## Comparison of Liquid Conductivity Probes with the MFIA

In the previous blog post, I have described how to use the MFIA impedance analyzer to measure the electrochemical impedance spectrum (EIS) of a liquid sample, and to derive the conductivity with EIS modeling. The post presented a 5-element equivalent circuit model, which required a relatively deep understanding of electrochemistry. However, it should be mentioned that, even 5 elements may not be enough to fully illustrate the physics. For instance, I did not include a series inductance, which is as commonly found due to the parasitics of the connecting cables, not because it does not exist, but because the inductive effect is barely visible, and forcing a fit will mathematically generate an unreliable result. For simplicity, this time we just treat the liquid sample as a black box. We will focus on the EIS measured with different third-party conductivity probes. We find that **a probe with a lower parasitic impedance is more stable against conductivity changes and helps also in the calibration process.**

**Experimental**

In this work, we choose 2 different conductivity probes, Probe A (a high-end one), and Probe B (a mid-range one), which can be connected to the MFIA in a 4-terminal configuration (details are described in the previous blog post). A conductivity probe is often made of a parallel plate capacitor. For a non-ideal capacitor, a 2-element circuit model, Rp||Cp, is usually a good representation, and it can be conveniently chosen in the impedance module in the LabOne software. Naturally, this oversimplification may only work within a narrow frequency range. So we will skip the low-frequency section which is limited by the electrochemical double-layer capacitance (EDLC), and start our EIS experiment from 1 Hz using the LabOne Sweeper module. The choice of the frequency range is arbitrary and may vary on different samples. But without the time-consuming measurements at low frequencies, the entire sweep lasts just a minute or so.

**EIS of tap water**

We measure the same tap water sample with two different conductivity probes (Probe A and Probe B), and can compare the Bode plots of impedance amplitude and phase as shown in Fig. 1. We can find the frequency of the phase peak, and then use the corresponding amplitude as the resistance of the sample. This method is equivalent to finding the ‘charge-transfer’ resistance from the intersection on the X-axis in the Nyquist plot. We can see that Probe A (in blue) has a characteristic frequency at ~1 kHz with a resistance of 3 kOhm. Probe B (in red) peaks at around 2 kHz and has a resistance at a slightly higher value, 3.2 kOhm. In both probes, the phase peaks slightly below 0 deg, with Probe A being closer. At this frequency, one may assume that the capacitive probe becomes nearly resistive. By further restricting the frequency to 1 kHz with a proper probe design, one can possibly run a simple, fixed-frequency resistance measurement with just an AC Ohmmeter. But we will see next that this simple approach may bear a significant error when the conductivity changes.

*Fig. 1 Bode plot showing the impedance amplitude and phase of a tap water sample, measured with Probe A (in blue) and Probe B (in red). Click the figure zoom.*

In addition, we can also look at the derived Cp extracted by the built-in model in LabOne, as shown in Fig. 2. If Rp||Cp truly represents (dominates) the impedance of the entire circuit, then the derived Cp value must be (close to) a constant, independent of frequency. Probe A (in blue) has such a flat band above 10 kHz, with 59 pF in Cp. Probe B (in red) shows a less flat behavior, with a thrice large Cp at 165 pF.

*Fig. 2 LabOne-derived Cp of a tap water sample, measured with Probe A (in blue) and Probe B (in red). Click the figure zoom.*

Regardless of the accuracy (will be discussed later), Probe A with a lower parasitic Cp and a wider flat band is better than Probe B, as it is more stable when the conductivity (salinity) changes.

**EIS of saline water**

To see a larger difference between the two probes, we add a small amount of salt into the tap water sample and repeat the measurements at the same testing condition. As shown in Fig. 3, the phase changes drastically in addition to the impedance amplitude. Particularly with Probe A (blue), the phase even exceeds 0 deg and becomes inductive toward the high-frequency end. If we use the same criteria to determine the characteristic frequency, the resistance becomes 137 Ohm for Probe A at 19 kHz and 130 Ohm for Probe B at 190 kHz. If we would stick to the frequency around 1 kHz instead, as in the tap water sample, the result on Probe A remains good (also 137 Ohm). But for Probe B (143 Ohm), the measurement error goes to ~10%. By referring to the conductivity values in the literature, the salt concentration is only about 0.45 wt% (very dilute). So if one works at a higher concentration, the situation will become even worse.

*Fig. 3 Bode plot showing the impedance amplitude and phase of a saline water sample, measured with Probe A (in blue) and Probe B (in red). Click the figure zoom.*

**How to properly compensate for the impedance in liquids?**

As shown in Fig. 1 and Fig. 2, the resistance (inverse conductance) measured from the same tap water sample differs by 200 Ohm (~6.7%) with two different probes. The main reason for this is the accuracy of the cell constant (distance over area of the parallel plates) of the probes, which deviates from the standard at 1/cm. To calibrate for the error, a KCl or phosphate-buffered saline (PBS) calibration solution in a known conductivity can be used. As measurements on the same probe carry the same systematic error in the cell constant, one may just imagine that scaling the conductance is good enough.

Electrical engineers may often hear of short-open-load (S-O-L) calibration, whose equation is:

, where Zdut, Zxm, Zo, Zs, Zsm, Zstd meaning the corrected impedance of the device under test (DUT), the measured impedance of the DUT, the measured impedance at open, the measured impedance at short, the measured impedance of the reference load, and the true value of that load, respectively.

As there are no short or open standards in liquid, we can drop these two terms by setting them to 0 or infinity, respectively. The equation then simplifies to:

This equation is also a proportional scaling. However, keep in mind that the impedance should be scaled, and it can only be reduced into resistance when the phase is (close to) 0 deg. This leads to an immediate question, what if the sample and calibration solution have very different phase responses, reaching 0 deg at different frequencies. For a better understanding, let’s make an analogy to the impedance measurement in solid-state samples, as listed in Table 1 below.

Most requirements are indeed similar in solid- and liquid states. But we should note in the last row that, calibration in liquid works in a narrow frequency range, where the solution impedance is close to resistive (0 deg). To have a more accurate result, it is certainly better to measure both the calibration solution and sample over a suitable frequency range, and carefully determine the characteristic values. With a low-parasitic probe (Probe A, for instance), the characteristic frequencies will be close. Alternatively, if one uses a calibration solution in a similar conductivity, the error will also be smaller. Do take care here, as otherwise, it may result in calibrating at a different frequency (range) from the measurement.

*Table 1: Comparison of calibration requirements in solid and liquid samples. *

**Conclusion**

In contrast to the previous blog post focusing on electrochemistry, this one describes how to measure and interpret liquid conductivity using different conductivity probes. A probe with a lower parasitic impedance is found to be better for such an application, more stable against salinity changes. This knowledge can also be particularly useful in the calibration process.

For further information on EIS measurements, please do not hesitate to get in touch.