Conductivity of saline solution

Before attempting the lab I was considering in Trying to measure ionic current through small holes, I decided to look at a simpler problem: measuring the resistance of a pair of electrodes in salt water.  As I cited in the earlier post, the resistivity of 1M NaCl should be about 0.12–0.14 Ωm.  Converting that bulk property into a resistance for a pair of electrodes is messy, unless a particularly simple geometry is used. There are two classical simple geometries: opposite faces of a cube and concentric cylinders.  I believe that concentric cylinders are the popular shape for commercial probes, as they are fairly easy to make and there is little stray field outside the probe.

I did not have the materials or energy to try to make a concentric probe, so I decided to make a simpler probe that just has two rods pushed through holes in a piece of plastic.  I got 1/8″ diameter 316L stainless steel welding rod from the local welding-supply store, choosing 316L because it was the most corrosion-resistant of the stainless steels they had (according to the guy behind the counter). The rod comes with a Material Safety Data Sheet, which gives the composition for 17 different stainless steels. The 316L composition is given as

element	composition %
C	<0.03
Cr	18–20
Ni	11–14
Mo	2–3
Mn	1.0–2.5
Si	0.3—0.65
Cu	<0.75
P	<0.3
S	<0.3
Fe	balance

I cut a couple of pieces about 5″ long—the stainless steel was a bit difficult to cut with a hacksaw, so I borrowed my wife’s Dremel tool and used a diamond cutoff wheel.  The noise it makes is too close to the sound of a dentist’s drill to be really comfortable. I then drilled a couple of 1/8″ holes about 2cm apart (2.02 cm, according to my calipers) in a scrap of plastic from an old cutting board (probably made of HDPE).  The rods were a tight fit, so I hammered them through until they protruded about 2.65 cm on one side.  (The length was arbitrary—the distance from the top of the jaw on my drill press vice to the steel base of the vice.)

I put the electrodes in a cup and added enough 1M NaCl solution to come part way up the plastic, so that the electrodes were covered just for the length below the plastic.  This is a lot cruder than putting together a concentric stainless steel tube and rod with epoxy to hold them together and a relief hole, but it was cheap and did not take long to make.

The electrodes in a red plastic beer cup (for that college verisimilitude). The tops of the electrodes stick out of the cup for attaching clip leads.  The brown crud on the bottom has accumulated after a few days of the electrodes being in the salt water. I don’t think that 316L stainless is surgical steel (I think that is type 304).

Stainless steel electrodes, unlike the Ag/AgCl electrodes used in the EKG labs, are polarizable—that means they behave more like capacitors than resistors.  The insulating layer comes from depletion of ions in a thin layer around the stainless steel.  The electrodes are not perfectly polarizable—there is some leakage current through the insulating layer.  The following schematic gives an approximate model for the electrolyte-electrode interface:

Equivalent circuit for a 316 stainless steel electrode in salt water, based on the generic model given in Figure 5.4 of Medical Instrumentation: Application and Design. The half-cell potential of -0.18v (relative to a calomel electrode) is from Galvanic Series in Flowing Sea Water.  Other sources give different half-cell voltages for 316 stainless steel.

Because we are using two identical stainless steel electrodes, the half cell potentials should cancel, and we are left with just a capacitor in parallel with a resistor and a series resistor.  The challenge for the circuits course is to try to determine the resistances and capacitances.

My first thought was to make a simple AC voltage divider, and measure the AC voltage across the unknown impedance and a known resistance (say a 1kΩ resistor).  The magnitude of the load impedance would then be Vload/V100*1kΩ.

AC voltage divider for determining an unknown impedance. If the impedance is a simple resistance, the frequency does not matter, but if there is capacitance or inductance, the magnitude of the voltage will vary with frequency.

By sweeping the frequency, I could get the magnitude of the load impedance as a function of frequency.  With luck, I could fit this to the theoretical curve , or since I’m not measuring phase but only amplitude , where .  Actually, there is no need for all this mess, as gnuplot is perfectly capable of defining j=sqrt(-1.0) and doing absolute values of complex numbers.  You can define Zc(f,c)= 1/(j*2*pi*f*c) and Zpar(z1,z2) = z1*z2/(z1+z2) and try to fit abs(R2+Zpar(R1, Zc(C1,f))), which is much easier to read and to type.

I tried making a series of measurements with a known load first (a 0.56µF capacitor in parallel with a 51kΩ resistor and a 100Ω series resistor), and then recovering the values from fitting the data with gnuplot.  Because my function generator has a strong DC bias, I put a 470µF electrolytic capacitor in series with its output, to provide an AC-coupled output (otherwise my multimeter might have been confused).  I used my Fluke 8060A multimeter to measure the frequency and a cheap Radio Shack multimeter to measure the voltage across the “unknown” load and across the 1kΩ resistor. The smallest voltage I measured was 22.7mV, which is plenty of resolution (the Radio Shack multimeter has trouble with voltages below about 1mV, and resolution is limited to 0.1mV).

