumber of zero; by itself this instability would lead to the formation of solute bands in the solid. This mode has its own absolute-stability limit determined by solute trapping and kinetics. Under certain conditions there exists a window of stability above the steady absolute-stability boundary and below the oscillatory-stability; here the planar, segregation-free state is restabilized.
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