The mysterious case of the CPE

In my previous article I explained how a 'convolution integral' can be used to simulate a constant phase element (abbreviation 'CPE', symbol 'Q') in the time domain. This is useful, but there is much more to this simple element than meets the eye. Specifically, when combined with a series resistance to form a very basic RQ network, it can produce two separate time constants. This counter-intuitive behaviour is the explanation for a baffling electrochemical illusion, one that I think would be sufficient to confuse many experienced researchers in this field.

For the full explanation, please see the following paper recently published in the International Journal of Hydrogen Energy:

Open access (valid until 2nd October 2020): https://authors.elsevier.com/c/1bZf91HxM4spH~

Restricted access: https://doi.org/10.1016/j.ijhydene.2020.06.029

EIS results for TiN coated 316SS electrode
EIS results for TiN coated 316SS electrode
Cyclic Voltammetry results for TiN coated 316SS electrode
Cyclic Voltammetry results for TiN coated 316SS electrode

The above two figures show the Electrochemical Impedance Spectroscopy (EIS) and Cyclic Voltammetry (CV) results for the same TiN-coated 316SS electrode in 0.5 M NaOH. These measurements are taken around open-circuit potential. According to my previous argument, these different methods will be measuring exactly the same impedance, and will therefore produce exactly the same results, but they don't.

To prove this, it's instructive to perform a best-fit of a specific electrical network to the empirical data. In the frequency-domain (EIS) this can be performed with an RQ network, and in the time-domain (CV) this can be performed with an RCR network (resistor and capacitor in parallel, in series with a resistor). The networks are different because (at the time) I could not perform a best-fit of an RQ network to CV data. I shall describe how to do this in a future post, but it's not actually relevant to this argument, for which an RCR network is sufficiently close.

As can be seen, the 'CV RCR' best-fit produces a poor match to the EIS results (the dashed orange line), and similarly the 'EIS RQ' best-fit produces a poor match to the CV results (the dotted green line). It therefore looks like the two analytical methods are in fact measuring a totally different impedance, and thus producing totally incompatible results. This is best illustrated on a Bode plot, as shown below.

Bode plot representation of the EIS results above
Bode plot representation of the EIS results above
Size of 'resistance anomaly' observed across a varied set of electrodes.
Size of 'resistance anomaly' observed across a varied set of electrodes.

The Bode plot shows that the observed impedance at high frequency of the 'EIS RQ' and 'CV RCR' best-fit networks disagrees by more than two orders of magnitude, a disagreement which I subsequently dubbed the 'resistance anomaly'. At this point it would be tempting to conclude that CV just cannot be used to measure impedance, but I don't like an unsolved mystery. I therefore plotted the anomaly across a varied set of electrodes, and was astonished to see that it followed a straight line (see right hand figure above).

Even more suspiciously, the slope of the line of best fit was 1:1. It seems the size of the anomaly is inversely proportional to the roughness factor of the electrode, which in turn is proportional to its double-layer capacitance. Was it possible that we had discovered a new form of resistance, one that no-one else had noticed? No. It turns out that it was all an illusion.

Later CV results for the TiN-coated 316SS electrode
Later CV results for the TiN-coated 316SS electrode
Close up view of bi-exponential CV response, together with the best-fit simulation
Close up view of bi-exponential CV response, together with the best-fit simulation

The breakthrough came when closer inspection of some of the later CV measurements for the same TiN coated 316SS electrode revealed that the electrode was exhibiting bi-exponential behaviour. This is as shown in the left-hand figure above, and in greater detail on the right. The word 'bi-exponential' means that the electrode is simultanesouly exhibiting two time-constants. The first, faster time-constant is associated with the rapid change from 0 to 10 micro-amps, and the second, slower time-constant with the more curved reponse thereafter. This is exactly the same behaviour that is observed (under the right conditions) with the simulations of a single RQ network.

To illustrate this, the curve-fitting technique used to fit an RCR network to the CV results was used on the transient results produced by simulating an RQ network, as shown below.

Apparent time-constant of RQ network whilst varying R
Apparent time-constant of RQ network whilst varying R
Apparent time-constant of an RQ network whilst varying alpha
Apparent time-constant of an RQ network whilst varying alpha

The figure on the left shows the 'apparent time constant' of the RQ network as R is varied through four orders of magnitude. This time-constant is measured by curve-fitting, and (quite understandably) is only able to measure the larger of the two time-constants produced by the RQ simulation, since this is by far the more dominant feature. Thus, over a wide range, the apparent time-constant is largely invariant. Crucially, this range includes the typical values of series electrolyte resistance seen in electrochemistry experiments.

The figure on the right shows how this invariant behaviour comes on extremely quickly as the value of alpha (the degree to which the double-layer capacitance exhibits non-ideal behaviour) is reduced below 1 (ideal capacitor). Again, the range 0.7 to 0.9 covers the range typically seen in electrochemistry experiments.

Thus the illusion is explained:

  • The cyclic voltammetry measurements of the electrode exhibit the bi-exponential behaviour of the RQ network formed by the double-layer capacitance and the electrolyte resistance.
  • The 'CV RCR' method is only able to lock onto the larger of the two time-constants, because it is the most dominant feature.
  • With typical values of electrolyte resistance and alpha, this time-constant is invariant.
  • Since the time-constant is equal to the product of R and C, this creates the illusion that R (the resistance anomaly) is inversely proportional to C (the double-layer capacitance).

Once this is understood, it can be seen that no such resistance anomaly exists. It is a deduced resistance that explains the leaky behaviour of the constant phase element, but has no physical basis. What is more interesting is that the CPE seems to have a built-in time-constant of its own, and this time-constant is about 0.3 seconds. Why 0.3 seconds? Why not 0.3 minutes, or 0.3 weeks? I'm still trying to think of an explanation for that one.

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