Undercooling

As stated previously, a melt can remain in equilibrium with its parent solid indefinitely, but melting and solidification are not equilibrium processes. To cause the sample to melt further, additional heat must be provided to supply the enthalpy needed to break the solid bonds in order to form the melt. To drive additional solidification, it is necessary to increase the difference between the free energy of the liquid and the solid, which is usually accomplished by lowering the temperature of the melt below the equilibrium melting temperature, a process known as undercooling. Define AT = Tm — T. Since AS AH/TM in the vicinity of Tm, 

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An important approach to the study of nucleation of solids is the investigation of small droplets of large molecular clusters. Years ago, Turnbull showed that by studying small droplets one could eliminate impurities in all except a few droplets and study homogeneous nucleation at significant undercoolings [13]. 

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. 

A larger number of smaller spherulites are produced at larger undercoolings, a situation suggesting nucleation control. Various details of the Maltese cross pattern, such as the presence or absence of banding, may also depend on the temperature of crystallization. 

Entrapolated to represent metastahle equihhrium with undercooled hquid. 

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. 

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. 

Fig. 6.7. How pearlite grows from undercooled y during the eutectoid reaction. The transformation is limited by diffusion of carbon in the y, and driving force must be shared between all the diffusionol energy barriers. Note that AH is in units of J kgn2 is the number of carbon atoms that diffuse from or to Fe3C when 1 kg of y is transformed. (AH/njKfT - 7]/TJ is therefore the free work done when a single carbon atom goes from or to Fe,C.

This sort of nucleation - where the only atoms involved are those of the material itself - is called homogeneous nueleation. It cannot be the way materials usually solidify because (usually) an undercooling of 1°C or less is all that is needed. Flomogeneous nucleation has been observed in ultraclean laboratory samples. But it is the exception, not the rule. 

It is easy to estimate the undercooling that we would need to get heterogeneous nucleation with a 10° contact angle. From eqns (7.11) and (7.3) we have... 

And it is nice to see that this result is entirely consistent with the small undercoolings that we usually see in practice. 

In order to get the iron to transform displaeively we proceed as follows. We start with f.c.c. iron at 914°C which we then cool to room temperature at a rate of about 10 °C s . As Fig. 8.6 shows, we will miss the nose of the 1% curve, and we would expect to end up with f.c.c. iron at room temperature. F.c.c. iron at room temperature would be undercooled by nearly 900°C, and there would be a huge driving force for the f.c.c. b.c.c. transformation. Even so, the TTT diagram tells us that we might expect f.c.c. iron to survive for years at room temperature before the diffusive transformation could get under way. 

The crystal structure of ice is hexagonal, with lattice constants of a = 0.452 nm and c = 0.736 nm. The inorganic compound silver iodide also has a hexagonal structure, with lattice constants (a = 0.458 nm, c = 0.749 nm) that are almost identical to those of ice. So if you put a crystal of silver iodide into supercooled water, it is almost as good as putting in a crystal of ice more ice can grow on it easily, at a low undercooling (Fig. 9.2). 

The final note is that pearlite and bainite only form from undercooled y. They never form from martensite. The TTT diagram eannot therefore be used to tell us anything about the rate of tempering in martensite. 

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).

Duncan, A.G. and Phillips, V.R., 1979. The dependence of heat exchanger fouling on solution undercooling. Journal of separation process technology, 1, 29-35. 

A. Milchev, I. Gutzow. Temperature dependence of the configurational entropy of undercooled melts and the nature of the glass transition. J Macromol Sci B 22 583-615, 1982. 

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. 

To be specific, we consider the two-dimensional growth of a pure substance from its undercooled melt in about its simplest form, where the growth is controlled by the diffusion of the latent heat of freezing. It obeys the diffusion equation and appropriate boundary conditions [95]... 

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,... 

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. 

Our main interest here is concerned with patterns which can grow at constant speed even at low undercoolings A < 1, because if they exist they will dominate the system s behavior. A two-phase structure must then exist behind the growth front, filling the space uniformly on sufficiently... 

The aim is to predict, for given undercooling A and anisotropy e, the type of the two-phase structure and its characteristic length scales and velocity that is, to calculate the functions/ and v in the relation (80). The results will be summarized in the morphology diagram shown in Fig. 6. As it turns out. 

Dendrites can grow at constant speed at arbitrarily small undercooling A, but usually a non-zero value of the anisotropy e is required. The growth pattern evolving from a nucleus acquires a star-shaped envelope surrounding a well-defined backbone. The distances between the corners of the envelope increase with time. For small undercooling we can use the scaling relation for the motion of the corners as for free dendrites [103-106] with tip... 

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 ... 

M. J. Uttormark, J. W. Zanter, J. H. Perepezko. Repeated nucleation in an undercooled aluminum droplet. J Cryst Growth 777 258, 1997. 

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