Crystal have typically an anisotropic surface energy (per area). Hence, you may increase the surface area while decreasing the total energy in all (upon keeping the total crystal volume constant; typically achieved via a Lagrange multiplicator approach). Nowadays wikipedia can serve as a good starting point for the discussion. https://en.wikipedia.org/wiki/Wulff_construction See also Porter-Easterling.

Thanks for your answer Dr. Andreas Leineweber

My specific question is why we would have whisker growth and Wulff construction in a similar condition with the same materials. I would appreciate it if you could provide me with a research article which is discussing the mechanisms, factors affecting, and differences of these two growth modes.

Hallo Alireza, the growth of metallic whiskers described on the following wikipedia page is indeed an interesting phenomenon:

https://en.wikipedia.org/wiki/Whisker_(metallurgy)

The Wulff-construction of the crystal shape can be based on anisotropic surface energies or anisotropic growth kinetics. You may have a look at our work on the formation of anisotropic theta’ precipitates in Al-alloys:

Article Simulation of the θ’ Precipitation Process with Interfacial ...

or our work on the formation of LiFePO4 nano particles:

Article Phase-field modeling of Li-insertion kinetics in single LiFe...

If you want to explain spontaneous growth of metallic whiskers using the geometrical Wulff-construction, you need a good idea on the atomistic origin of the anisotropy of the growing crystal. In my opinion, the most important aspect is to explain the spontaneous appearance such a surprisingly strong anisotropy: In order to get an anisotropic Wulff-shape with an aspect ratio of 1000 you roughly need 1000 times lower surface energy or 1000 times faster growth kinetics in the one crystallographic direction as compared to the others.

Dear Mr. Ghorbani,

Whisker shapes are, in my understanding, usually not equilibrium shapes of crystals but a (partial) consequence of the growth kinetics. Hence I would not look for explanations based on an anisotropy of the surface energy to explain the quite anisotropic shapes of whiskers. Only perpendicular to the whisker growth axis one may reasonably assume a 2d Wulff shape.

Whisker growth is typically explained either by growth from bottom (like from stressed metal like Sn) or from the top (upon formation of whiskers from the gas phase; then you may typically have a chemically distinct whisker cap). Whiskers typically have some crystallaphic direction [uvw] as axis and the mantle planes are accordingly (hkl) fulfilling the Weiss zone rule with the [uvw]. Popular is also the screw dislocation mechanism which is generally popular for crystal growth for many system and was popularized by Frank in the 1950s. I remember, however, statements that in several cases, search of this magic dislocation inside the whisker did not succeed.

However, I have to disappoint you, I cannot currently offer nice reviews/books covering the whole issue. I loosely remember some books and review articles which cover (unfortunately only) aspects of all this. I would have to do a google scholar search on this issue to put together the pieces, but this you can also do by yourself....

Thank you, Michael Fleck Michael Fleck. I greatly appreciate your introduction of these papers, and I hope that they will be helpful to me, particularly the Li insertion paper.

Thank Dr. Andreas Leineweber for your detailed response to my question. unfortunately, I was also unable to find a complete answer to my question. If you happen to come across any additional materials, I would greatly appreciate it if you could share them with me.