How to read an ESBD

An electrochemical species band diagram packs several different kinds of line and marker onto a single voltage axis. This page is the field guide to all of them: every creature you will meet as we go. There is no need to memorize it: every figure in the book carries a small link back here, so whenever a diagram starts to look like a thicket, one click returns you to the key.

The axis

Up is voltage, in volts. That sounds unremarkable until you recall that the energy band diagrams of semiconductor physics plot energy upward, and for electrons energy runs opposite to voltage. So an ESBD is an energy band diagram flipped top to bottom: the conduction band sits at the bottom, the valence band at the top (more on that in semiconductors). The horizontal axis is usually position across a device, though now and then it is a radial coordinate or just an abstract "here versus there".

The whole diagram can be slid up or down freely: only vertical differences carry meaning. That freedom is deliberate, and we lean on it throughout.

The lines

Species voltage $V_i$ — a solid, species-coloured line, one for each mobile carrier (an electron, a lithium ion, a chloride ion, and so on). It is the star of the show. A flat $V_i$ usually means that carrier is in equilibrium and carries no net current, while a sloping $V_i$ usually means current is flowing and dissipating; the real link between slope and current is the conductivity, a tie that loosens in several entertaining ways.^[1] Several $V_i$ can run through the same place at once. (→ species voltage)

Standard state $V^\circ_i$ — a thinner line of the same colour, the carrier's "band edge" or reference level. A carrier's $V_i$ floats away from its $V^\circ_i$ by a distance set by the logarithm of concentration, so the gap between the two reads as a concentration. Which side the carrier floats on depends on the sign of its charge, so a faint hatching marks the rung's concentrated side: a carrier below standard concentration sits in the clear, and a carrier line entering the hatching means the concentration has passed $1~\mathrm{mol/L}$, the same visual as a Fermi level entering a band. The texture even carries the charge itself, being drawn compressed by the diagram's own $1/(z_i F)$ scaling: stripes run at slope $1/z_i$, so an anion's stripes mirror over and a divalent ion's flatten to half height. The $V^\circ_i$ lines shift together as one rigid ladder. (→ solutions, semiconductors)

Implied redox level $V_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red})$ — a dashed, electron-coloured line drawn inside a solution: the level where one of the solution's redox couples "wants" the electrons to sit, even with no electrode present. A solution out of equilibrium can show several at once. (→ half-reactions)

Standard redox level $V^\circ_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red})$ — the standard-state sibling of the line above, drawn thin and dashed: a redox couple's level at unit activities, floating along with the rest of the $V^\circ$ ladder. It is the "standard electrode potential" $E^\circ$, before any reference has been subtracted off. (→ half-reactions)

Vacuum level $\phi_{\mathrm{vac}}$ — a dashed line wearing the electron's colour, because by convention it is the electron's vacuum rest level. Drawn only out in a vacuum or insulator, stepping down from a metal's $V_{\mathrm{e}^-}$ by the work function; we never thread it through the bulk of a conductor. (In the master figure above, and in most figures, that step is compressed: a real $\Phi/e$ is $4$–$5\ \mathrm{V}$ and would fall clean off the plot. It always steps down, though.) (→ capacitance, vacuum levels)

The markers

The ⇌ symbol marks a reaction. With two species, this fixes the step $V_i - V_j$ between their lines (including the step from $V_{\mathrm{e}^-}$ to an ion's $V_i$ at an electrode). Its ends always sit on carrier levels (solid $V_i$, or a dashed implied level like $V_{\mathrm{e}^-}(\mathrm{Ox}/\mathrm{Red})$), never on a $V^\circ$ rung: a reaction ties what the carriers actually feel. The same reaction driven (running net, out of equilibrium) wears its overpotential openly: a species-coloured dot marks where the tied level would sit at equilibrium (its ghost), and a thick coloured stub runs from the ghost to the actual level, tipped with an arrow that always points down, the direction of the conventional current falling through the reaction. Which rail the stub touches tells you where that current lands or departs. When the reaction is a plain same-carrier hop (an electrode exchanging electrons with a redox couple's implied level, or electron–hole recombination), the ⇌ bubble is dropped and the bare arrow is the marker. At equilibrium the stub shrinks away and only the ⇌ remains. (→ equilibrium, kinetics)

A capacitor symbol (‖) between two lines marks a chemical capacitance: the storage that lets their gap flex as charge is banked, the elastic counterpart to the rigid ⇌. It may bridge a carrier and its own standard state $V^\circ_i$, or two different carriers' rails $V_i$ and $V_j$; those are two portrayals of the same banked charge. (→ capacitance)

An ↕ in a bubble marks a plain vertical distance: a labelled gap between two levels (a work function, a band gap, a Nernst activity term), asserting no reaction and no coupling, just how far apart they sit. Hover it for what the gap is. Cousins you will meet: a circled V is a voltmeter reading the gap it bridges, and a circled Δ names a fixed step defined in the surrounding text.

A small zigzag break (⌇) across a line (as on the generic ion in the master figure above, always accompanied by "per-species offsets" in the axis label) flags a species drawn with a display offset: all of its levels have been shifted together by a constant (hover the mark for the exact value) so that widely separated species can share one readable plot. Everything about the flagged species itself is still to scale; only its raw vertical distance to other species (including any step a ⇌ marker bridges to them) is not. This is less of a cheat than it may sound: the distance between species that share no reaction was set by bookkeeping convention in the first place. (→ solutions)

A slider on a figure lets you move something (a concentration, an applied voltage, the overall offset) and watch the lines respond. Reach for it; the diagrams are built to be played with.

Lost?

Those are the core players. If a later figure ever shows a line or symbol you don't recognise, the link in its corner brings you straight back to this page.

NEXT TOPIC: Solutions