Kinetic undercooling

In spite of this dominance of heat flow, the solidification speed of pure metals still obeys eqn. (6.15), and depends on temperature as shown in Fig. 6.6. But measurements of v(T) are almost impossible for metals. When the undercooling at the interface is big enough to measure easily (T, -T 1°C) then the velocity of the interface is so large (as much as 1 m s 0 that one does not have enough time to measure its temperature. However, as we shall see in a later case study, the kinetics of eqn. (6.15) have allowed the development of a whole new range of glassy metals with new and exciting properties. 

In this section we discuss the basic mechanisms of pattern formation in growth processes under the influence of a diffusion field. For simphcity we consider the sohdification of a pure material from the undercooled melt, where the latent heat L is emitted from the solidification front. Since heat diffusion is a slow and rate-limiting process, we may assume that the interface kinetics is fast enough to achieve local equihbrium at the phase boundary. Strictly speaking, we assume an infinitely fast kinetic coefficient. 

In Eq. (76) we neglect the kinetic effects, that is, the dependence of the interface temperature on the growth velocity v . The approximation holds at sufficiently small undercoolings and velocities. 

In the perspective discussed in the present contribution, bundle formation occurs within the amorphous phase and in undercooled polymer solutions. It does not imply necessarily a phase separation process, which, however, may occur by bundle aggregation, typically at large undercoolings [mode (ii)]. In this case kinetic parameters relating to chain entanglements and to the viscous drag assume a paramount importance. Here again, molecular dynamics simulations can be expected to provide important parameters for theoretical developments in turn these could orient new simulations in a fruitful mutual interaction. 

This relation is consistent with previous results observed experimentally [49]. Although the kinetic theory of Lauritzen and Hoffman predicts the same law as Eq. 16, it predicts a divergence in L at lower undercoolings. The simulations do not show any evidence for such a catastrophe. 

For an alloy droplet, the post-recalescence solidification involves segregated solidification and eutectic solidification. 619 Droplet cooling in the region (1),(2) and (6) can be calculated directly with the above-described heat transfer model. The nucleation temperature (the achievable undercooling) and the solid fraction evolution during recalescence and post-recalescence solidification need to be determined additionally on the basis of the rapid solidification kinetics. 154 156 ... 

With the above-described heat transfer model and rapid solidification kinetic model, along with the related process parameters and thermophysical properties of atomization gases (Tables 2.6 and 2.7) and metals/alloys (Tables 2.8,2.9,2.10 and 2.11), the 2-D distributions of transient droplet temperatures, cooling rates, achievable undercoolings, and solid fractions in the spray can be calculated, once the initial droplet sizes, temperatures, and velocities are established by the modeling of the atomization stage, as discussed in the previous subsection. For the implementation of the heat transfer model and the rapid solidification kinetic model, finite difference methods or finite element methods may be used. To characterize the entire size distribution of droplets, some specific droplet sizes (forexample,.D0 16,Z>05, andZ)0 84) are to be considered in the calculations of the 2-D motion, cooling and solidification histories. 

An additional effect of undercooling on the kinetics and microstructure of spinodal decomposition arises from the temperature dependence of d2fhom/dc2B, which... 

Useful insights into the kinetics of a phase transformation that proceeds by nucle-ation and growth can be obtained by observing the fraction transformed, , under isothermal conditions at a series of different temperatures. This is usually done by undercooling rapidly to a fixed temperature and then observing the resulting isothermal transformation. The kinetics generally follows the typical C-shaped behavior described in Exercise 18.4. If a series of such curves is obtained at different temperatures, the time required to achieve, for example, ( = 0.01, 0.50, and... 

The experiments of Kovacs et al. also demonstrate beautifully that although the extended chain crystal is the thermodynamically most stable form, when the undercooling is sufficient, kinetics favors folded chain lamellae. As we have seen, long chain polymers only crystallize at finite rates at high undercoolings, so only form folded chain structures. 

Nucleation kinetics are experimentally determined from measurements of the nucleation rates, induction times, and metastability zone widths (the supersaturation or undercooling necessary for spontaneous nucleation) as a function of initial supersaturation. The nucleation rate will increase by increasing the supersaturation, while all other variables are constant. However, at constant supersaturation the nucleation rate will increase with increasing solubility. Solubility affects the preexponential factor and the probability of intermolecular collisions. Furthermore, when changes in solvent or solution composition lead to increases in solubility, the interfacial energy decreases as the affinity between crystallizing medium and crystal increases. Consequently, the supersaturation required for spontaneous nucleation decreases with increasing solubility, ° as shown in Fig. 7. 

---