The results of fitting the parameters of the electrical model to the measured data. The fitting of R1 (the parallel resistor) is terrible—small changes in initial estimates result in large changes in the final value. One would probably need much lower frequency measurements to determine R1, but the Fluke 8060A doesn’t go down to such low frequencies. I could probably use the Arduino with an interrupt on the rising edge of the square-wave output to get the period (and hence frequency) for lower frequencies, but the Radio Shack multimeter stops giving stable AC voltage readings for too-low frequency as well, so I didn’t bother. Both R2 and C1 were recovered to within the precision of the components used, so this method looks fairly viable for those parameters.

I then tried the same setup using the electrodes in 1M NaCl in place of the known load. The measured voltages across the electrodes got quite small, particularly at the higher frequencies (above about 2 kHz), and so are not very trustworthy, since we are approaching the resolution of the meter. I repeated the measurements with a 100Ω resistor instead of 1kΩ resistor, but the voltages still got quite small at higher frequencies.

The curve fitting for the model is a bit dubious, as I see no evidence of flattening at the higher frequencies (which is what determines R2). Perhaps there needs to be a small capacitance in parallel with the load, to provide the high-frequency rolloff. I only fit with data below 3kHz, as the higher frequency points are dubious, because of the limitations of the Radio Shack multimeter for measuring small voltages. The two sets of measurements were fairly consistent, at least.

I tried doing a 4-parameter fit for the data, and got fairly decent results:

Doing a 4-parameter fit for (C2 || (R2+ (C1 || R1)) gives a fairly decent fit to the data. There are still problems with the resolution of the measurements at the higher frequencies.

I tried making the same measurements again, this time using a known impedance on the bottom of the voltage divider consisting of a 4.7µF capacitor in parallel with a 24kΩ resistor. The fit is quite similar, though different starting points for the optimization result in slightly different values for the resistors.

Fitting a model of the form (C2 || (R2+ (C1 || R1))) to the data collected from an AC voltage divider whose lower branch was (4.7 µF || 24kΩ) gives similar results to the voltage divider with a purely resistive lower branch. I could go up to about 7kHz with this circuit, but beyond that the Radio Shack multimeter gave fluctuating voltage readings. I used more data points and got a better fit with the R||C lower branch, so I think I trust the results here a bit more than the earlier ones,

Some problems I see with this lab

It takes a lot of data to fit a 4-parameter model, and I’ve not found a good model with fewer parameters.  It may be asking a lot of the students to fit a 4-parameter model.

Students will need to know how to use a good model-fitting program.  Gnuplot is nice, but installing it will be too painful for those with Macs (unless the gnuplot community has finally gotten around to making a Mac installer that doesn’t require gigabytes of downloads).  Whatever curve-fitting program we use, I’d probably have to scaffold nearly the whole fitting process for them, as few students have had any experience with fitting parameters of models.  It would be a good thing for them to learn, though perhaps best with a simpler model. It might also be possible for them to take the frequency response and use some sort of filter-design software to fit the model.  That might be a bit tricky to set up, though, and would not give them a general-purpose method for fitting models.

The students will have the advantage of using better equipment than I have at home, so their function generator will report frequency directly and their multimeter will probably be able to handle higher frequency AC input.  Whether it will also handle very low frequency inputs is unclear.

My original idea, to have students make salinity measurements using electrodes like these, may not be so great, as the value for R2, which would be expected to vary inversely with the salinity, is hard to recover from the measurements with any accuracy.  Perhaps the capacitance (which is more easily estimated) varies with salinity also.  Maybe I need to do some experiments on that also, trying 0.1M and 0.01M NaCl as well.

A non-polarizing electrode would almost certainly be better, so perhaps I should get some fine silver wire, and either electroplate on some AgCl or treat it with chlorine bleach to form AgCl chemically.  It might be worthwhile to have two sets of electrodes, both stainless steel and Ag/AgCl, for the students to measure, so they can see the difference between polarizing and nonpolarizing electrodes.  If we do use Ag/AgCl electrodes, it might be a good idea to get some sintered ones, rather than coating silver wires.  The E215P from Invivometric looks suitable, but costs $19.40 per electrode.  I’m not sure that they are enough better than having the students coat silver wires to justify the price (and learning to make Ag/AgCl electrodes could be a useful skill for projects in other labs).