Burton-Cabrera-Frank theory

Burton-Cabrera-Frank theory, explanation of growth rate dispersion, 4-5... 

A detailed theory of the spiral growth of crystals has been developed by Burton, Cabrera, Frank, Mott and Levine [4.33-4.41] who have considered the case of spiral growth from a supersaturated vapor phase. Later the theory was adapted to electrocrystahization by Vermilyea [4.44] and Fleischmann and Thirsk [4.48] and developed further and verified experimentally by Budevski, Bostanov, Staikov, Nanev et al. [4.28, 4.69-4.75] (see also the pioneering work of Kaischew, Budevski and MaUnovski [4.76]). 

The classical crystal growth theory goes back to Burton, Cabrera and Frank (BCF) (1951). The BCF theory presents a physical picture of the interface (Fig. 6.9c) where at kinks on a surface step - at the outcrop of a screw dislocation-adsorbed crystal constituents are sequentially incorporated into the growing lattice. 

The screw-dislocation theory (sometimes referred to as the BCF theory because of its development by Burton, Cabrera, and Frank) is based on a mechanism of continuous movement in a spiral or screw of a step or ledge on the crystal surface. The theory shows that the dependence of growth rate on supersaturation can vary from a parabolic relationship at low supersaturations to a linear relationship at high supersaturations. In the BCF theory, growth rate is given by ... 

Burton, Cabrera, and Frank [56] and Bennema and Gilmer [57] have developed a theory to predict the crystal growth rate for screw dislocations. The growth rate will depend on the shape of the growth spiral. For an Archimedian spiral, shown in Figure 6.14 [58], the distance between the steps of the spiral yo is... 

The first theoretical treatment of the metal deposition mechanism via surface diffusion was given by Lorenz [2.24], who adapted the theory of Burton, Cabrera and Frank [2.1] to the case of electrolytic metal deposition. Later the problem was treated by Vermilyea [2.29], Despic and Bockris [2.25], Fleischmann and Thirsk [2.28], and Damjanovic and Bockris [2.26], who brought the theory of surface diffusion to its present state. 

The results of the experiments described can be taken as a very good quantitative verification of the theory of Burton, Cabrera, and Frank. 

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... 

Concluding we should point out that Burton, Cabrera and Frank [4.35, 4.53] (see also Lorenz [4.54, 4.55]) have solved also the problem of stationary surface diffusion towards steps with circular symmetry. The theory of the more complex case of combined surface diffusion and bulk diffusion limitations can be found in the works of Fleischmann and Thirsk [4.48], Damjanovich and Bockris [4.50] and Gilmer et al. [4.56]. Note, however, that all these theoretical considerations do not accoimt for difficulties coimected with the so-called Ehrlich-Schwoebel barrier [4.57-4.62]. Elastic [4.63, 4.64] and entropic [4.65] interactions between growing steps are also neglected. 

W.K Burton, N.Cabrera and F.C.Frank, Phil.Trans.Roy.Soc., A243( 1951)299. Ja.I.Frenkel, Kinetic Theory of Liquids, Dover, New York, 1955. 

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