Understanding conductivity at nanoscale and validity of Ohm's law Nanoparticles such as gold and silver are becoming more and more used to print circuit or enhance electrical properties of another material. But I have not been able to find sources that clarify how current transport takes place between nanosized particles. In particular, what I am trying to learn is the following:
Suppose we have printed a layer of metallic (such as gold) nanoparticles on a substrate, the size of particles may range from 10-50 nm. What is the dominant form of electron transport at such scale? I know that for instance if the particles are suspended in an insulating medium, then electron tunneling becomes the relevant transport mechanism. But what if we just have the metallic nanoparticles? Are applied voltages linearly related to induced currents? (i.e. Ohm's law holds).
Reiterated questions:

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*If there's no junction, how does current transport take place between metallic nanoparticles? Do two adjacent particles need to have their surfaces in contact for electron transfer to take place? Is contact resistance the relevant mechanism in this case?


*For such nanoscopic systems, do we still expect Ohm's law to hold? that is, if we apply a voltage difference at two junctions of the system, is the measured current related linearly (with conductance being the proportionality constant)? Is there a clear breaking point, in terms of particle sizes, below which we know Ohm's (linearity) law not to hold anymore?
 A: Here is a somewhat dated review article that you may find interesting (it is obviously worthwhile looking up more recent articles citing this one).
Much will depend on the size of the particles (e.g., whether they are in Coulomb blockade regime or not), the distance between them, the substrate, whether they are ordered, whether there is variation of particle size and shape, etc.
To give a bit more specific answers to your questions:

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*Particles do not have to physically touch themselves to conduct current: if they are sufficiently close to each other, the electrons may be able to tunnel from one particle to the other - the rate of tunneling obviously would decay exponentially with separation of particles. It is also significantly affected by the Coulomb blockage.

*Ohm's law is an empirical law, which obeys for most materials at macro-scale in normal conditions. In other words, it does not work in most situations of interest on nanoscale. Here one has to learn a whole new chapter, involving Coulomb blockade, conductance quantization, weak and strong localization, etc. One possible introduction to the field of nanoscale transport in the book by Joe Imry.

A: At least in silver epoxy, it is contact resistance. That is an enormous subject in itself (see the classic work of Holm and the conferences that bear his name). They are probably mesoscale and not nanoscale. It depends an awful lot on the details of the of the particles. At some point they are no longer metallic.
At any, Ohm‘s law is not appropriate.
