Species voltage: a real voltage for real charges

Our key player is the species voltage $V_i$, a potential defined for every type of charge carrier $i$ (e.g., electron $\mathrm{e}^-$, or hydrogen ion $\mathrm{H}^+$, or sulfate ion $\mathrm{SO_4}^{2-}$, ...). It's derived from a more fundamental thermodynamic quantity called the electrochemical potential, $\bar\mu_i$, by simply normalizing for the species' charge $q_i$:

$$V_i = \frac{\bar\mu_i}{q_i}.$$

The electrochemical potential $\bar\mu_i$ is a deep subject in its own right, and a later topic, Understanding electrochemical potential, explores it properly (making the case that $\bar\mu_i$ is the real, indivisible chemical potential). But none of that depth is needed here, because you likely already know how $V_i$ works:

The electronic quantity $V_{\mathrm{e}^-}$ is precisely the voltage seen in basic electronic circuits.

That is,

• $V_{\mathrm{e}^-}$ equalizes when we short metal wires together,
• $V_{\mathrm{e}^-}$ drops across a resistor according to $\Delta V = IR$,
• $V_{\mathrm{e}^-}$ is thermodynamically analogous to an electronic 'pressure',
• $V_{\mathrm{e}^-}$ differences are what we measure with voltmeters,^[1]
• it is a $V_{\mathrm{e}^-}$ difference that you measure across a battery's terminals,
• $V_{\mathrm{e}^-}$ is conventionally assigned to $0$ for the electrical ground,
• and so on, you get the idea.

We only have to make a couple more conceptual leaps to arrive at a voltage for ions too:

• Any charged species deserves a quantity that plays the role 'voltage' plays for electrons, and consequently,
• there can be multiple voltages in the same place. Semiconductor physicists already accept coexisting voltages in non-equilibrium situations, under the name "quasi-Fermi levels"; for ions, coexisting voltages are everyday reality even at equilibrium.

Voltage or potential? Some may object to labelling $V_i$ as a "voltage" rather than "potential"; technically a voltage should be a potential difference. However, there are way too many things called "potential" in electrochemistry, most of which are potential differences; if voltages are so heavily conflated with potentials then we might as well go with it. Moreover, in electronic circuits, an unreferenced single-point voltage simply means a potential difference versus a common electronic ground point, a convention which we will tend to adopt. More technically: $V_i$ are all unreferenced (gauge covariant) quantities, and we are free to spend our one global gauge freedom to assign 0 to electronic ground (just the one: we do not get a separate ground per species).

What about electrostatic potential $\phi$?

It is commonly taught that the electrostatic potential $\phi$ (ostensibly from Maxwell's equations) is what defines voltage. I had to unlearn that when doing nanoelectronics,^[2] and it already goes wrong in something as ordinary as a copper wire joined to a blob of solder, where $\phi$ takes a step that the circuit voltage does not:

Worse, the in-material $\phi$ is not accessible to any measurement made with the ions and electrons themselves. Its overall offset is free, of course, but the deeper trouble is that even differences in $\phi$ (the Galvani potentials) are generally unmeasurable. So $\phi$ ends up assigned by convention, material by material, at which point it is no longer an electrostatic potential at all, and $-\nabla\phi$ need not be any real electric field. The $\phi$ picture can be made to work once every convention is reconciled and every ambiguity hedged (in the end it cancels out of all observables), but that fragile apparatus is exactly what has bred so many errors and misconceptions.

So we will simply not use an in-material $\phi$. We stick to $V_i$, and later to $V^\circ_i$, which does $\phi$'s job with far less ambiguity; you will see as we go that we never actually need it, a $\phi$-less philosophy that semiconductor device physics already lives by.^[3] The full case against $\phi$ has its own topic, and the Offsets galore topic lets you see for yourself just how arbitrary it is.

With $\phi$ set aside, let's take stock of the key principles that make $V_i$ a genuine voltage.

Differences in $V_i$ are available work

The fundamental rule of thermodynamics is that particles flow from high to low electrochemical potential ($\bar\mu_i$) to release free energy. The difference, $\Delta \bar\mu_i$, is the maximum work that can be extracted from this flow. (To be precise, this is only true when both bodies have equal temperatures,^[4] but we will generally assume isothermal conditions).

