# Surface energy for different crystallographic planes

For a given crystal structure, say body centred cubic, on what factors does surface energy of each plane depend? A basic approach considers the atomic density on each crystallographic plane, the number of bonds broken if the surface is cleaved along that plane and combines these two.

Is this a realistic and reasonable approach if I want to compare the surface energies of two crystallographic planes, say (100) and (111)? What are the other factors, if any, that should be included in the comparison?

Edit : Background : My problem related to the wettability of a crystal surface by a liquid. The contact angle that the liquid makes with the crystal surface is determined by the interfacial energies between solid-liquid, liquid-vapour and solid-vapour. Keeping all other factors, I only change the nature of the solid crystal surface, by presenting a different crystallographic plane. I would like to theoretically estimate which crystallographic plane would have the highest energy, and which would have the lowest, so that a prediction can be made about the extent of wettability of the surface.

Thank you!

• As you speak of a solid/liquid interface, obviously you must take into account the liquid phase as well. But I think you have overgeneralised your problem. Be more specific! – Karl Jan 14 '20 at 9:13
• Thanks @Karl, I have removed the bit about the interface, and now am asking only about the surface energy. – AppliedAcademic Jan 14 '20 at 10:05
• Background info is generally good! What I meant was that you should be more specific about your actual problem. You ask a very general question, those rarely have a clear answer. – Karl Jan 14 '20 at 18:22
• Thanks @Karl, now I understood. Hopefully with the background, the problem is now more specific. :) – AppliedAcademic Jan 15 '20 at 6:21

## 1 Answer

You can think about this problem from the definitions of surface energy.

The surface energy quantifies the disruption of inter-molecular bonds that occurs when a surface is created

• The composition or structure of a surface depends on how the bulk crystal has been cut.
• The electronic structure (or number of dangling bonds) depends on surface composition
• The reactivity of a surface depends on the electronic structure.

Imagine a battenberg cake: • The cake has been cut horizontally and the surface is comprised of tasty red and yellow squares.
• If however we had cut the cake vertically straight down the middle, you would only see a red line next to a yellow line (Cut 1).
• If you cut vertically and diagonally across the cake, you might only see either a purely red or purely yellow surface. In the case of the example image, you would only see a red surface (Cut 2).

I hope this example shows that (by relating the coloured squares to atoms), different cuts result in different surfaces with different properties.

When calculating the surface energy of a system:

• You need to calculate the bulk crystal energy.
• You need to calculate the energy of the surface slab. Make sure your model is converged with respect to slab thickness. The central layers should be representative of the bulk crystal. Look at magnetic change, charge difference, degree of ionic relaxation.
• You then calculate the difference between the slab energy and the bulk crystal energy.

Note: In periodic calculations, a surface slab is constructed by cutting the bulk in a specified plane and introducing a vacuum gap. This vacuum gaps should be sufficient that the surface does not interact with its periodic mirror images. Around 15 Angstroms.

There are a number of codes that can help in the construction of surface slabs (METADISE, Materials Studio, Vesta)

If you wish to calculate the energy of a surface (and assuming you are using a periodic DFT code), I would highly recommend the paper by Wenhao Sun and Gerbrand Ceder (https://www.sciencedirect.com/science/article/pii/S003960281300160X)

Note: be careful of magnetic influences.

• Thanks @Wychh, you explained this beautifully. Now, I'm not familiar with DFT packages, so I'm thinking of using a simplified broked-bond model, such as that described here (sciencedirect.com/science/article/abs/pii/S0022369799004151). Do you think this is a reasonable approximation, or will I make major deviations if I dn't do an exact DFT calculation? – AppliedAcademic Jan 15 '20 at 17:49