Mullins-Sekerka stability

Favier used the USMP-1 opportunity to explore the interfacial breakdown in Bi-doped Sn, which, like most metals, solidifies as a plane front with little kinetic undercooling. His U.S. co-investigator, Abbaschian, investigated interfacial stability on the other side of the phase diagram i.e., Sn-doped Bi, which solidifies with a faceted interface. The purpose was to test the extension of the Mullins-Sekerka stability criterion to include the effects of anisotropy, which acts to stabilize the interface against breakdown into cellular and dendritic growth. " ... 

W.W. Mullins and R.F. Sekerka. Stability of a planar interface during solidification of a dilute binary alloy. J. Appl. Phys., 35(2) 444-451, 1964. 

Mullins and Sekerka (17) were the first to construct continuum descriptions like the Solutal Model introduced above and to analyze the stability of a planar interface to small amplitude perturbations of the form... 

Normally, it is not possible to obtain analytical solutions for this transport problem and so we cannot a priori calculate the reaction path. Kirkaldy [J. S. Kirkaldy, D. J. Young (1985)] did pioneering work on metal systems, based on investigations by C. Wagner and the later work of Mullins and Sekerka. They used the diffusion path concept to formulate a number of stability rules. These rules can explain the facts and are predictive within certain limits if applied properly. One of Kirkaldy s results is this. The moving interface in a ternary system is morphologically stable if... 

Mullins and Sekerka (88, 89) analyzed the stability of a planar solidification interface to small disturbances by a rigorous solution of the equations for species and heat transport in melt and crystal and the constraint of equilibrium thermodynamics at the interface. For two-dimensional solidification samples in a constant-temperature gradient, the results predict the onset of a sinusoidal interfacial instability with a wavelength (X) corresponding to the disturbance that is just marginally stable as either G is decreased... 

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... 

Mullins, W.W. Sekerka, R.F. Morphological stability of a particle growing by diffusion or heat flow. J. Appl. Phys. 1963, 34, 323-329. 

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. 

Mullins, W. W. Sekerka, R. F. 1964. Stability of a Planar Interface During Solidification of a Dilute Binary Alloy, J. Appl. Phys. 35, 444-451. 

Later, Mullins and Sekerka (1964) developed a more rigorous theory based on a stability analysis that included the liquid-solid interfacial energy, which can provide a stabilizing effect on the interface. However, the difference between the two theories is so small that, for the most part, the more conservative CS criteria can be used for experiment design. 

A quite different approach to the problem of interface breakdown i lliat of Mullins and Sekerka, who considered the stability of the mict race in the presence of shape fluctuations. This treatment has some advantages, particularly since it predicts some of the parameters, e.g. cell size, which were not treated in the original an ysts i>r constitutional supercooling. The treatment is mathematically complex and details can be obtained from the original paper. 

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