The available work $\Delta\bar\mu_i$ is free energy per unit of particle count (e.g. kJ/mol or eV/particle). By normalizing $V_i = \bar\mu_i / q_i$, the corresponding $\Delta V_i$ is available work per unit charge (volts). Mind that this is the work for charge transferred via that particular species, and no other route.

Differences in $V_i$ drive currents

Since differences in $V_i$ are available work, currents from high to low $V_i$ occur spontaneously (such flows increase entropy). The simplest form of this is Ohm's law:

$$J_i = -\sigma_i \nabla V_i$$

Note this is simpler than the common split seen in electrochemistry and solid state physics, where the driving force $-\nabla V_i$ is split into drift (from $-\nabla\phi$) and diffusion (from $-\nabla c_i$), then reunited by the Einstein relation because both are really the one $-\nabla V_i$ (worked out in basic transport).

Not all currents are so simple as Ohm's law, of course. Interfaces often have a nonlinear current-voltage relationship; there may be cross coupling where electrochemical potential gradients in one species drive another species (including neutral species); there may be other driving forces like magnetic induction or thermoelectricity; there may be convection/advection.

Differences in $V_i$ are measurable

The previous two sections were about what a difference $\Delta V_i$ does in nature (it drives currents and delivers work), and those hold for every species alike. Being measurable is a claim of a different kind: it is about what we can reach, and reach depends on the instrument. A difference $\Delta V_i$ is physically defined for any species; but only for electrons is it easy to get at.

A common voltmeter has metal probes and reads differences in $V_{\mathrm{e}^-}$, letting a tiny, ideally negligible electron current flow in. It works so cleanly because so many of our conductors are purely electronic: join wires of any assortment of metals and, since electrons are the only mobile carrier, $V_{\mathrm{e}^-}$ carries across all of them perfectly.

Ions are harder. We have no good "ion wires" (an electrolyte carries at least one cation and one anion) and no ionic voltmeter, so we reach $V_{\mathrm{ion}}$ only indirectly, by coupling it to $V_{\mathrm{e}^-}$. A carefully prepared ion-reversible electrode chemically locks $V_{\mathrm{e}^-}$ to the $V_i$ of exactly one ion, producing a fixed step $V_{\mathrm{e}^-} - V_{\mathrm{ion}} = \Delta$ that an ordinary voltmeter can then read.

But $\Delta$ is a gap between different species (an electron and an ion), so, like any $V_i - V_j$ below, it carries a chemical convention, and a single electrode does not hand you an absolute $V_{\mathrm{ion}}$. What it gives cleanly is a same-ion difference: put a matched electrode in each of two solutions and $\Delta$ cancels, leaving exactly $V_{\mathrm{ion}}(\text{B}) - V_{\mathrm{ion}}(\text{A})$, a fully physical voltage obtained without ever knowing $\Delta$.^[5]

So an ionic $V_i$ is no less real than the electronic $V_{\mathrm{e}^-}$: it is defined the same way and carries the same physical difference. It is only less reachable: what an ordinary instrument hands you is its same-ion difference, not its absolute value. This is all I mean by saying we can access a $V_i$: that we can measure some voltage difference involving it. (A more general recipe, using ion-selective membranes, comes later; it reaches every ion in principle, at the cost of being far less practical.)

Gaps $V_i - V_j$ store charge

The electrode's fixed step $\Delta$ was our first meeting with a gap between different species, though there it reached across an interface. Cross-species gaps have a physics of their own even at a single spot: they store charge. Two carriers sharing one pocket of material behave like the two plates of a capacitor. Lay $+Q$ on the $i$ population and $-Q$ on the $j$, so the pocket stays neutral, and the rails answer: the gap $V_i - V_j$ opens in proportion to the charge laid down. That is a capacitance, $C = \mathrm{d}Q / \mathrm{d}(V_i - V_j)$, wired straight between the two $V_i$ rails, referring to the species voltages alone and never to a standard state or an electrostatic $\phi$; it is known as the chemical capacitance.^[6] The storage can be large (a dilute salt solution banks kilofarads per cubic centimetre) but, like most things chemical, it is also nonlinear: $C$ depends on the concentrations themselves, so it drifts as charge accumulates. Indeed, an intercalation electrode is exactly such a capacitor charged deep into the nonlinear regime: the flatter its charging curve, the larger its capacitance. (With three or more carriers the bookkeeping grows into a matrix.)

