Ambiguity on the notion of potential in electrical circuits? As everybody else I have been taught elementary electrical circuits from secondary school to engineering level in analog electronics at university. Invariably, the notion of potential used to described what is going on in these circuits is seemingly that of electric potential that would follow then Maxwell's equations.
Yet, when learning a bit more about fundamental elecrokinetics in solid state physics, a current arises whenever there is a difference/gradient of chemical potential of the charge carriers and not necessarily a gradient of electric potential.
Basically, the chemical potential can probably be expressed as the sum:
$\mu = \mu_{id} + \mu_{exc} + \mu_e$
where $\mu_{id}$ is an ideal (albeit quantum) contribution, $\mu_{exc}$ an excess part comprising the interactions with the lattice, interactions/correlations with other charge carriers, quantum effects etc... and $\mu_e \sim q \phi$ an electric part containing a mean field contribution from the rest of the charges and that follows Maxwell's equations.
If the piece of metal is always the same, retrieving elementary laws for electrical circuits is kind of easy according to the above definition.
The only problem I have is that in most circuits there is more than one type of material, there are in fact many different materials that contribute to the overall current flow. 
At the junction between two pieces of materials for instance, current will flow for various reasons related to electric potential differences but also to differences in the overall chemical potential (where the ideal and the excess non electrostatic part would also differ).
A good example of that is of course NP junctions that develop an electric potential difference at the junction to make uniform the chemical potential of the charge carriers and as a consequence there is no flow of charges in a diode.
Also, from what I understand, chemical batteries based on oxydo-reduction reactions impose a chemical potential difference for electrons only (it is chemistry driven). Hence although it is easy to claim that the potential difference for such a chemical battery will be given by Nernst equation, this potential difference can only make electrons flow and not for instance ions in an electrolyte solution. I may be mistaken here but, from what I understand, you can't close a circuit using a chemical battery by bridging the two cells with a material that would not conduct electrons (even if it were to conduct other stuff). Note that I do know that there is a salt bridge in chemical batteries to ensure that the chemical potential difference is not balanced precisely by an electric potential difference.
So here is my question(s):


*

*Am I overthinking it by believing that there is some ambiguity going on in the use (and potentially misuse) of the notion potential as being always interpreted as an electrical potential in circuits?

*Why then would the concept of potential in circuits be so ubiquitous? Are all my worries simply solved by the gauge freedom of the electric potential?

*Are there cases where the notion of potential would indeed be misleading if understood as electrical potential? I think I have mentioned some but I may be misunderstanding something... 
 A: *

*There is an ambiguity. Although I did not understand your analysis of the problem completely, charge carriers certainly can run against the (averaged) electric force due to difference in available bands and other particle statistics effects.

*The gauge freedom is irrelevant. There are two cases for the “ubiquity”. First, these non-Maxwellian deviations usually have the order of 1 volt or even less, so for such technologies as the mains electricity or electrical transport they are negligible. Second, even where they are not negligible (such as in semiconductors and heterogeneous metallic circuits) they usually almost cancel themselves when form a closed circuit. Thermocouples form a notable exception.

*Inside a piece of usual atomic matter, even if it is an electricity conductor, there is no such thing as value of the electric potential. Nuclei emits their Coulomb fields, electronic shells emits theirs, electrons in conduction bands are also changed, and even worse, all this things (especially electrons) obey the laws of quantum mechanics and hence have not any certain spatial position. Do you imagine a Maxwellian field in these circumstances? I do not. Maxwell’s theory is hardly usable inside conductors. Insulators provide some emulation of it on the macroscopic scale, although not unperturbed vacuum Maxwellian fields (I refer to such things as electrical permeability and optical density), but also have aforementioned hell inside them on the molecular level.

A: 
The only problem I have is that in most circuits there is more than one type of material, there are in fact many different materials that contribute to the overall current flow. 

It's not an ambiguity.  Instead it's an unstated traditional over-simplification, as follows.
A closed metal circuit may have many junctions between differing materials (e.g. copper wires, tin/lead solder, aluminum IC bond-wires, steel screw terminals, borate electrolyte in capacitors.)  Even at zero current, each material will be at a different potential relative to some reference point.  Yet if all junctions between differing materials remain at the same temperature, then the net potential difference across all junctions sum to zero around the entire circuit. So, charge-carriers which make the trip around the complete circuit will end at the same potential where they started.
We then perform the following oversimplification:  just assume that the charge carriers never change potential at all!
See what's going on? Charge-carriers may move from copper to tin-plate, and must change energy doing so.  But then they move from tin-plate to copper in some other part of the circuit.  They may move from copper to solder to copper, and from steel to tin-plate to steel.
For all these changes in potential, we simply ignore their presence, since the carriers end up at the same potential where they started.
Of course all of the above only applies to connecting wires.   Inside the various components (batteries, resistors, pn junctions) we don't ignore the changes in potential.   In other words, we treat a diode pn junction potential as real, yet we ignore the quite large potentials located at the nonrectifying junctions where copper connects to the diode's silicon.
And, of course all of the above is no longer ignored whenever thermocouple or thermo-pile or Peltier generators/coolers are present, or when millivolt/microvolt DC signals are significant.
In Electrical Engineering language, all circuits are filled with potential hills; with parasitic thermocouples and with Peltier heater-junctions and cooler-junctions.  But traditionally we sweep them all under the carpet and pretend that the entire circuit is made of copper.   When this habit causes unexpected artifacts, traditionally we then throw a 4-junction "kelvin probe" at the problem and hope it goes away.
:)
One topic of confusion typically appears:  students try to measure the absolute potential of a thermocouple junction, and find that it's always zero.  This of course is caused by the metal of their voltmeter probes, which create new junctions at the two thermocouple wires.  The actual potentials of various circuit conductors sum to zero and are impossible to measure with a normal DVM.  We'd need some sort of electrostatic field-mill, or other voltmeter device which detects e-fields directly and doesn't close the circuit.
Yet also our digital voltmeters are reinforcing the above traditional oversimplification, and telling us that these potentials are actually zero.  We're pretending that these potentials don't exist, and our measuring devices are pretending the same thing.  Two artifacts fortunately cancel each other, so beginners may never encounter them at all, while experienced experts can gain full understanding of what's going on in circuits, and then keep it as part of their "bag of tricks," the Black Arts held proprietary within the medieval guild of analog design engineers!
