Practical Guide to Large Amplitude Fourier-Transformed Alternating Current Voltammetry—What, How, and Why

Fourier-transformed alternating current voltammetry (FTacV) is a technique utilizing a combination of a periodic (frequently sinusoidal) oscillation superimposed onto a staircase or linear potential ramp. The advanced utilization of a large amplitude sine wave induces substantial nonlinear current responses. Subsequent filter processing (via Fourier-transformation, band selection, followed by inverse Fourier-transformation) generates a series of harmonics in which rapid electron transfer processes may be separated from non-Faradaic and competing electron transfer processes with slower kinetics. Thus, FTacV enables the isolation of current associated with redox processes under experimental conditions that would not generate meaningful data using direct current voltammetry (dcV). In this study, the enhanced experimental sensitivity and selectivity of FTacV versus dcV are illustrated in measurements that (i) separate the Faradaic current from background current contributions, (ii) use a low (5 μM) concentration of analyte (exemplified with ferrocene), and (iii) enable discrimination of the reversible [Ru(NH3)6]3+/2+ electron-transfer process from the irreversible reduction of oxygen under a standard atmosphere, negating the requirement for inert gas conditions. The simple, homebuilt check-cell described ensures that modern instruments can be checked for their ability to perform valid FTacV experiments. Detailed analysis methods and open-source data sets that accompany this work are intended to facilitate other researchers in the integration of FTacV into their everyday electrochemical methodological toolkit.


SCRIPTING & DATA PLOTTING: (i) GENERATING A POTENTIAL INPUT
The method by which the potential-time input is defined will depend on the potentiostat being used.Many manufacturers provide the option to apply a user-defined sequence of time-potential values, and as such we provide the below python code.The input frequency is often selected on the basis of the expected value of the electron-transfer rate constant, as discussed in the main paper.The scan rate should be slower than the time-constant of the frequency for two reasons.If the scan rate is too fast then the potential input doesn't spend enough time in the Faradaic window, reducing the amount of information obtainable from harmonic analysis.In addition, the faster the scan rate, the broader the width of the 0 Hz harmonic (referred to as aperiodic DC component in the main text) in the frequency spectrum; at inappropriate combinations of the scan rate and the frequency the 0 Hz harmonic can overlap with the fundamental harmonic, making analysis challenging.Ac_amplitude=0.15# Amplitude of the sine wave in volts.Check with the manufacturer as to how they define this value

SCRIPTING & DATA PLOTTING: (ii) CHECK-CELL VALIDATIONS
In the main paper we show data in Figure 5 that compares the consistency with which a sinusoidal oscillation is achieved throughout an FTacV experiment by the three different instruments tested.We extract the phase of each sinusoid in the applied potential-time dataset using a lock-in amplifier method to determine the phase of each sinusoid in a potential input and the corresponding current output, as detailed below.

SCRIPTING & DATA PLOTTING: (iii) PLOTTING TIME-POTENTIAL-CURRENT
This is trivial to do in any plotting software, but it is the opportunity for some important checks: • That the potential has been applied properly, for the correct period of time, over the correct potential range and with the correct amplitude.
• That there is not significant noise in the applied potential.
• Usually (unless operating at very low concentrations/high resistances/high background currents) it will be possible to observe a Faradaic peak at the midpoint potential in the total experimental current.

SCRIPTING & DATA PLOTTING: (iv) APPLYING WINDOWING FUNCTIONS
When plotting harmonics, "ringing" artefacts of large amplitude at the start and end of the time series can be observed, as shown in Figure S4.These artefacts can be large enough to obscure the actual harmonic signal, and arise from aperiodicity in the current signal.Such ringing effects can be removed by applying the Hanning window, as shown in Figure S4, although care must be taken; the window suppresses the magnitude of the current linearly, such that the beginning and end of the signal are set

SCRIPTING & DATA PLOTTING: (v) PLOTTING THE FOURIER SPECTRUM
Investigating the Fourier spectrum of the total current is another useful check of the validity of the FTacV experiment.Visually inspecting the Fourier spectrum allows an experimentalist to verify the presence of harmonics, determine the true input frequency applied by the potentiostat (by checking the frequency at which the fundamental harmonic occurs in the Fourier spectrum), and assess at which harmonic number the S9 Fourier spectrum peak can be meaningfully separated from the baseline noise level in the spectrum.To plot in python:

SCRIPTING & DATA PLOTTING: (vi) APPLYING FILTERS TO EXTRACT FOURIER-DOMAIN HARMONICS
In FTacV analysis, we wish to ultimately obtain a signal that is the same length as the number of timesteps/applied potential values.To do so, we create an m*n array of complex zeros, where m is the number of harmonics we want to plot, and n is the length of the input current.We then select the appropriate "box" around each desired harmonic (at this point you may wish to apply more complex filters) and assign this to the appropriate position in the m*n array.

OPTIMISING THE NYQUIST FREQUENCY OF A FTacV EXPERIMENT
As detailed in the main paper, both the frequency of the maximum recoverable harmonic and the signal-to-noise resolution of an FTacV experiment is limited by the Nyquist frequency, which is half the sampling frequency.Amongst many other limitations which may impact an FTacV experiment, an instrument will have a maximum sampling frequency, and a maximum number of data points which it can S11 store in memory.The total number of datapoints required for an FTACV experiment is given by the sampling frequency multiplied by the experiment time; in FTacV the experiment time is determined by the time taken to complete the DC potential cycle.
To illustrate this and assist researchers in considering the maximum harmonic they can theoretically access according to the Nyquist frequency, Figure S5 shows the number of points required to access the 30th harmonic as a function of input frequency and scan rate for a theoretical experiment measured over a linear potential window of 1 V.The number of points required for a particular set of parameters can be easily calculated using the code for potential generation found above.As noted above, we emphasise that caution must be taken if using a fast a scan rate in FTacV; it is important that the Fourier spectrum is always carefully analysed to ensure that a combination of fast linear scan rate and high frequency does not cause an overlap in current contributions from the 0th/aperiodic DC component and the 1st/fundamental harmonic.

Figure S1 .
Figure S1.(A) Circuit diagram and (B) photograph of the check-cell used in this study.Cables are used to connect the instruments tested to the check-cell in either a series RCideal circuit or a series RCnon-ideal circuit, as described in the main paper.
In this section, the methods used to proceed through the workflow of an FTacV experiment are described; the overall process is summarised in FigureS2.

Figure S2 .
Figure S2.A schematic of the overall workflow needed to conduct an FTacV experiment and visualise the data in the manner shown in the main paper.
5000 #Delta T in seconds.The minimum value of this will be defined by the instrument desired_Hz=10 #Input frequency in Hz desired_scan_rate=22.5e-3 # Scan rate in V/s E_reverse=0.5 # Switching potential in V E_start=-0.2 # Start potential in V phase=0 # Phase of the input sinusoid strictly_periodic=True # For Fourier analysis, you may want to enforce periodicity of your sine wave.

Figure S3 .
Figure S3.Plots showing the apparent variation in the input potential phase as a function of time.Experiments were conducted on the ideal check cell circuit.The FTacV parameters used are different for the two instruments, since they were selected to elicit very large apparent phase shifting.(A) Data obtained from the Gamry instrument using a voltage window of -0.5 -0.5 V, an input frequency of 120 Hz and an amplitude of 100 mV.(B) Data obtained from the Ivium instrument using a voltage window of 0 -1 V, an input frequency of 72 Hz and an amplitude of 100 mV.

ACS
Figure S4.Effect of applying a Hanning window (orange line) versus no Hanning window (blue line) for harmonics 0-5 (top to bottom).The Hanning window is used to reduce the size of the ringing artefacts at the beginning and end of the harmonics, by suppressing the start and end of the total current to zero.However, it also has the effect of suppressing harmonic magnitudes, and introducing asymmetry into the harmonic lobes.The harmonics are obtained from the data plotted in Figure1in the main paper.

Figure S5 .
Figure S5.Number of points required to access the 30th harmonic as a function of input frequency and scan rate for a theoretical experiment measured over a linear voltage window of 1 V.The maximum number of points collectable by the Gamry potentiostat (1 million) is shown as a horizontal line.