$V_i$ are thermodynamic state variables

Alongside temperature $T$, pressure $P$, and the chemical potentials $\mu_i$ of neutral species, the $V_i$ (or $\bar\mu_i$) are the proper intensive state variables for charged species. Since $T$, $P$, and the solvent $\mu$ tend to be constant or implicitly clear, our plots of varying $V_i$ will be a complete visualization of the spatial variations of the thermodynamic state in many cases.

Note that in general if we have $N$ charged species then there are $N$ independent $V_i$ values, but only $N-1$ independent bulk concentrations, because the bulk has to (generally) be charge neutral. That extra $V_i$ degree of freedom looks redundant, and as far as the bulk goes it is; but it is not truly redundant, because it carries the electrical state of the body. If we raise all of one body's $V_i$ together, $V_i \rightarrow V_i + \delta$, relative to its surroundings, the body is in a genuinely distinct state: we have charged it, though usually with an inconsequential amount of charge.^[7] The bulk stays neutral, so that added charge sits on the surface, but the electrical state has really changed. Shifting everything by the same $\delta$, on the other hand, changes nothing physical, only our bookkeeping. This brings us to our next point...

$V_i$ floats, and $V_i - V_j$ carries a convention

Two last subtleties round out the picture; here is the short version.

We have in fact already met the first of these in the figure just above. The whole set of $V_i$ shares one global float: raise every $V_i$ (along with $V^\circ_i$, $\phi_{\mathrm{vac}}$, and the rest) by the same amount, and nothing observable changes, exactly as the global float slider does. It is a single gauge degree of freedom for the entire universe, so there is no meaningful absolute voltage; only differences carry meaning. (The thermodynamic origin of this float is derived in Understanding electrochemical potential.)

While I could appoint a firm reference to nail this down, I find it actually advantageous to work reference-free.^[8] To actually draw a diagram we of course pin the zero somewhere; I put the negative electrode at $0$, as is usual in electronics.

Comparing different species, $V_i - V_j$, inherits a chemical-potential convention: a globally-consistent but somewhat arbitrary offset, set by how we reference each element. Comparing the same species across places, $V_i(x) - V_i(y)$, has no such ambiguity; it is always physical, even between different solvents. (This is the same fact we just met when measuring ions: a matched pair of electrodes cleanly reads exactly these same-ion differences.) The specific convention we adopt is introduced in the next topic, which opens on exactly this point: once chemical reactions enter, each $V_i - V_j$ offset takes a concrete value, set by how we reference the neutral elements.

Takeaways

So, we've seen that this electrochemical species voltage $V_i$ is no stranger to us, and in fact it just rigorously generalizes the familiar notion of electronic circuit voltage. Going forward, we will see $V_i$ do triple duty: a hands-on voltage, a visual tool, and a quantity with deep thermodynamic and chemical meaning.

As we go along through the next topics, we're going to follow a 'top-down' approach, starting with pure thermodynamics ($V_i$ only), and keeping $V_i$ as our reliable lifeline as we later dive down into microscopic concepts. This is the opposite of the usual 'bottom-up' solid-state and electrochemistry teaching, which starts with idealized microscopic concepts like independent electrons and infinite crystalline solids, or ideal solutes and homogeneous solutions, and gradually adds on complications like nonideality, inhomogeneity, and junctions. In the bottom-up pedagogy, when we finally arrive at the full thermodynamic picture, we are often left clinging to strained and bandaged microscopic concepts, and the unifying power of $\bar\mu_i$ is left unappreciated.

Now that we've established $V_i$ as our core quantity, we're ready to put it to work. Next we turn to equilibrium and the electrochemical reactions that link different charged species (connecting our electronic and ionic circuits) and thereby set $V_i - V_j$.

NEXT TOPIC: Equilibrium