Wulff point

The volume of a crystal, thought of as the sum of the pyramids formed on the crystal faces with a common apex in the Wulff point, is related to hi and A,- by... 

Figure 4.2 The white arrows represent the normal distances of the Wulff point, located in the center of the crystal, ftnm the respective faces.

As seen, the influence of the substrate is restricted to the thickness of the crystal only. A higher value of p only makes the crystal flatter, leaving all distances except the distance hj unchanged. With increasing p, the distance of the contact face to the Wulff point decreases, taking negative values zip > oj. 

The contact face j usually lies parallel to one of the faces belonging to the equilibrium form, so that the top face of the crystal very often lies parallel to the substrate surface, as illustrated in Fig. 4.4. A thickness d can be ascribed to the crystal in this case d = hj + hj, where hj is the distance of the top face from the Wulff point and hj is that to the crystal-substrate interface. With eq. (4.10) one obtains for the thickness of a crystal lying with its contact face parallel to the plane j on a substrate ... 

Figure 4.4 Influence of the adhesion energy P on the shape and thickness 4 of a crystal staying in contact with a substrate. The distances of the side faces from the Wulff point remain unaffected by the presence of the substrate. The top face also remains unaffected in shape and dimensions.

For the equilibrium form, the Gibbs-Wulff-Kaischew theorem can be applied. It states that the normal distances hi of all faces from the Wulff point, including the top face and the contact face /, are proportional to the specific surface energies (cf. eq. (4.10)). Hence, the specific surface energies in eq. (4.21b) can be replaced by the respective distances, hi/A = ov, so that... 

This is a general form of the well-known Gibbs-Thomson (Lord Kelvin) equation applied to the case of electrochemical metal deposition. It gives the size of the critical nucleus and its equilibrium form in terms of the normal distances of the equilibrium form faces from the Wulff point, hi, as a function of the overvoltage. When this form is a regular polyhedron (cr,- = const.), the size of the nucleus can be given by the radius of the inscribed sphere, perit = h, so that... 

From Eq. (27), using the Gibbs-Wulff rule hi/ei = const, the size of the critical nucleus, described by the distances hi from the Wulff point, is obtained hi/si = s/zeoVc- When the equilibrium form is a regular polygon, the distance h is identical with the radius of the inscribed circle p, used very often as a measure for the size of the nucleus. Hence... 

Figure 19. Illustration of the Gibbs-Wulff-Kaischew theorem, o-i to as are equal. The faces 1 to 6 are equidistant from the Wulff point. In the presence of the substrate, the distance h to the contact plane is reduced as indicated, becoming negative as ag >0-4.

The linear dimensions hi of the critical cluster are determined according to Eq. (60) by the overpotential. The presence of the substrate affects only the value of hf, i.e., the distance of the Wulff point from the contact interface. 

In equations (1.50) and (1.51) are the specific free edge energies, Ij are the lengths of the edges and fj are the distances from the Wulffs point inside the polygon to its j-th side. Note that the quantity is defined as the work done to create a unit edge length. It does not coincide with the line tension k ... 

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