Undercooling critical

Fig. 14.11. Typical data for recrystallised grain size as a function of prior plastic deformation. Note that, below a critical deformation, there is not enough strain energy to nucleate the new strain-free grains. This is just like the critical undercooling needed to nucleate a solid from its liquid (see Fig. 7.4).

Unter kom, n. undersize (grain), -korrektur, /. undercorrection, undercompensation, -kreide, /. (Geol.) Lower Cretaceous, imter-kriechen, v.i, crawl under, -kriegen, v.t. get the better of. -kritisch, a. below-critical (temperature, speed, etc.), -kiiblent v.t. supercool, undercool. 

The critical condition for the absence of undercooling can then be written... 

The critical growth velocity, v, above which the planar interface in a two-component melt becomes unstable is related to the undercooling, A Tc, by an equation given by Tiller ... 

It follows that the rate of nucleation is zero at the melting temperature. With decreasing temperature the critical size is smaller, which means that smaller nuclei become stable. For that reason more nuclei will be formed at lower temperatures, and thus the nucleation rate is larger at stronger undercooling. 

Nucleation was first observed by Mandell, McTague, and Rahman in simulations of small (128-particle) Lennard-Jones systems. They monitored the nucleation process by following the magnitude of the structure factor, the increase in temperature associated with the release of the heat of fusion, and the apparent absence of diffusion. In subsequent work they examined the effect of an increase in system size, which led to larger undercoolings before crystallization was seea In addition, they determined the structure of the resulting solid to be bcc, as opposed to the thermodynamically stable fee phase. They also introduced a method for locating the critical nucleus at a series of times the velocities of the particles were randomized and it was then determined whether the nucleation had disappeared or whether it still took place. In the former case the intervention time was taken to be precritical, while in the latter it was postcritical. In this way they estimated the critical nucleus to contain 40-70 atoms. 

The free energy 0 of formation of a rectangular nucleus of critical size at an undercooling AT is proportional to a "specific interfacial energy difference Ay (38. ... 

It is now appropriate to ask what is meant by the free energy of a nonequilibrium phase. For example, what is meant by the liquid curve L in Fig. 2.1 when T < T p Strictly speaking, thermodynamic functions can only be defined for equilibrium states, and yet it is natural to consider an undercooled liquid. Turnbull has discussed the fact that liquid metals can be extensively undercooled for extended observable times [2.12] without crystallization. The explanation for this observation lies in the resistance of the undercooled liquids to formation of crystalline nuclei of critical size. The timescale for nucleation of crystals, tn, depends strongly on undercooling as described by Turnbull. Near Tm, tn can be... 

There are two approaches to fabrication of nanostmctured materials based on liquid-to-crystal or crystal-to-liquid transitions. The first approach is crystallization of a melt. This way is not widely spread now. Meantime, it is rather promising. Recently i t was sh own t hat i t i s p ossible to f orm b ulk n anostracured a Hoys with grains of 3-6 nm by rapid solidifying of the melt [18]. This effect is connected with the formation of a lot of clusters in the undercooled melt and their transformation into the critical nuclei. 

Figure 6 The volume free energy of homogeneous nucleation of undercooled water at —40°C as a function of the cluster radius at —40°C (see Equation (2)). According to Equation (3),r = 1.85 nm. The critical nucleus then contains ca. 200 molecules. The value for a was obtained from the nucleation data of Michelmore Franks ...

Apart from the dimensions of critical nuclei under varying conditions, it is also important to consider the kinetics of nucleation, especially relative to the kinetics of crystal growth. The nucleation rate (/) is commonly estimated by means of transition state theory. With the aid of the best available values for transport properties of undercooled water,it has been estimated that in the neighbourhood of —40°C, J increases rapidly with decreasing temperature, by about a factor 20 per degree nucleation is thus a well-defined event that is hardly affected by the rate of cooling. ... 

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