Step free energy

A final issue of importance is the role of energetic repulsions between the steps. From eqn. (6), expanding about the average position of the step, we find the leading correction to the step free energy discussed while treating eqn. (1) is cx, where c = bg/f Then in the EC case, pi)2x/3y2 (implicitly) in eqn. (1) is replaced by 2cx, leading ultimately to... 

Figure 6 The step free energy per unit of projected length along the x-direction as function of 1/ tan( ). Steps with orientations close to ( ) = nil are unstable as can be seen from the cusp in the figure.

The quantity under the square root in (5) becomes negative at small values of P5x, and P5, this happens at temperatures above the roughening temperature of the (110) surface where the solid-on-solid approximation for the step free energy f ( ) is not positive definite and the simple theory considered here breaks down. 

The equilibrium shape of the (110) facet can be found by applying a one-dimensional Wulff construction to the step free energy as function of orientation (Van Beijeren and Nolden, 1987). The result of this construction, for a representative choice of step energies and temperature, is shown in Fig. 7. Steps with orientations close to ( ) = Jt/2 are unstable and would phase separate into combinations of two steps of orientations ( )o and-( )o the shape of the (110) facet resembles that of an almond, with cusps along the [110] direction. 

For the calculation of the step free energy we consider first the partition functions G, and G, of straight segments of steps along the x and y direction. The segments have energies per unit length equal to 6 and 8y. In Fourier space one has ... 

Impurities and the solvent component, which affect step free energy y. 

In this subsection we shall add further aspects to this transition, emphasizing in particular that the step free energy k Ts T) vanishes at 7r (Weeks, 1980 van Beijeren and Nolden, 1987). This fact has particular implications for vicinal (i.C., high index-) surfaces, e.g. Cu(llf), with i odd at low temperatures such a surface can be viewed as a dense regular array of steps relative to a (100) surface. 

The step-free energy k Ts(T) can be conveniently introduced by considering the interfacial tension of interfaces which are tilted through an angle 8 relative to a low-index lattice plane (fig. 65). For small 9 the angular-dependent surface tension takes the form... 

The growth mechanism of a crystal will vary depending on factors that include the surface and step free energies of the crystal faces, the nature and density of defects within the crystal, and most importantly the degree of supersaturation, S, of solution. Figure 11 shows the relationship between S and growth mechanism. In this figure theoretical... 

Simulations on the effect of step free energy on grain growth behaviour have also been made. Figure 15.11 shows the result of a Monte Carlo simulation made by Cho. For the simulation, Cho assumed that the grain network was a set of grains with a Gaussian size distribution (standard deviation of 0.1) located on vertices of a two-dimensional square lattice. Deterministic rate equations, Eq. (15.15) for v/> and Eq. (15.29) for v j, were... 

Figure 15.11. Simulated developments of microstructure with step free energy The initial size distribution was assumed to be Gaussian.

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