Burton-Cabrera-Frank model

All of the above models assuming two-dimensional surface nucleation predict crystal growth rates at low supersaturation which are much lower than those observed in practice. A possible solution to this problem was put forward by Frank (1949), who proposed a self-generating step creation process involving a screw dislocation. The model was further formalized by Burton et al. (1951) and became known as the Burton-Cabrera-Frank (BCF) model of crystal growth. Detailed discussion of the BCF model can be found in Ohara and Reid (1973) and Nyvlt et al. (1985). 

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... 

Slow Growth Regime (a < 5). The surface is atomically smooth and growth is controlled by the movement of steps emerging from dislocations. Slow growth kinetics and parabolic dependence on supersaturation results, and growth behavior is as predicted by the Burton-Cabrera-Frank (Burton, W.K., et al. 1951) model... 

Crystal growth by the layer growth mechanism describes the formation of steps (i.e., layers) by two different mechanisms—2-D nucleation and screw dislocation. The model for 2-D nucleation was developed by Volmer[ ° l and Stranski. The screw dislocation model was first described by Burton, Cabrera, and Frank (BCF). The details of the derivations for these models have been summarized in a number of other references. ° ° ... 

Some authors have observed that the growth rate at very small supersatirration is greater than predicted by the nucleation models. This can be explained by the so-called BCF model (Bruton et al. 1951). The authors assume that the presence of spiral dislocations which end somewhere on the crystal surface creates steps, which are thus a continuous soirrce of favorable integration sites. The soirrce of such screw dislocations is a lattice imperfection which prevents an ideally smooth crystal surface. The steps of these spiral dislocations are remote from the centers and considered to be parallel and the same distance apart from each other. The linear displacement rate of a face is controlled by surface diffusion. With the siuface diffusion coefficient the growth rate Vg p according to Burton, Cabrera, and Frank is... 

Burton, Cabrera and Frank (1951) investigated quantitatively the flux of adsorbed molecules firom surface to steps caused by surface difliision, starting from the Ising model of a two-dimensional lattice gas (BCF theory). According to BCF theory, the velocity of movement of the straight step is estimated as... 

Analogous to the formahon of two-dimensional nuclei/hUlocks for crystal growth, in dissoluhon the rate of step movement from a pit of radius r can be obtained from treatments similar to the model of Burton, Cabrera and Frank [82],... 

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