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Zinc oxide comes as a powder that is insoluble in water but soluble in most acids.
I once received a clear liquid containing zinc oxide from an Italian pharmacy as a deodorant.
What could they have used to make the $\ce{ZnO}$ soluble and safe for use on the skin?
As I previously stated in a comment, I think that the zinc oxide is not actually dissolved. That would just leave ions which you would hardly consider zinc oxide any more. $$\ce{ZnO + 2HCl <=> Zn^2+ (aq) + 2Cl- (aq) + H2O}$$
Instead I think that there are nanoparticles in use, that appear to be dissolved. In deodorant or other skin products it is likely to have quite a low concentration.
An admittedly short search turned up a patent by Marc Paye, in which is stated:[1]
[...]
(f) 0.01% to 1%, more preferably 0.03% to 0.5% of Nanoparticulates of zinc oxide which functions as an anti-irritant system for anionic surfactants,
[...]
Or from another patent of the same company:[2]
[...]
(c) 0.5-10% (particularly 0.5-7.0, and more particularly 0.5-5.0%) of a small particle size zinc oxide, particularly a micronized zinc oxide or a nanoparticle size zinc oxide having a particle size in the range of 20 nanometers-200 microns;
[...]
I also found, but have not completely read an article that confirmed my theory.[3]
As an ingredient in dry deodorants to reduce wetness under the arm, ZnO can be used between 0.05 and 10% by weight with average particle size in the range of 0.02–200 microns. ZnO may be used to provide a pH range desirable for deodorants designed for use on sensitive skins [174].
At least in the last article you should also find some information about how to synthesise these particles. Spoiler: It's nothing you would try at home.
ZnO ia an inorganic compound that is oil soluble. It does react with some fatty acids which are responsible for bad odor, thus the use of it in some deodorants (articles below).
The fact the it is used on many baby powder products (for rushes) shows that it is safe on the skin. By making a quick search you can find that ZnO particles are within the range of 70 nm to 100 nm (http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2714038/).
These values show that the ZnO particles do not actually penetrate the skin. They usually also say coated or uncoated ZnO. This is done in order to make ZnO less reactive (coated ones).
Another quick search lead me to this site which has very informative stuff about ZnO and how to make your won deodorant.
