If a phase transition requires a number of particles that is in the TD-limit, can nanoparticles (~10 atoms) have phase transitions? What kind of phases and transitions nanoparticles have?
We can't really talk generally about the phase of an individual nanoparticle, as we aren't in the thermodynamic limit, as the question you linked to suggests.
Think about what a gas is, for example: gases have a higher average kinetic energy than a liquid, which causes it to be less dense and occupy a greater volume than liquid. But if we look at the particles comprising the gas at a microscopic level, we see a distribution of velocities - there will be some proportion of the molecules that have less kinetic energy than the average energy of, say, a liquid. But in a liquid, there will be some small proportion of molecules that have more kinetic energy than the average kinetic energy of a gaseous phase. So it makes sense to talk about phases in terms of an average over a large number of particles and not in terms of an individual particle. The number of particles comprising a nanoparticle is many orders of magnitude less than that required for the thermodynamic limit.
I suppose you could describe a single nanoparticle in a Bose Einstein Condensate state, since the BEC phase is described by all particles taking on a common wavefunction which is a different requirement than some sort of average over many particles.
You can, however, talk about the phase of a large collection of nanoparticles; this is no different than how we describe the solid, liquid, or gaseous state of any other material.