Understanding electrochemical potential

This is an optional deep dive. The main thread treats $V_i$ as a working tool and tells you, quite deliberately, that you don't need to understand electrochemical potential to use it. That is true, and if you are happy taking $V_i$ at face value you can skip this page entirely.

But some readers, especially those arriving from semiconductor physics or physical chemistry, will want to know what is underneath. And there is a genuine claim worth defending here, so let me separate two things this project asserts, because they are not equally bold:

1. That $V_i$ behaves like a voltage and can be plotted on one shared axis. This is a visualization. It is a good one, but I will not ask you to believe that $V_i$ is "really" a voltage.
2. That the electrochemical potential $\bar\mu_i$ is the genuine chemical potential of a charged species: one indivisible quantity, and not a sum of a "chemical part" and an "electrostatic part" that merely happen to travel together.

The first is a convenience. The second is the claim I actually care about, and it is the one that makes $V_i$ feel inevitable rather than invented. This page is about earning it.

What particles want

How does a mobile particle know where to go? For any single particle that is a hopeless tangle of microscopic forces, but on average the statistical tendency is captured by one number, the chemical potential $\mu_i$ for species $i$. Chemical potential is to particles what temperature is to energy: two bodies in contact equilibrate by exchanging particles until their $\mu_i$ match, exactly as they exchange heat until their temperatures match. Particles drift from high $\mu_i$ to low $\mu_i$ (at a fixed temperature), and the spontaneous flow stops precisely when $\mu_i$ is everywhere equal.

That last sentence is the whole reason chemical potential is worth defining. The equality of $\mu_i$ at equilibrium is a kind of zeroth law for particles, and everything downstream (reactions, phase changes, diffusion, the readings on a voltmeter) is a statement about $\mu_i$ trying to level out.

Thermodynamically, $\mu_i$ is just a derivative of a free energy: take a body with free energy $G$, add one particle of species $i$ while holding temperature, pressure, and every other species fixed, and the cost is

$$\mu_i = \frac{\partial G}{\partial N_i}.$$

Simple enough for uncharged particles. The trouble begins when species $i$ carries charge.

Ions are weird

Charged species behave differently in one specific way: their chemical potential is sensitive to the electrostatic state. If we hold the material composition fixed and merely shift the surrounding electrostatic potential by $\Delta\phi$, the chemical potential of each charged species moves by $z_i F \Delta\phi$, a different amount for each, depending on its charge.

Watch what that does. Here are some hypothetical charged species together in one body, and a slider that only changes the electrostatic offset $\Delta\phi$. The levels scatter every which way: positive species rise, negative species fall, doubly charged ones move twice as fast. It looks like chaos, and it makes comparing $\bar\mu_i$ values across different materials or conditions genuinely awkward.

There is a subtler problem hiding underneath, and it is the crux of everything that follows. Go back to the definition $\mu_i = \partial G / \partial N_i$. To take that derivative for an ion, you must add one ion to the body, and that changes the body's total charge, whatever it was to begin with. The very act of defining a single ion's chemical potential is an act of charging. That much is unavoidable.

(And there is a fair question we are skating past: when you add that ion, where did it come from? That turns out to matter, and we will come back to it. For now, just notice that you cannot even ask for a single ion's $\mu_i$ without disturbing the charge.)

So a single-ion chemical potential is inescapably a charged-body quantity. It cannot avoid the electrostatics. This is why, for charged species, I write the chemical potential as the electrochemical potential $\bar\mu_i$, with a bar, as a reminder that it carries this electrical sensitivity that an ordinary neutral $\mu_i$ does not.

A natural split, and why it dissolves

The classic way to tame the scattering levels is to subtract off the electrostatic part by hand. Posit an electrostatic potential $\phi$ inside the material, declare that an ion's electrostatic energy is $z_i F \phi$, and define what is left as an internal chemical potential:

$$\mu_{\mathrm{int},i} = \bar\mu_i - z_i F \phi.$$

Now repeat the experiment. The full $\bar\mu_i$ still scatter under $\Delta\phi$, but the $\mu_{\mathrm{int},i}$ sit perfectly still:

This looks like a triumph. The $\mu_{\mathrm{int},i}$ contain ordinary, well-behaved chemistry, and all the annoying electrical motion has been quarantined into one tidy $z_i F \phi$ term. From here it is a short step to the usual textbook stance: that $\bar\mu_i$ and $\mu_{\mathrm{int},i}$ are equally valid descriptions, two sides of one coin, take your pick.

I want to argue that they are not equally valid, and that $\bar\mu_i$ is the real one. Two reasons, an easy one and a deep one.

The easy one is decisive on its own. Ask what nature actually equalizes at equilibrium. When an ion is free to move between two bodies and settles into equilibrium, the quantity that becomes equal across them is $\bar\mu_i$, the full thing, bar and all. Not $\mu_{\mathrm{int},i}$, not $\phi$, not any reshuffling of the two. The zeroth law from the first section, applied to ions, picks out $\bar\mu_i$ and nothing else.^[1] If that settles it for you, skip to the payoff.

The deep reason is that the tidy split was an illusion to begin with. We assumed a clean electrostatic potential $\phi$ and a clean $z_i F \phi$ energy, but nothing the ions and electrons do measures $\phi$.^[2] In a clean model system (a dilute ideal solution, say) you can write down a sensible-looking $\phi$, but it comes from the model, not from thermodynamics, and the moment the system is non-ideal the split becomes only nominal. What survives, untouched, in every case is $\bar\mu_i$ itself. The decomposition is something we add; $\bar\mu_i$ is something that is there.

