Two-Dimensional Nucleation Models

It has also been shown quite definitively that, within the precision of the kinetic data that are available, no decision can be made regarding whether a two-or three-dimensional nucleation process is operative [138]. This conclusion is true for virtually all polymers that have been studied. This is admittedly a rather frustrating situation, since nucleation plays such an important role in crystallization of polymers. In analyzing the kinetic data we shall for convenience utilize the Gibbs two-dimensional-nucleation model. The same general conclusions are reached if instead three-dimensional nucleation is assumed. In what follows, no assvunptions are made with respect to the chain structure within the nucleus. 

In dissolution, the sequence of steps is reverse to that in growth. Therefore, as in growth, we have the following two types of models two-dimensional nucleation models, and surface diffusion models. 

Models used to describe the growth of crystals by layers call for a two-step process (/) formation of a two-dimensional nucleus on the surface and (2) spreading of the solute from the two-dimensional nucleus across the surface. The relative rates at which these two steps occur give rise to the mononuclear two-dimensional nucleation theory and the polynuclear two-dimensional nucleation theory. In the mononuclear two-dimensional nucleation theory, the surface nucleation step occurs at a finite rate, whereas the spreading across the surface is assumed to occur at an infinite rate. The reverse is tme for the polynuclear two-dimensional nucleation theory. Erom the mononuclear two-dimensional nucleation theory, growth is related to supersaturation by the equation. 

A drastic departure from nucleation theory was made by Sadler [44] who proposed that the crystal surface was thermodynamically rough and a barrier term arises from the possible paths a polymer may take before crystallizing in a favourable configuration. His simulation and models have shown that this would give results consistent with experiments. The two-dimensional row model is not far removed from Point s initial nucleation barrier, and is practically identical to a model investigated by Dupire [35]. Further comparison between the two theories would be beneficial. 

Typical surfaces observed in Ising model simulations are illustrated in Fig. 2. The size and extent of adatom and vacancy clusters increases with the temperature. Above a transition temperature (T. 62 for the surface illustrated), the clusters percolate. That is, some of the clusters link up to produce a connected network over the entire surface. Above Tj, crystal growth can proceed without two-dimensional nucleation, since large clusters are an inherent part of the interface structure. Finite growth rates are expected at arbitrarily small values of the supersaturation. 

Figure 3.10. Schematic to show the layer-by-layer growth model due to two-dimensional nucleation. This figure assumes the mode of nucleation to be the mononuclear model. Other models, such as the poly-nuclear or birth and spread models, as explained in the text, may also be considered.

In this theory, since the growth rate of a face is controlled by the rate of two-dimensional nucleation, we should expect the presence of a critical driving force, only above which growth can take place. Below this value, there will be no growth. As possible modes of two-dimensional nucleation, three different models may be considered. 

Figure 3.15. Areas where rough and smooth interfaces are expected. The growth rate versus the driving force relations expected for the three models of growth are indicated on the growth rate, R (vertical), axis versus the driving force (A/x/kT) diagram. Curve A shows the spiral growth mechanism B represents the two-dimensional nucleation growth mechanism C denotes the adhesive-type mechanism.

Fig. 11.13. Calculated Fe3C>4 —>Fe metal TPR peaks for six reduction models using E = 111 kj/mol (TPR on dry H2/ Ar) 0.2 K/min (a) three-dimensional nucleation according to Avrami-Erofeev, (b) two-dimensional nucleation according to Avrami-Erofeev, (c) two-dimensional phase boundary, (d) three-dimensional phase boundary, (e) unimolecular decay, (f) three-dimensional...

Several significant developments in polymer crystallization theory have been made more recently, e.g., the two-dimensional nucleation and sliding diffusion theory of Hikosaka,231 the two-stage crystallization model by Strobl,232 and a number of simulation studies have been carried out and are in progress. Past and future experiments on model compounds will undoubtedly benefit these developments. 

The introductory section of this chapter cites texts showing full development of the crystal growth models described above. Those models describing continuous growth, two-dimensional nucleation, and screw dislocation (BCF and variants), with or without diffusional limitation, predict values of kinetic order (exponent r) between 1 and 2 in... 

Barradas and Bosco [34] presented three novel models of two-dimensional nucleation. They involve the coupling of nucleation with diffusion in the electrolyte, with and without additional metal dissolution from the electrode surface. 

Burton-Cabrera-Frank (BCF) Model. The models discussed in the previous section all require two-dimensional nucleation events for a new layer to start. These models fail to account for observed crystal growth rates at low supersaturations and are unsatisfying in the sense that they make crystal growth a noncontinuous process with the formation of a critical size two-dimensional nucleus the rate-determining step. A basis for a model in which the steps are self-perpetrating was put forward by Frank (1949). Frank s idea was that dislocations in the crystal are the source of new steps and that a type of dislocation known as a screw dislocation could... 

The Cabrera-Vermilyea model also implies the existence of a critical supersaturation below which no growth would take place in the presence of impurities. Mathematically, this is given by the condition < 2pc, indicating that the density of impurities has reached the point where step movement by two-dimensional nucleation is prohibited. The size of the critical two-dimensional nucleus is relatable to supersaturation through the expression (Cabrera and Vermilyea 1958 Black et al. 1986)... 

Several growth models based on crystal surface (two-dimensional) nucleation, followed by the spread of the monolayers have been developed in recent years (O Hara and Reid, 1973 van der Eerden, Bennema and Cherepanova, 1978). The term birth and spread (B + S) model will be used here, but other names such as nuclei on nuclei (NON) and polynuclear growth may also be seen in the literature to describe virtually the same behaviour. As depicted in Figure 6.13, growth develops from surface nucleation that can occur at the edges, corners and on the faces of a crystal. Further surface nuclei can develop on the monolayer nuclei as they spread across the crystal face. 

The two-dimensional Ising model permits the evaluation of the nucleation rate, beyond an unpredictable proportionality factor, expressed in the unit of numbers of critical nuclei formed per Monte Carlo step and lattice site. Figure 10.12 compares both theoretical and experimental results for crystallization of lysozyme. The values of span over two orders of magnitude from 9.1 x 10 /cycle site at low roughness (r = 1.2) to 5.2 X 10" /cycle site at r = 1.6. As a reference. Sear obtained a nucleation rate on an impurity of 6 spins on the order of 10 /cycle site, compared to a homogeneous nucleation rate of 10 /cycle site (Sear 2006). 

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