# Constant Phase Elements

In electrochemistry it is common to come across behaviour that can only be modelled using a constant phase element (CPE). For example, a measurement taken using electroimpedance spectoscopy (EIS) often produces a flattened or depressed semi-circle on a Nyquist plot. This indicates that a form of capacitance is present (coupled with a resistance), but it is non-ideal. Were it ideal, it would produce a perfect semi-circle, perfectly centred on the axis, but in electrochemistry, to my knowledge this never happens.

There are several explanations for this, such as electrode porosity, non-uniform current density and reactant concentration gradients, and this reflects the fact that several causes can produce the same effect. Your situation might be one of them, or a mixture of all of them. However, pragmatism dictates that explanations are ultimately less important than accurate predictions. A famous example of this arises with quantum physics, which raises fundamental questions about the nature of reality, and prompted David Mermin to tell his colleagues to just “shut up and calculate”.

The mathematics of the response of a CPE in the frequency domain are relatively simple. It’s just like a capacitor, but as its name implies, it maintains a constant phase angle which is less than 90 degrees. With my background in electronics, I still find this behaviour mysterious. I try to imagine the products and reactants in the electrolyte sloshing back and forth in response to the sinusoidal driving force. The lower the frequency, the more time they get, and the further they can move. Thus, instead of the phase angle (the time delay between driving force and product/reactant response) disappearing at low frequencies, it is maintained. In the case of a porous electrode, the more time the products are given to get down into the pores and the reactants to get out, and thus the larger the amount of the pore that is in use. That explanation may not be quite right, but I suspect that a better one involves wrestling with some pretty difficult mental imagery.

However, EIS has a limitation, and that limitation is typically about 0.1 Hz. Below this frequency, not only does the acquisition time get very long, but in my hand-waving fashion, the products and reactants have enough time to slosh entirely from one end of your electrochemical cell to the other. There is also time for diffusion to be utterly disrupted by natural convection. It may be possible with a body of electrolyte the size of a swimming pool to take EIS measurements at lower frequencies, but who wants to build an electrochemical cell that big?

What seems to be far less common in the literature is much discussion about the response of a CPE in the time domain. This seems strange to me, because EIS is not the only method to measure impedance, even though I’ve heard it referred to as ‘impedimetry’, as if it has sole claim to this ability. Cyclic voltammetry (CV) also measures impedance, and I simply will not accept any assertion that the impedance the two methods measure is somehow different. Impedance is impedance, and if the two methods disagree with each other (which they do) then it is simply not valid to declare one of them right and the other one wrong.

However, it is not possible to convert a transient response measured using CV into an equivalent circuit containing a CPE, unless the transient response of a CPE is known. Fortunately it is, and has been calculated using Laplace transforms, producing a convolution integral (Athanasiou et al. DOI: 10.1002/cta.2474):

${V}_{\mathrm{CPE}}\mathrm{\left(t\right)}=\frac{1}{Q\mathrm{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\mathrm{\left(t}–\mathrm{u\right)}}^{\alpha –1}\mathrm{I\left(u\right)}\mathrm{du}$

where Q and α are the magnitude and phase-argument of the CPE, t is time, u is an auxilliary variable, and Γ is the gamma function. This is computationally expensive, and even more inconveniently expresses the voltage as a function of the current. Since cyclic voltammetry involves controlling the voltage and measuring the current, this is the wrong way round.

Nevertheless, the Euler method can be used to solve this numerically, and I have written some PHP code to do this here. This code is able to simulate the response of an RQ network (resistor and CPE in series) to a voltage ramp, and produces results like those shown below: