Wulff construction for surface energy dependent on depth

[decart]

Hello,

Using surface energy dependence on surface orientation it is possible to predict equilibrium crystal shape by applying Wulff construction. But I faced the problem when the surface energy depends not only on the surface orientation but also on the surface depth or surface termination and depends significantly.
So, usual Wulff relation
∑γ_is_i=min
becomes
∑γ_i(r_j)s_i=min
where γ_i(r_j) is periodical function of surface energy γ_i upon the depth r_j.
How it is possible to correctly apply Wulff construction to this case?

Thank you!

 

[Useful nucleus]

I do not know the answer but I think that there will be inifitely many Wulff constructions for the same material depending on the surrounding thermodynamic conditions. For example, the surfaces of the metal oxide MO can be terminated by M, O, depending on oxygen chemical potential. Thus, for a give value, of T, P, μ_O, there will be a Wulff construction. Just a thought.

 

[decart]

Thank you for the answer!
Yes,I agree with you. But how should I choose the γ_i(r_j)? Should I choose maximal or minimum value (which will depend on P,T, etc.), or something else?

 

[Useful nucleus]

I'm not sure I understand the depth part of your question. Do you mean the the thickness of the thin film (or the slab model in a computational model)? Or let's say the size of the nano particle you are trying to study?

At any rate systems try to minimize their free energy. And in order to do this properly, you need to compute/measure the surface energy (or better free energy) for each surface (h,i,k) and use the termination that minimizes the surface free energy in the sum you indicated above for the Wulff construction.

Again my responses are based on primitive understanding of surface science.

 

[decart]

Thank you for the answer and sorry for the misleading (I expected that term "depth" is vague). I talk about surfaces, not slabs. The surfaces with same orientation (a⋅h,a⋅i,a⋅k) for a=1...N has different γ_a, whereas application of usual Wulff relation requires γ_a=const for the same surface (h,i,k). Please find the graphical representation of the problem attached to the message.

 

[Useful nucleus]

Thank you for explaining this by a schematic, it made it much clearer. I always thought that the surface energy of (001) is the same as the surface energy of (002) or (003),... Please could you give an example, where this is actually not the case. A specific material with certain crystal structure will help orienting the discussion.