Interfacial Instability during Solidification

Example 6.2 Characteristics of Interfacial Instability DURING Solidification 

Find the wavelength of the fastest growing disturbance and its rate of growth for solidification of water when (a) the rate of crystal formation is vray fast b oo) and (b) b = 10 cm/sec K. Assume that the temperature gradient (- G) in the water at the interface is 10°C/cm and use the following values  

By differentiating this equation with respect to a and setting the derivative equal to zero, we find 

With the above values, is 639/cm, i.e., the wavelength 2jila of the fastest growing perturbation is 98.3 pan. Note that VIDjOl = 1.9 x 1(H, so that the assumption made above is justified. The time factor for growth of this disturbance is 0.069/sec (i.e., the time constant is 14.7 sec). 

Note that for akIbXp to have a magnitude of order unity, a must be about 0.57/cm, or the wavelength 2jt/a must be about 11 cm. As seen in (a), the wavelengths of interest are much shorter than this value, so that the assumption made in deriving Equation 6.E2.3 is valid. Note further that with V= 1.75 x 10 cm/sec, the temperature decrease at the interface is 17.5 C, according to Equation 6.60. 

---

S. Coriell, M. Cordes, W. Boettinger, R. Sekerka. Convective and interfacial instabilities during unidirectional solidification of a binary alloy. J Cryst Growth 49 13, 1980. 

---