Basic transport

$V_i$ hands us transport almost for free: back in species voltage we saw that a carrier simply flows down its own $V_i$ slope, one of the quiet beauties of the picture, and we have even glimpsed it in action already, in a battery's discharge and a driven double layer. A working device is never quite at equilibrium; the moment current flows the $V_i$ lines tilt, and the appeal of the picture is that those tilts are precisely the dissipation. Where the previous topic treated the storage-side response, charge banked against a voltage difference, this one takes up its flow-side twin: current in proportion to a voltage gradient, and everything that one simple law implies once the lines are free to slope.

One driving force per species

The thermodynamic force on a carrier is the gradient of its electrochemical potential, $-\nabla\bar\mu_i$. Dividing by charge, as we always do, the force per unit charge is just

$$-\nabla V_i.$$

That is the whole of what drives a current: each species is pushed down its own $V_i$ slope. On the band diagram the reading could not be more direct. A flat $V_i$ ordinarily carries no net current, the species sitting at equilibrium; a sloping $V_i$ marks a driving force, and the steeper the slope, the harder the dissipation. Resistance is visible as gradient.^[1]

Drift and diffusion are one law

Because the force is $-\nabla V_i$, the current of species $i$ obeys a per-species Ohm's law,

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

with $J_i$ the charge current density and $\sigma_i$ the species conductivity.^[2] This one equation already holds the drift-diffusion machinery that electrochemistry and semiconductor physics usually present as two separate mechanisms. Dividing through by the molar charge $z_i F$ recasts it as the Nernst–Planck equation in its familiar form, a number-current density carrying a drift term and a diffusion term:

$$\begin{aligned} N_i &= \frac{J_i}{z_i F} = -\frac{\sigma_i}{z_i F}\,\nabla V_i \\ &= \underbrace{-\,u_i\, c_i\, \nabla\phi}_{\text{drift}} \;\underbrace{-\;D_i\, \nabla c_i}_{\text{diffusion}}, \end{aligned}$$

where the split uses $\nabla V_i = \nabla\phi + \tfrac{RT}{z_i F}\tfrac{\nabla c_i}{c_i}$ in an ideal homogeneous medium. The two coefficients are the ion's diffusion coefficient $D_i$ and its (signed) electrical mobility $u_i$, and because both descend from the single conductivity they are not independent:

$$\sigma_i = z_i F\, u_i\, c_i = \frac{z_i^2 F^2 D_i\, c_i}{RT}.$$

Reading off the last equality, $u_i$ and $D_i$ are tied by the Einstein relation, $u_i = z_i F D_i/RT$. The Einstein relation is therefore automatic here, a consequence of drift and diffusion being two faces of the one $\sigma_i\nabla V_i$.^[3]

There is a subtlety hiding in the drift term. We assumed $\nabla V^\circ_i = \nabla\phi$, but that holds only inside a uniform medium. Where the medium itself changes, at an interface or through a graded material, the standard state carries its own gradient, and that extra piece is a real driving force with no electrostatic origin: a "quasi-electric field," the reason the textbook drift-diffusion split quietly fails across interfaces.^[4] The inhomogeneities topic takes this up properly, graded densities of states and all. Beyond conduction, currents can also be driven by advection, thermoelectric gradients, and magnetic induction, but plain $-\nabla V_i$ conduction is our concern here.

The "ohmic current" shortcut

Engineering practice often collapses all of this into a single bulk current driven by one electrostatic potential,

$$\begin{aligned} J_{\mathrm{tot}} &= -\sigma\,\nabla\phi, \\ \sigma &= \sum_i \sigma_i. \end{aligned}$$

It is a handy shortcut, and its origin is easy to see: sum the per-species drift currents and assume uniform concentrations, so that every $\nabla V_i$ reduces to the shared $\nabla\phi$. The weakness is that very assumption. As soon as concentration gradients appear, the species' $V_i$ slopes part ways, the diffusion currents start contributing to the charge flow, and $J_{\mathrm{tot}} = -\sigma\nabla\phi$ no longer holds. Tracking the individual $V_i$ keeps us honest in exactly the situations where the lumped law breaks down, and those situations are the next two effects.

