Wulff construction

Wulff construction ( to George V. Wulff ) is the method for determining the form of a crystal in the thermodynamic equilibrium. The thus determined form minimizes the free energy of the surface at a fixed volume.

The construction

For Wulff construction is required for each possible orientation of the surface, the surface energy, and more specifically, the free energy per unit area. Each orientation of the surface is defined by the direction perpendicular to it. Man wearing now as a function of the direction; This is the red curve in the figure. For example, the length of the red arrow to the surface free energy for a surface perpendicular to the direction of the arrow.

We construct now for each direction (dark red lines), a perpendicular 't' by that point where this line meets the curve of the surface energy. Some of those vertical lines are drawn in dark blue in the picture. The crystal shape results from the innermost of all possible blue lines, and the thick blue line in the image. The blue lines 't' so are tangents to the crystal form.

In contrast to the figure to be designed for three-dimensional shape of a crystal in place of the blue lines 'T' levels; the crystal shape results from the innermost of all thus constructed (tangential) planes.

This construction can be understood mathematically as a Legendre transformation between the direction-dependent surface free energy and the crystal form.

Application to crystals

In crystals often have certain directions, a particularly low surface energy, such as the image perpendicular to the surfaces of the black lines. In this case, there are large flat surfaces. It is often true that the constructed from the surface energies of the Wulff shape largely consists of such surfaces at low temperature; many other directions of the surface does not occur, that is, that the additional planes or lines t lie anywhere outside the crystal form in the Wulff construction.

At higher temperatures, but also occur many other crystal directions, although they have an unfavorably high surface energy, but because of their higher entropy not so high free energy. Therefore, the equilibrium shape at high temperatures has rounded edges and corners (as in the picture above).

In practice, crystals will only achieve the equilibrium shape ( according to the Wulff construction ) when the atoms can diffuse rapidly across the surface sufficient to achieve each of the binding site with the best (ie lowest ) free energy. That for many metals in small crystals, and high temperatures is possible, for example, metal clusters, such as those used in catalytic converters. However, the shape of the crystals will be determined at the most minerals by the rate of growth of the various crystal faces and not by the Wulff construction.