Liquid junctions: the voltage where two solutions meet

Wherever two different solutions touch (the tip of a patch pipette, a salt bridge, the frit of a reference electrode), their ions diffuse into each other at different speeds, and the faster ones pull ahead until the field they leave behind holds them back: a junction potential, from a few millivolts to more than 15, inside every such measurement. It's usually estimated with Henderson's formula; Planck's is exact for a junction held in a steady state; a junction growing freely from first contact is neither. Here it's simulated ion by ion, and the formulas sit alongside it. (More about the model below; the Daniell cell has a salt bridge at work, leaking ion by ion.)

junction sample NaCl frit charge show

About this simulation

Each ion moves by Nernst–Planck, down its own electrochemical potential, in strictly neutral water: the field is whatever keeps every point neutral, so it's the field the junction's faster ions build. The junction potential is φ_right − φ_left between the two solutions' bulk. Its classical estimates differ in what they assume the mixing zone looks like:

Henderson: at every point the junction is the two solutions mixed in some proportion, a "continuous mixture", so every ion's mixing fraction is the same curve (whatever its shape). A closed form, and what JPCalc and most electrophysiology corrections use.
Planck: the steady state of diffusion through a zone held between the two solutions (a frit or a bridge, renewed at both ends). Exact for that junction; driftlet's steady state between two baths is it, checked against a direct numerical (shooting) solution and the stationary Nernst–Planck codes LJPcalc and JLJP.
Free diffusion: a junction growing from first contact. It spreads as √(Dt), but its shape, and so its potential, stays the same: here it's the simulation, grown from a sharp boundary, and for these mixtures it lies between the other two.

For a single salt they all agree (for a 1:1 salt, (t₊ − t₋)V_T ln(c₁/c₂)). For mixtures they differ by tenths of a millivolt to a millivolt: about what the corrections claim to be good to. For the patch pipette the published values are 15.6 mV (JPCalc, Henderson) and 16.05 mV (LJPcalc, Planck), for Figl et al.'s solutions at 20 °C; driftlet gets 15.61 and 16.05, and 15.74 for the free junction. As a free junction's zone reaches whatever holds the two solutions (here the baths, 0.5 mm away, after some tens of seconds), it turns into Planck's: watch the trace move from one to the other. On the way it overshoots. A junction's potential depends only on the path it takes through compositions, not on its width, which is why a free junction's is constant. Show each ion's mixing fraction, (c − c_left)/(c_right − c_left), to see the paths: Henderson assumes every ion's is the same curve. In the free junction they sort by mobility (each ion spreads by its own √(Dt), gluconate's front the narrowest); held between the baths they sort by charge instead (in the steady state each ion's profile is set by the field and its charge, its D setting only its flux). The fast ions reach the baths and rearrange a few times sooner than slow gluconate, so for a while the junction holds a mix of the two orderings that neither junction has, and a potential beyond both. With gluconate as fast as Cl⁻, the overshoot all but vanishes, and so does most of the gap between the three junctions.

The minority ions show the junction's field best, since they hardly shape it and only feel it. In the patch junction, Mg²⁺ (1 mM in the pipette, 2 in the bath) diffuses toward the pipette, and the field across the junction's core (the bath 16 mV above the pipette) drives it the same way, doubly so for its double charge: there drift carries up to five times what diffusion does, taking Mg²⁺ from the bath side of the core to the pipette side, so its rise through the core flattens, and its mixing fraction holds a plateau between two steps, which unwinds as the junction becomes Planck's and the field spreads out. In ZnCl₂ | KCl, the Cl⁻ runs ahead of the slow Zn²⁺, and at the leading edge only K⁺ from the KCl side can keep it neutral: the field draws K⁺ in, and a bulge of it, above its own 3 mM, rides the front (the stacking isotachophoresis puts to use).

Why patch-clampers correct for it. The amplifier is zeroed with the pipette in the bath, where the junction between pipette solution and bath adds its potential, and zeroing absorbs it. In whole-cell recording, once the membrane under the tip is broken through, the pipette solution faces the cytoplasm, which it resembles, and that junction is gone; but the offset that cancelled it stays. So every reading is off by the junction potential: the true membrane potential is the recorded one minus it (here about 16 mV more negative). Strictly it can't be measured at all (a φ difference between two different solutions never can be); in practice it's computed, or measured against a 3 M KCl bridge whose own junctions are taken to be small. Either way it's an estimate, which is why the choice of theory matters.

Salt bridges and frits. A salt bridge of 3 M KCl keeps its own junctions small, because K⁺ and Cl⁻ move almost equally fast and the bridge's salt dominates both junctions. That works until the sample is very dilute, and less well if the bridge's frit carries a surface charge: against a dilute sample the charged pores keep the co-ions out and the frit turns into a permselective membrane, adding a Donnan potential that drifts with the sample (a known trouble with pH and ion-selective electrodes in pure water). Against 0.1 mM NaCl, a frit with 0.1 M of charge in its pore water is almost perfectly cation-selective: its junction falls by nearly 59 mV for each tenfold dilution, a Nernstian membrane, not a liquid junction. That much charge is nanoporous glass (Vycor, with pores of a few nanometres); ceramic frits with micron pores carry far less, and matter only in much purer water. The frit here is 1 mm of strictly neutral porous medium, its charge a density in the pore water, with ions moving a tenth as fast as in water; the sample beside it is stirred, so the frit meets it as it is (unstirred, the bridge's KCl leaking through would fill it: another way bridges contaminate samples).

Data: equivalent conductivities from Lange's Handbook (15th ed., Table 8.32), and for gluconate and HEPES relative mobilities from JPCalc (Barry and Lynch, J. Membr. Biol. 121, 101 (1991)), as tabulated by LJPcalc; the patch solutions are Figl et al.'s (2003).