Progressive Freezing

The results have implications for the structures in products. For example, for quiescently frozen products, the data produced here can be used to identify possible separation of solute components as freezing progresses. This implies that the frozen phase will change in composition and properties, dependent on the distance from and temperature of the cold sink. 

When glass is at an elevated temperature, its structure is at equilibrium and is that of the melt so that Ti=T and we get the VFT expression for the viscosity. Close to glass transition, glass freezes progressively, Tf departs from T and viscosity is affected. After each increment of time, the fictive temperature (structure) can be calculated following the Markovsky and Soules (1984) algorithm that suggests the use of partial fictive temperatures T with the Prony series ... 

To derive the concentration profile for progressive freezing, a material balance is employed for solidification of a small fraction dg of melt, as shown in Figure 1. Integration from the beginning of solidification gives (1,4,8) ... 

Fig. 3. Impurity concentration profiles resulting from progressive freezing with different values of distribution coefficient k (from eq. 2).

Component Separation by Progressive Freezing When the distribution coefficient is less than I, the first solid which ciystaUizes contains less solute than the liquid from which it was formed. As the frac tion which is frozen increases, the concentration of the impurity in the remaining liquid is increased and hence the concentration of impurity in the sohd phase increases (for k < 1). The concentration gradient is reversed for k > 1. Consequently, in the absence of diffusion in the solid phase a concentration gradient is estabhshed in the frozen ingot. 

FIG. 22-4 Cl irves for progressive freezing, showing solute concentration C in the solid versus fraction-solidified X (Pfann, Zone Melting, 2d ed., Wileij, New York, 1966, p. 12. )... 

The ultimate distribution represents the maximum separation that can be attained without cropping the ingot. Equation (22-5) is approximate because it does not include the effect of progressive freezing in the last zone length. 

As in progressive freezing, many refinements of these models have been developed. Corrections for partial liquid mixing and a variable distribution coefficient have been summarized in detail (Zief and Wilcox, op. cit., p. 47). 

Falling-film crystalhzation utilizes progressive freezing principles to purify melts and solutions. The technique estabhshed to practice the process is inherently cyclic. Figure 22-15 depicts the basic working concept. First a ciystalline layer is formed by subcoohng a liquid film... 

---