Mullins-Sekerka condition

For wavenumbers oa for which the thermal steady state approximation is valid [11], we obtain for the non-oscillatory marginal state the Mullins-SeKerKa condition with additional terms related to the mass density difference between liquid and solid ... 

The classical linear stability theory for a planar interface was formulated in 1964 by Mullins and Sekerka. The theory predicts, under what growth conditions a binary alloy solidifying unidirectionally at constant velocity may become morphologically unstable. Its basic result is a dispersion relation for those perturbation wave lengths that are able to grow, rendering a planar interface unstable. Two approximations of the theory are of practical relevance for the present work. In the thermal steady state, which is approached at large ratios of thermal to solutal diffusivity, and for concentrations close to the onset of instability the characteristic equation of the problem... 

A more stringent condition was derived by Mullins and Sekerka in their now classical linear stability analysis. It reads... 

A modified version of this criterion was developed by Mullins and Sekerka (1964). The authors attribute the incorporation of impurities essentially to the nonplanar growth of the crystalline layer and analytically describe this by superposing a planar layer with a sinusoidal disturbance. Afterwards, they determine the conditions under which this disturbance is damped and derive a criterion which generally guarantees such conditions during the process. The applicability of this stability criterion was proven for a variety of metallic and organic compounds. 

---