Supercooling, constitutional

Fig. 7. Constitutional supercooling, (a) impurity concentration profile during solidification (b) actual temperature T and equilibrium freezing temperature T...

At the onset of constitutional supercooling, the melting-point gradient exceeds the temperature gradient. Equating these gradients leads to the criterion for constitutional supercooling ... 

Figure 9.2. Constitutional supercooling in alloy solidification (a) phase diagram (b) solute-enriched layer ahead of the solid/liquid interface (c) condition for a stable interface (d) condition...

Earlier, a special issue of Materials Science and Engineering (Jones and Kurz 1984) to mark the 30th anniversary of the identification of constitutional supercooling includes 21 concise survey papers which constitute an excellent source for assessing the state of knowledge on solidification at that stage. Another source is a textbook (Kurz and Fisher 1984) published the same year. 

Figure 3.19. Schematic illustration of constitutional supercooling (C ). Tj is the liquid temperature gradient, is the growth temperature gradient, and is the actual temperature gradient.

The phase-field simulations reproduce a wide range of microstructural phenomena such as dendrite formation in supercooled fixed-stoichiometry systems [10], dendrite formation and segregation patterns in constitutionally supercooled alloy systems [11], elastic interactions between precipitates [12], and polycrystalline solidification, impingement, and grain growth [6]. 

Connection between Transport Processes and Solid Microstructure. The formation of cellular and dendritic patterns in the microstructure of binary crystals grown by directional solidification results from interactions of the temperature and concentration fields with the shape of the melt-crystal interface. Tiller et al. (21) first described the mechanism for constitutional supercooling or the microscale instability of a planar melt-crystal interface toward the formation of cells and dendrites. They described a simple system with a constant-temperature gradient G (in Kelvins per centimeter) and a melt that moves only to account for the solidification rate Vg. If the bulk composition of solute is c0 and the solidification is at steady state, then the exponential diffusion layer forms in front of the interface. The elevated concentration (assuming k < 1) in this layer corresponds to the melt that solidifies at a lower temperature, which is given by the phase diagram (Figure 5) as... 

Constitutional supercooling resulting from boundary layer formation. 

From equation 7, it may be seen that the tendency toward constitutional supercooling increases as the freezing rate increases, the temperate gradient G decreases, the impurity content w increases, the separation (w i) between Hquidus and soHdus in the phase diagram increases, and the stirring decreases (5 increases). This explains why zone melting is limited to purification of materials with low impurity contents, and why substantial temperature gradients and low zone-travel rates are necessary. 

Several authorshave conceptually related the transition from planar to cellular freezing of ice to the condition of constitutional supercooling Here the theoretical framework of morphological stability theory for a planar interface is combined with simple models of diffusion and free convection to derive the transition salinity in dependence on growth conditions. 

The concept of constitutional supercooling is the basic explanation of cellular freezing of alloys and saline solutions. Tiller and coworkersonce postulated an appropriate quantitative condition... 

The present model approach has combined three equations to predict the onset of cellular growth during freezing of natural waters (i) constitutional supercooling from morphological stability theory, (ii) an exact diffusive solute redistribution and (iii) an intermittent turbulent solutal convection model. The main results are ... 

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