Two-dimensional nucleation

After complete formation of each successive monolayer of atoms, the next layer should start to form. This requires two-dimensional nucleation by the union of several adatoms in a position 1. Like three-dimensional nucleation, two-dimensional nucleation requires some excess energy (i.e., elevated electrode polarization). Introducing the concept of excess linear energy p of the one-dimensional face (of length L) of the nucleus, we can derive an expression for the work of formation of such a nucleus (analogous to that used in Section 14.2.2). When the step of two-dimensional nucleation is rate determining, the polarization equation becomes, instead of (14.39),... 

Let us discuss main ideas of the above mentioned methods on an example of thin film growth kinetics for structureless gas flux under the conditions when diffusion processes are faster than those of adsorption-desorption (see. (8.3.7)). On the initial stage of film formation a role of diffusion is restricted to the promotion of two-dimensional nucleation (two-dimensional vapor of adatoms is assumed to be supersaturated). The study of growth kinetics at all stages of film formation can be done on the base of (8.3.7). On the other hand, if one is interested in the film growth description on large time scale (when nuclei are large and can be treated thermodynamically) methods of linear thermodynamics of irreversible processes can... 

Those exponents which we have discussed expUcitly are identified by equation number in Table 4.3. Other tabulated results are readily rationalized from these. For example, according to Eq. (4.24) for disk (two-dimensional) growth on contact from simultaneous nucleations, the Avrami exponent is 2. If the dimensionality of the growth is increased to spherical (three dimensional), the exponent becomes 3. If, on top of this, the mechanism is controlled by diffusion, the... 

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. 

Equation 16—18 can be simplified considerably by recognizing that in many systems the quantity s is much less than 1. In that case, ln(l -H 5) is approximately s. Making this substitution, the growth rate from the mononuclear two-dimensional nucleation theory becomes... 

At higher temperatures, other degrees of freedom than the radius R must also be considered in the fluctuation. However, this becomes critical only near the critical point where the system goes through a phase transition of second order. The nucleation arrangement described here is for heterogeneous or two-dimensional nucleation on a flat surface. In the bulk, there is also the formation of a three-dimensional nucleation, but its rate is smaller ... 

As already stated, we shall not explain the details but refer the reader to the literature for further developments. However, we would stress that there are very large intrinsic differences between one- and two-dimensional nucleation, and these are likely to be important for highly mobile phases such as the hexagonal phase in polyethylene. 

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. 

Fig. 6. Two reaction models which result in obedience to the power law [eqn. (2), n = 2 ] at low a, or the Avrami—Erofe ev equation [eqn. (6), n = 2 ] over a more extensive range of a. In (a), there is growth of semi-circular nuclei in a thin plate of reactant in (b), there is cylindrical growth of linear internal nuclei. In both examples, rapid nucleation (0 = 0) is followed by two-dimensional growth (X = 2).

A difiiculty with this mechanism is the small nucleation rate predicted (1). Surfaces of a crystal with low vapor pressure have very few clusters and two-dimensional nucleation is almost impossible. Indeed, dislocation-free crystals can often remain in a metastable equilibrium with a supersaturated vapor for long periods of time. Nucleation can be induced by resorting to a vapor with a very large supersaturation, but this often has undesirable side effects. Instabilities in the interface shape result in a degradation of the quality and uniformity of crystalline material. 

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. 

Obretenov W, Bostanov V, Popov V (1982) Stochastic character of two-dimensional nucleation in the case of electrocrystallization of silver. J Electroanal Chem 132 273-276... 

Mathematical expressions for the N G model can be derived from the classical theory for the nucleation and growth of two-dimensional hlms [Schmickler, 1996]. Two regimes are distinguished ... 

Johans et al. derived a model for diffusion-controlled electrodeposition at liquid-liquid interface taking into account the development of diffusion fields in both phases [91]. The current transients exhibited rising portions followed by planar diffusion-controlled decay. These features are very similar to those commonly observed in three-dimensional nucleation of metals onto solid electrodes [173-175]. The authors reduced aqueous ammonium tetrachloropalladate by butylferrocene in DCE. The experimental transients were in good agreement with the theoretical ones. The nucleation rate was considered to depend exponentially on the applied potential and a one-electron step was found to be rate determining. The results were taken to confirm the absence of preferential nucleation sites at the liquid-liquid interface. Other nucleation work at the liquid-liquid interface has described the formation of two-dimensional metallic films with rather interesting fractal shapes [176]. 

The calculation for the important case of two-dimensional nuclei growing only in the plane of the substrate will be based on the assumption that these are circular and that the electrode reaction occurs only at their edges, i.e. on the surface, 2nrhy where r is the nucleus radius and h is its height (i.e. the crystallographic diameter of the metal atom). The same procedure as that employed for a three-dimensional nucleus yields the following relationship for instantaneous nucleation ... 

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