The Theory of Spiral Growth

The Period of Rotation and Step Distance of a Spirai of Growth 

The most important factors for the process of growth in the presence of screw dislocations is the Burgers vector of the screw dislocation and the rotation period Tr of the spiral of growth. In the steady state, when the form of the spiral remains unchanged with time, the current density is given by 

The case of polygonized spirals has been discussed by Cabrera, Kais-chew et and by Chapon and Bonissent. In all these treatments the 

The assumption i = Uoo for / = Ic and u = 0 for / /c is an oversimplification. It is obvious that the time lapse Tc and hence the period of rotation should be in reality larger because the step adjacent to the new one propagates with a speed smaller than Doo. The calculation is complicated because the propagation rates of all the adjacent steps are also dependent on / and hence the propagation rate of the spiral is given by a system of differential equations. A numerical calculation on this system of equations was given by Budevski et who found that irrespective of the form of the spiral, the period of rotation of a polygonized spiral is 

The growth pattern becomes even more complicated if more than two dislocations are interacting and especially if the component of the Burgers vector normal to the face is higher than the monatomic step height. 

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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]). 

It became necessary to understand how crystals grow at the atomic level so as to form a deeper understanding of why crystals can take a variety of forms. This was achieved through the layer growth theory put forward in the 1930s by Volmer, Kossel, and Stranski on the structure and implication of the solid-liquid interface, the spiral growth theory by Frank in 1949, and the theory of morphological... 

He started to work at the Chemical Faculty of Sofia University where he became a professor and the head of the Department of Physical Chemistry, in 1947. Kaishev founded the Institute of Physical Chemistry of the Bulgarian Academy of Sciences in 1958, and helped to establish the Central Laboratory of Electrochemical Power Sources [i]. Kaishev started to collaborate with - Stran-ski in Berlin in 1931 [iii] and became his assistant in Sofia in 1933. They laid the fundamentals of the crystal growth theory. They proposed the first kinetic theory of the two-dimensional nucleation and growth. The spiral type growth during electrocrystallization was first observed by Kaishev on silver [iii]. On the history of the creation of the molecular-kinetic theory of crystal growth see [iv]. 

The concept of dislocations was theoretically introduced in the 1930s by E. Orowan and G. I. Taylor, and it immediately played an essential role in the understanding of the plastic properties of crystalline materials, but it took a further twenty years to understand fully the importance of dislocations in crystal growth. As will be described in Section 3.9, it was only in 1949 that the spiral growth theory, in which the growth of a smooth interface is assumed to proceed in a spiral step manner, with the step serving as a self-perpetuating step source, was put forward [7]. 

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 ... 

The Davies and Jones derivation makes some fundamental assumptions concerning the surface concentrations of the lattice ions and the BCF theory is only applicable to very small supersaturations. Thus, both theories have limitations which affect the interpretation of the results of growth experiments. Nielsen [27] has attempted to examine in detail how the parabolic dependence can be explained in terms of the density of kinks on a growth spiral and the adsorption and integration of lattice ions. One of the factors, a = S — 1, comes from the density of kinks on the spiral [eqns. (4) and (68)] and the other factor is proportional to the net flux per kink of ions from the solution into the lattice. Nielsen found it necessary to assume that the adsorption of equivalent amounts of constituent ions occurred and that the surface adsorption layer is in equilibrium with the solution. Rather than eqn. (145), Nielsen expresses the concentration in the adsorption layer in the form of a simple adsorption isotherm equation... 

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 investigation of the spiral growth mechanism of faces with a low screw dislocation density represents the first quantitative confirmation of Frank s theory. First, the parabolic dependence of the normal growth rate on the supersaturation was... 

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