Predicting Crystal Shapes: Wulfs Theorem

[aaaa202]

Is it possible to predict, at least in theory, which shape a crystal of some combination of atoms would have. If so, what are the principles for such a calculation. I have seen a theorem in classical thermodynamics called wulfs theorem about the equilibrium shape of crystals. Does this explain why we see different crystal shapes in nature?

 

[OhYoungLions]

There are two parts to your question:

First, one has to ask how the atoms arrange themselves with respect to one another at the microscopic level. This information is what people usually refer to as the "crystal structure". Usually, the arrangement of atoms is periodic - it repeats in a pattern, so we typically visualize the crystal structure in terms of the smallest repeat unit, which is called the "unit cell". The crystal structure that nature chooses will usually be the one that minimizes the free energy, and as such it can be predicted in theory. It turns out for simple materials (say pure Copper), this can be done pretty straightforwardly as long as you have a good way of computing the energy of various configurations, and make some assumptions about the number of atoms and size of your repeat unit. For more complicated systems, it turns out to be difficult in practice to predict the crystal structure, because there are so many possibilities to try. This is an active topic of research.

The second part of your question concerns the macroscopic "shape" of a crystal, which depends mostly on the growth dynamics, which could depend on many factors, one of which is the crystal structure. You can imagine when you are growing a crystal that it starts out small, and then additional atoms or molecules (say in the gas or liquid phase) have to come nearby and stick to the surface. Some surfaces may be more attractive to the nearby atoms or molecules, so a crystal may grow faster in particular directions, and this is what determines the overall shape. Some of the factors that affect the growth can be controlled - it is a real art to be able to grow a crystal with a certain shape. As far as I know, Wulf's construction predicts the equilibrium shape of the crystal, which is often not the way the crystal comes out when you first grow it. One thing you can say for sure is that crystals will have some symmetries at the microscopic level. The growth along any two directions related by symmetry should occur at roughly the same rate, so often if a crystal has a macroscopic symmetry, this will be related to a microscopic symmetry. The usual example of this is snowflakes, many of which have hexagonal symmetry (they look the same if you rotate them by 360/6 = 60 degrees). This is because the crystal structure of ice has the same symmetry.

Hopefully this has answered your difficult question.

 

[DrDu]

Usually one distinguishes between crystal form an crystal habit. The form describes the set of planes (or their Miller indices) occurring in a crystal, while the habit describes their relative size. Both can be influenced by the composition of the solution from which a crystal is grown. For example, some additives will adsorb preferentially on some specific crystal planes and hence slow down their growth.

 

[aaaa202]

So when the wulff theorem talks about equilibrium shape, are you saying that it means equilibrium shape of the microscopic structure, i.e. of the unit cell? Is it not that given we know that a crystal for instance grows by the repetition of hexagonal unit cells the Wulffs theorem predicts the macroscopic equilibrium shape of the crystal? I mean after all the minimization of surface energy uses the idea of unsaturated bonds on macroscopic crystal planes.

 

[ZapperZ]

So when the wulff theorem talks about equilibrium shape, are you saying that it means equilibrium shape of the microscopic structure, i.e. of the unit cell? Is it not that given we know that a crystal for instance grows by the repetition of hexagonal unit cells the Wulffs theorem predicts the macroscopic equilibrium shape of the crystal? I mean after all the minimization of surface energy uses the idea of unsaturated bonds on macroscopic crystal planes.

I'm a bit confused here. When you use the phrase "crystal shape", are you talking about the macroscopic shape of a lump of "crystal", or are you talking about crystal structure? The former is undefined, the latter is well-defined in terms of what it is.

A macroscopic crystal shape can be anything. I can take diamond and have a huge range of shapes, depending on how I cleave it. A crystal structure of diamond, on the other hand, is well-defined.

Secondly, as has been alluded to, simply knowing the atoms that make up the crystal does not tell you the crystal structure it will form. How it is formed plays a significant role. I can have a bunch of carbon atoms, and I can end up having carbon, graphite, or diamond, three very distinctly different entities with wildly different properties.

Zz.

 

[aaaa202]

Okay so the wulff shape is a statement about the equilibrium shape of the unit cell?

 

[DrDu]

Okay so the wulff shape is a statement about the equilibrium shape of the unit cell?

No, the Wulff theorem is about the fraction of the different surfaces of a crystal in equilibrium with its solution or melt. It is only that the shape of most crystals is the result of non-equilibrium kinetics.