Wulff construction
The Wulff construction (according to George V. Wulff ) is the method for determining the shape of a crystal in thermodynamic equilibrium . The shape determined in this way minimizes the free energy of the surface with a fixed volume.
The construction
For the Wulff construction, the surface energy is required for every possible orientation of the surface , more precisely, the free energy per unit area. Each orientation of the surface is defined by the direction perpendicular to it. One now plots as a function of direction; this is the red curve in the figure. For example, the length of the red arrow indicates the surface free energy for a surface perpendicular to the direction of the arrow.
For each direction (dark red lines) one now constructs a perpendicular 't' through the point where this line passes through the curve of the surface energy. Some of these vertical lines are drawn in dark blue in the picture. The crystal shape results from the innermost of all possible blue lines and is the thick blue line in the picture. So the blue lines 't' are tangents to the crystal shape.
In contrast to the illustration, for the three-dimensional shape of a crystal 't' planes must be constructed instead of the blue lines; the crystal form results from the innermost of all so constructed (tangential) planes.
This construction can also be understood mathematically as a Legendre transformation between the direction-dependent free energy of the surface and the crystal shape.
Application to crystals
With crystals, certain directions often have a particularly low surface energy, such as the surfaces perpendicular to the black lines in the picture. In this case, large flat areas occur. It is often the case that the Wulff shape constructed from the surface energies consists largely of such surfaces at low temperatures; many other directions of the surface do not appear at all, that is, the corresponding planes or lines t in the Wulff construction lie outside the crystal form everywhere.
At higher temperatures, however, there are also many other crystal directions that have an unfavorably high surface energy, but because of their higher entropy a not so high free energy. Therefore, the equilibrium shape has rounded edges and corners at high temperatures (as in the picture above).
In practice, crystals only reach equilibrium (according to the Wulff construction) if the atoms can diffuse across the surface sufficiently quickly to reach the bond site with the most favorable (i.e. lowest) free energy. This is possible for many metals with small crystals and high temperatures, for example for metal clusters such as those used in catalysts . In contrast, the shape of the crystals in most minerals is determined by the growth rate of the different crystal surfaces and not by the Wulff construction.
literature
- G. Wulff: On the question of the speed of growth and the dissolution of the crystal surfaces. In: Journal for Crystallography and Mineralogy. Volume 34, 1901, pp. 449-530.
- Salvador Miracle-Sole, Wulff-shape of crystals , Scholarpedia 2013