[fsonnichsen]

The common mantra is that when measuring conductivity in solutions the electrodes will build up a polarization layer and disturb the measurement, thus a 4 electrode measurement is used and an alternating current impressed up on the outer electrodes, usually around 1000 hz.

I have been trying to learn more about the dynamics of these layers but my search of the journals and my somewhat limited chemistry library reveal nothing on this issue other than a brief mention. I would expect the theory of Kolrausch et. al. for ion mobility would play into this but I cannot find either theoretical information on the evolution of the layer with time, nor any experimental data.

Does anyone here know of a journal or textbook resource that deliberates on this, especially mathematically?

Thanks
Fritz

[Enthalpy]

Not only for electrolytes. At pretty much any conductivity measurement, including metals, semiconductors... four electrodes are better, because
- one expects uncontrolled processes at the contacts with the electrodes
- the shape of the current is difficult to impose with two electrodes, but is well known near the two central electrodes if using four.

I can't tell in detail what happens at the electrodes with electrolytes, sorry. I suppose a bad mess: radicals making temporary bonds with the surface of the electrode and migrating at the surface until they meet an other radical to form molecules, things like that. It explains why some cells with identical anode and cathode need a minimum voltage to conduct when redox computations don't predict it. Half-cell equations only consider the initial and final compounds, not the intermediates which must be pretty much unknown.

Is there, can there be, a good theory common to all electrolytes and electrodes here? I've no opinion.

Anyway, expecting trouble at the electrodes, the natural choice is four electrodes.

AC avoids the buildup of desequilibria. Perhaps more important, even with "identical" electrodes, you often observe a DC voltage of several mV in an electrolyte, supposedly due to impurities on the electrodes or uneven electrolyte concentration. Measuring in DC cancels this off. Not only for conductivity measurements: electromagnetic meters for boat speed do it too. A setup using induction would have the same advantages, say with a toroidal tank, but for dilute electrolytes the effect of the conductivity is small.

My guess is that 1kHz in uncritical, it just makes electronics simpler. A much higher frequency would reveal other effects that ion movements, and in dilute electrolytes, the permittivity may overshadow the conductivity. For very pure water, 1kHz is already 100* too high.

[fsonnichsen]

All excellent points! Regarding "I suppose a bad mess:.",  I suppose this is why a good analysis of polarization effects is not well reported. I think the closest one might come to finding something is with regards to layers built up on nice spherical microparticles in colloidal suspension.
   My original question was rooted in the "best" oscillator frequency for this type of work. On the one hand one must "out run" the traveling ions to prevent their accumulation at the electrodes. At the other extreme one does not want the device to become a permittivity measuring device, subject to the capacitance of the system. You alluded to this. I have made the point with my colleagues here that there is  a rather vague line between various types of chemical potentiometry, impedance spectrometry, and the art of measuring conductivity--all of which fold into your responses.
   I have dabbled with both capacitance based devices and torroidal/inductive methods in the lab. The latter in particular works well at ocean conductivity levels (~56,000uS) and is offered by some commercial vendors for marine use- but it falls apart in lakes and ponds at around 1000uS. 
  In the final analysis I have designed two circuits, one with a constant current source, and the other supplying a constant voltage. I will compare them operating over a set of very small "printed" platinum electrodes to see what produces the best results. I think best case is the uncomfortable solution of having 2 sensors, one 4-pole, the other inductive.

Thanks!
Fritz

[Enthalpy]

I'm confident that you can use either a four-electrodes or an induction setup over a wide range of conductivity. The limits of the meters you tested must rather result from their intended range, which the designer didn't bother to widen, or from the skills of the designer, since this needs both electromagnetism and analog electronics.

For the inductive measure:

Observe only the losses created by water conductivity. At 1000µS/m, 1% accuracy accepts j100µS/m from permittivity, and this effect is easily computed away afterwards. ε_r=80 allows F=22kHz, for which ferrite pots are excellent and let make coils with Q>100.

Build a resonant LC circuit at 22kHz, excite it and measure the decay time, or measure the voltage amplification when injecting a permanent excitation through a small capacitor. I take a water toroid of arbitrary h=20mm R_i=10mm R_o=15mm: at 1000µS/m its resistance is 785kΩ. If loading with the water sample shall drop its Q to 50, the necessary reactance is j16kΩ obtained at 22kHz by 120mH and 470pF, and the coil's equivalent series loss resistance must be << 320Ω while the parallel excitation and measurement circuit must apply >> 785kΩ.

Using an RM14 gapped for 1µH/turn² (warning, this is not correct nor consistent, just for a quick check)
https://en.tdk.eu/download/519704/069c210d0363d7b4682d9ff22c2ba503/ferrites-and-accessories-db-130501.pdf
it takes 346 turns which, if Litz wire occupies 30% of 20mm², have 28Ω resistance << 320Ω.
The N41 material offers Q=100 at 22kHz without gap, so the gap dividing the inductance by 6.8 brings Q=680 which is also >> 50. For accuracy better than 10% you must compare the Q loaded with water and unloaded.
The Q margin should allow to iterate the dimension design into something consistent.

For the resistive four-electrode measure, which makes simpler electronics:

Platinum isn't a must for water... Thin gold can be deposited on the copper of a printed circuit but is weak against abrasion. I guess graphite would do it too, and nickel as well - looks better than graphite, easy to clean, can be deposited by any subcontractor. Nickel is non-magnetic if its phosphorus content is >11% (or 15%?), which can be better at high frequencies due to the skin effect.

You could switch among a set of constant currents, or build a feedback: inject and measure the current needed to develop 100mV between the central electrodes.

I don't see a fundamental limit to the resistivity range here. It's more a question of design choices.

You might consider a capacitive measure too.

Build a capacitor whose dielectric is the water. A coaxial shape is less sensitive to EM perturbations and geometric tolerances. Operate at a frequency where the expected water is capacitive and the conductivity introduces losses, for instance 2MHz.

Build a resonant LC circuit for 2MHz, feed it through a small capacitor, measure the voltage amplification, deduce the losses and the conductivity.

I believe to have data about how constant water's conductivity is up to a few GHz, but didn't check it.

The capacitive design seems much more convenient than the inductive one: drop the capacitor in the water, instead of filling a toroid.

[Enthalpy]

Alas, I botched the inductance necessary for the inductive measure. It would need 120mH at the single loop of water, not at the many-turns copper coil.

I checked other coil shapes and bigger sizes, but all were impossible or cumbersome for 1000µS/m.

So prefer the resistive four-electrodes measure or the capacitive one.