Concentration polarization

Many interfaces are selective, letting one carrier through and blocking the rest: a lithium electrode passes $\mathrm{Li}^+$ while the salt anion is turned away, or an electrode reaction consumes one ion and ignores the others. The blocked species cannot leave, so it banks up on one side and is drawn down on the other, building a concentration gradient in front of the interface. The conventional telling then places the blocked ion in a delicate standoff, its drift in the field cancelled point by point by back-diffusion down its own gradient. On the diagram there is no standoff to arrange: a species carrying no current has an exactly flat $V_i$, and everything else follows from that one line. The standard-state ladder bends away from it to track the changing concentration, and the conducting species, tied to the same rigid ladder, picks up a concentration-driven slope on top of its ohmic one, the added "mass-transport" or polarization resistance. Pushed hard enough, the supply of the active ion at the interface runs dry and the current can climb no further, a limit we give its own topic shortly. The usual remedy is a swamping excess of inert supporting electrolyte, which carries the drift and leaves the active ion to move by diffusion alone.^[5]

Liquid junction potentials

Hold two different solutions in lasting contact, through a porous frit or a constricted opening, and a steady transition zone forms between them where the ions interdiffuse. Wherever those ions have unequal mobilities a new tension appears: the nimbler one tries to pull ahead, neutrality forbids any real charge separation, and a steady field builds across the zone that hurries the slow ion and reins in the fast one until their currents come into balance. That field is a diffusion potential, and summed across the junction it is the net liquid junction potential between the two solutions, a small but stubborn voltage that dogs careful electrochemical measurement. Its exact value is convention-dependent: in a nonideal solution it inherits the single-ion activity convention (see non-ideal solutions), so it is only ever as sharp as that convention.

For a single binary salt the gradients (and hence the steps) divide as a ratio between the ions by their transference numbers according to $\partial_x V_+/\partial_x V_- = -t_-/t_+$.^[6] For a 1:1 salt the transference ratio $t_+/t_-$ is simply the mobility ratio $D_+/D_-$ that the figure's slider sweeps. It gets significantly more tricky with three or more ions.^[7]

The classic suppression trick is the salt bridge: join the two solutions through a concentrated solution of an equitransferent salt, usually KCl, whose cation and anion happen to move at nearly equal speeds ($t_+ \approx t_-$). The bridge salt's swamping concentration lets it dominate the interdiffusion at both of its junctions, and with matched mobilities neither end builds much of a step. Note what kind of trick this is: the junction potentials are quietly minimized by a steadily maintained leak of bridge salt into both solutions, a nonequilibrium arrangement rather than any true equilibration, and one whose success also presumes the same solvent on all sides. This is the fine print behind letting a reference electrode act as if it dipped straight into the test solution.

Beyond steady state

Steady state is not the only place these ideas live. Picture a blob of salt left to spread on its own, its profile relaxing into a widening Gaussian. The two ions would each diffuse at their own rate, but the faster one cannot simply leave the slower behind without breaking neutrality, so the same diffusion-potential field we met at the steady-state junction reappears here, transient though the situation is, holding the quick ion back and urging the slow one along. The salt therefore spreads as a single entity, with an effective diffusion coefficient that is neither $D_+$ nor $D_-$ but a blend of the two.^[8]

The coupling reaches past the spreading salt itself. A third, dilute species drifts in that same diffusion-potential field, so a tracer ion or a charged colloid can be carried along by another salt's gradient, even drawn up it, an effect known as diffusiophoresis. The same mechanism runs inside a lithium-ion electrode: there the electrons are so much faster than the $\mathrm{Li}^+$ that it is the electrons who are held back, and the pair migrates as though neutral lithium were diffusing through the material, with no need to name a $\phi$ anywhere inside it.

Two cases of one channel

Concentration polarization and the liquid junction are, in the end, the same steady-state calculation seen from two sides. Take a channel with a fixed set of $V_i$ at one end and, in steady state, a spatially constant current $J_i$ for each species; march across, re-imposing local neutrality at every step so that the $V^\circ_i$ ladder stays slaved to the local composition, and the $V_i$ at the far end follow. Concentration polarization is the case where every current vanishes but one; the liquid junction potential is the open-circuit case, total current zero, with the individual currents set by the fixed end concentrations. In practice, though, we usually fix the voltage instead and let the device settle on the currents, and if we push past the point where the active carrier runs dry, no steady profile can span the channel at all. That breakdown is the current limit, and chasing it down is the climax this chapter builds toward in saturation.

Takeaways

Giving each species its own Ohm's law, $J_i = -\sigma_i\nabla V_i$, folds drift and diffusion back into the single quantity that actually drives them, and turns every transport resistance into a visible slope. The lumped "ohmic" law is just the special case of uniform concentration, while concentration polarization and diffusion potentials are what the per-species picture captures and the lumped one misses.

NEXT TOPIC: Saturation