Liquid, undercooled

The value of N o in Equation (10.9) has been estimated to be about 1039 nuclei/m3s for most metals. This leads to the prediction that a very large subcooling is necessary to produce any nuclei in any reasonable time. For copper at subcoolings of AT = 100 °C, Equation (10.9) predicts that N = 1039 exp[—5.58 x 10 18/(13 x KT24 x 1266)] = 1 x 1(T147 nuclei/m3s. At this rate of nucleation it would take 3 x 10138 centuries to form one nucleus in a cubic meter of liquid. Undercoolings of AT as 0.18 Tm have been reported for many liquids. See Table 10.3. 

Seed Crystals. Crystals inside droplets may occasionally stick out of the surface over several nanometers. If such a drop encounters another one by Brownian motion (see Section 13.2.1), the protruding crystal may occasionally pierce the surface of that droplet. If the latter is still fully liquid, the crystal may act as a seed and induce crystallization. This has been observed to occur in emulsions of hexadecane in water, where part of the drops were solid and part liquid (undercooled). It is a slow process, for instance taking two weeks for completion. It has been calculated that about one in 107 or 108 encounters was effective in such a case. 

The greater the undercooling, the more rapidly the polymer crystallizes. This is due to the increased probability of nucleation the more supercooled the liquid becomes. Although the data in Fig. 4.8 are not extensive enough to show it, this trend does not continue without limit. As the crystallization temperature is lowered still further, the rate passes through a maximum and then drops off as Tg is approached. This eventual decrease in rate is due to decreasing chain mobility which offsets the nucleation effect. 

In metals the situation is quite the opposite. The spherical atoms move easily from liquid to solid and the interface moves quickly in response to very small undercoolings. Latent heat is generated rapidly and the interface is warmed up almost to T, . The solidification of metals therefore tends to be heat-flow controlled rather than interface controlled. 

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).

Here U = T — T )Cp/L is the appropriately rescaled temperature field T measured from the imposed temperature of the undercooled melt far away from the interface. The indices L and 5 refer to the liquid and solid, respectively, and the specific heat Cp and the thermal diffusion constant D are considered to be the same in both phases. L is the latent heat, and n is the normal to the interface. In terms of these parameters,... 

It has been discovered recently that the spectrum of solutions for growth in a channel is much richer than had previously been supposed. Parity-broken solutions were found [110] and studied numerically in detail [94,111]. A similar solution exists also in an unrestricted space which was called doublon for obvious reasons [94]. It consists of two fingers with a liquid channel along the axis of symmetry between them. It has a parabolic envelope with radius pt and in the center a liquid channel of thickness h. The Peclet number, P = vp /2D, depends on A according to the Ivantsov relation (82). The analytical solution of the selection problem for doublons [112] shows that this solution exists for isotropic systems (e = 0) even at arbitrary small undercooling A and obeys the following selection conditions ... 

The rate of growth of polymer-salt complexes can provide fundamentally important information that is difficult to determine otherwise. The rate of crystal growth of (PEO)3 NaSCN from its undercooled liquid was measured and used to determine values for the diffusion coefficients of Na" " and SCN (Lee, Sudarsana and Crist, 1991). Also it was shown that the rate of the salt diffusion is independent of the molecular weight of the polymer for PEO molecular weights above 10. This result is fully consistent with the concept that ion motion is due to local segmental motion of the polymer. 

P and a represent, respectively, the undercooled liquid, the b.c.c. solid solution and the amorphous phase. ( ) are results from enthalpy of crystallisation experiments. Horizontal bars represent amorphous phase (I) interdifliision reaction and (2) by laser-quenching. 

Figure 2-13 Schematic drawing of (a) density as a function of temperature, and (b) entropy as a function of temperature for glasses with different cooling rates and hence different glass transition temperature (Martens et al., 1987). The entropy of the undercooled liquid is estimated assuming constant heat capacity.

At the glass transition temperature, the glass melts or an undercooled liquid freezes (Zallen, 1983 Elliott et al, 1986). 

Vapor-phase deposition of the sputtered or evaporated layer-forming material avoids the undercooling problems associated with liquid phase epitaxy, but it coats everything in the vaporization chamber unselectively. Sputtering is usually done by forming a plasma (ionized gas) in an electrical discharge in the vapor at low pressure. 

For undercooled liquid below normal F. P. grams/100 grams solvent... 

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