That is the claim in one line: $\bar\mu_i$ is the real $\mu_i$. The next section is why it holds up even when we are careful.

Charge conservation and the float

We hit a wall back when ions got weird: taking $\partial G / \partial N_i$ for a single ion charges the body. Let me be precise about what that wall is, because it is tempting to overstate it. You can formally violate charge conservation, and the mathematics will hand you a definite number. But it immediately raises a question of well-foundedness, the very question we skated past earlier: where did that charge come from? If you do not answer explicitly, your mathematical setup answers for you (the zero of some potential, a boundary condition at the edge of the problem), and its answer may be nonsense.^[3] Until the books are settled you are holding a toxic quantity, its value part physics and part hidden declaration. And it does clean up, the moment you pay back the charge debt: return the charge somewhere definite, and the declaration cancels out of the total. So restrict to operations that conserve charge, which is simply being explicit about where the charge comes from, and ask which derivatives of $G$ come out clean.

The elementary charge-conserving move is a transfer: take a little charge from species $j$ over there and redeposit it as species $i$ over here, holding the total $z_i F N_i(\text{here}) + z_j F N_j(\text{there})$ fixed. Per unit of charge moved, the energy cost is

$$\begin{aligned} \frac{\partial G}{\partial(\text{charge transferred})} &= \frac{\bar\mu_i(\text{here})}{z_i F} - \frac{\bar\mu_j(\text{there})}{z_j F} \\[2pt] &= V_i(\text{here}) - V_j(\text{there}). \end{aligned}$$

There it is. The transfer is a settled transaction: the charge that appears here is the charge that left there, the debt is paid within the operation itself, and no hidden declaration survives into the answer. (These combinations still feel the real electrostatics inside the system, double layers and all; that is physics they are supposed to feel.) And the combination is exactly a difference of species voltages. So $V_i$ differences are not a visualization gimmick: they are the charge-conserving derivatives of the energy, immune by construction to the gremlins of unaccounted charge. They knit all the $\bar\mu_i$ into one self-consistent web of relationships, with exactly one offset left undetermined, because no charge-conserving operation can ever pin the overall level. That single leftover freedom is the float.^[4]

There is a tempting mistake here. If you only ever look at $N_i$ and $N_j$ at the same place (a local cluster of ions), then internal chemical potentials seem just as good as the floaty electrochemical ones: locally, both are self-consistent. But nothing stops us from transferring charge at a distance, between here and there. The moment we do, every chemical potential across the whole domain is chained to every other by differences like the one above, and there is still only one undetermined offset. Widening our view from one spot to many does not grant each spot its own private float; it locks them together, leaving a single common-mode float for the entire domain. That is the precise difference between $\bar\mu_i$ and $\mu_{\mathrm{int},i}$: $\bar\mu_i$ differences are consistent across the whole domain, while $\mu_{\mathrm{int},i}$ only pretends to a per-spot freedom that long-range charge transfer forbids.

Why a band diagram works

The real content, we just found, is the web of $V_i$ differences, with a single common float left free. Now put space back in: let the voltages vary with position across a real device. The float becomes one slider, and watching it is the whole payoff.

The two panels share a position axis and a single float slider. The top is an ordinary energy band diagram of the $\bar\mu_i$; the bottom is the ESBD of the same physics, $V_i = \bar\mu_i / (z_i F)$. Drag the float: up top each $\bar\mu_i$ moves by $z_i F$ times the float, so the levels scatter apart by charge; down below every $V_i$ glides by the same amount, in unison. The relative shapes and gaps stay rigid in both.

That property (one global vertical freedom, with every relative level fixed and comparable across the whole domain) is not an analogy to a band diagram. It is what a band diagram is. A semiconductor band diagram works for exactly this reason: the whole thing can float, but inside it the Fermi levels and band edges keep rigid relationships you can read off and trust. So the question "is $\bar\mu_i$ a real, single thing?" turns out to be the same question as "do band diagrams mean anything?" In both cases the answer is yes.

That is the foundation under species voltage: $V_i$ is natural because $\bar\mu_i$ is real, and a $V_i$ landscape is a band diagram because both are pictures of one globally consistent, gauge-free quantity.

A closing attitude, rather than a result: by now you can see that the urge to nail down an absolute reference never gets satisfied and never needs to be. A material is too early to crown a zero; a disconnected device is too early; a finished circuit is a reasonable place to set $V_{\mathrm{e}^-} = 0$ at ground if you like, but you have also learned that you can keep deferring, and that joining two circuits just means choosing how their grounds relate. Hold off all the way to $\phi_\infty$ at the edge of the universe and the crowning buys you nothing you didn't already have. The float is not a loose end to be tied off. It is the freedom that lets diagrams compose.

What we can physically do with these voltages, and where each species' zero really sits (an open question even for neutral species), are the threads the rest of this chapter picks up, expanding species voltage much as this page expanded $\bar\mu_i$. First, reaching any $V_i$ carries $V_i$ from a bare rescaling of $\bar\mu_i$ to something we can get at in the lab: what it takes to measure an arbitrary species' voltage, and why electrons are the easy case. Then offsets galore makes the conventions concrete, touring every arbitrary offset in the framework and sorting the ones that move the picture from the ones that change nothing.

NEXT TOPIC: Reaching any $V_i$