The term neopolarogram refers to mathematical derivatives of polarograms or cyclic voltammograms that in effect deconvolute diffusion and electrochemical kinetics. This is achieved by analog or digital implementations of

fractional calculus Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D :D f(x) = \frac f(x)\,, and of the integration o ...

.Keith Oldham, Jerome Spanier; ''The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order'' (Dover Books on Mathematics) The implementation of fractional derivative calculations by means of numerical methods is straight forward. The G1- (

Grünwald–Letnikov derivative In mathematics, the Grünwald–Letnikov derivative is a basic extension of the derivative in fractional calculus that allows one to take the derivative a non-integer number of times. It was introduced by Anton Karl Grünwald (1838–1920) from P ...

) and the RL0-algorithms (

Riemann–Liouville integral In mathematics, the Riemann–Liouville integral associates with a real function f: \mathbb \rightarrow \mathbb another function of the same kind for each value of the parameter . The integral is a manner of generalization of the repeated antide ...

) are recursive methods to implement a numerical calculation of fractional differintegrals. Yet

differintegral In fractional calculus, an area of mathematical analysis, the differintegral (sometime also called the derivigral) is a combined differentiation/integration operator. Applied to a function ƒ, the ''q''-differintegral of ''f'', here denoted by ...

s are faster to compute in discrete fourier space using

FFT A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in th ...

.Jun-Sheng Yu, Zu-Xun Zhanga; "Differentiation, semidifferentiation and semi-integration of a digital signals based on Fourier transformations"; ''Journal of Electroanalytical Chemistry''; Volume 403, Issues 1-2, 21 February 1996, Pages 1-9;

Applications

The graphs below show the behaviour of fractional derivatives calculated by different algorithms for

ferrocene Ferrocene is an organometallic compound with the formula . The molecule is a complex consisting of two cyclopentadienyl rings bound to a central iron atom. It is an orange solid with a camphor-like odor, that sublimes above room temperature, ...

in acetonitrile at 100mV/s, the

reference electrode A reference electrode is an electrode which has a stable and well-known electrode potential. The high stability of the electrode potential is usually reached by employing a redox system with constant (buffered or saturated) concentrations of each ...

is 0.1M Ag⁺/Ag in acetonitrile (+0.04V vs. FcVitaly V. Pavlishchuk and Anthony W. Addison; "Conversion constants for redox potentials measured versus different reference electrodes in acetonitrile solutions at 25 °C"; ''Inorganica Chimica Acta'' Volume 298, Issue 1, 30 January 2000, Pages 97–102; ).

1st derivative of the "Semiderivative" or 1.5th order derivative in voltammetry

1.5th order derivative of a voltammogram hits the abscissa exactly at the point where the formal potential of the electrode reaction is found.

"Semiderivative" or numerical Grünberg-Letnikov derivative in voltammetry

The G1 algorithm produces a numerical derivative that has the shape of a bell curve, this derivative obeys to certain laws, for example the G1 derivative of a cyclic voltammogram is mirrored at the

abscissa In common usage, the abscissa refers to the (''x'') coordinate and the ordinate refers to the (''y'') coordinate of a standard two-dimensional graph. The distance of a point from the y-axis, scaled with the x-axis, is called abscissa or x coo ...

as long as the electrochemical reaction is diffusion controlled, the planar diffusion approximation can be applied to the electrode geometryMasashi Goto, Keith B. Oldham; "Semiintegral electroanalysis. Shapes of neopolarograms"; ''Anal. Chem.'', 1973, 45 (12), pp 2043–2050; and ohmic drop distortion is minimal. The

FWHM In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve mea ...

of the curve is approximately 100 mV for a system that behaves in the described manner. The maximum is found at the value of the formal potential, this is equivalent to the 1.5th order semiderivative hitting the abscissa at this potential. Moreover the semiderivative scales linearly with the scanrate, while the current scales linearly with the square root of the scanrate (

Randles–Sevcik equation In cyclic voltammetry, the Randles–Ševčík equation describes the effect of scan rate on the peak current ''ip''. For simple redox events such as the ferrocene/ferrocenium couple, ''ip'' depends not only on the concentration and diffusional pro ...

). Plotting the semiderivatives produced at different scanrates gives a

family of curves In geometry, a family of curves is a set of curves, each of which is given by a function or parametrization in which one or more of the parameters is variable. In general, the parameter(s) influence the shape of the curve in a way that is more ...

that are linearly related by the scanrate quotient in an ideal system.

"Semiintegral" or numerical Riemann-Liouville integral in voltammetry

The shape of the semiintegral can be used as an easy method to measure the amount of ohmic drop of an electrochemical cell in

cyclic voltammetry Cyclic voltammetry (CV) is a type of potentiodynamic electrochemical measurement. In a cyclic voltammetry experiment, the working electrode potential is ramped linearly versus time. Unlike in linear sweep voltammetry, after the set potential is ...

. Essentially the semiintegral of a cyclic voltammogram at a planar electrode (an electrode that obeys to the rules of planar diffusion) has the shape of a

sigmoid Sigmoid means resembling the lower-case Greek letter sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς) or the Latin letter S. Specific uses include: * Sigmoid function, a mathematical function * Sigmoid colon, part of the l ...

while the original data is gauss-sigmoid convoluted. This enables the operator to optimize parameters necessary for positive feedback compensation in an easy manner.Alan M. Bond, Keith B. Oldham, and Graeme A. Snook; "Use of the Ferrocene Oxidation Process To Provide Both Reference Electrode Potential Calibration and a Simple Measurement (via Semiintegration) of the Uncompensated Resistance in Cyclic Voltammetric Studies in High-Resistance Organic Solvents"; ''Anal. Chem.'', 2000, 72 (15), pp 3492–3496 If ohmic drop distortion is present the two sigmoids for the forward and the backward scan are far away from congruence, the ohmic drop can be calculated from the deviation from congruence in these cases. In the example shown slight distortion is present, yet this does not have adverse effects on data quality.

Merits of FFT techniques

The implementation differintegral calculation using fast fourier transform has certain benefits because it is easily combined with low pass quadratic filtering methods.Eric E. Aubanela, Janice C. Mylanda, Keith B. Oldham and Cynthia G. Zoskia; "Fourier smoothing of electrochemical data without the fast fourier transform"; ''Journal of Electroanalytical Chemistry and Interfacial Electrochemistry''; Volume 184, Issue 2, 25 March 1985, Pages 239-255; {{doi, 10.1016/0368-1874(85)85531-3 This is very useful when cyclic voltammograms are recorded in high resistivity solvents like

tetrahydrofuran Tetrahydrofuran (THF), or oxolane, is an organic compound with the formula (CH2)4O. The compound is classified as heterocyclic compound, specifically a cyclic ether. It is a colorless, water- miscible organic liquid with low viscosity. It is ...

toluene Toluene (), also known as toluol (), is a substituted aromatic hydrocarbon. It is a colorless, water-insoluble liquid with the smell associated with paint thinners. It is a mono-substituted benzene derivative, consisting of a methyl group (CH3) ...

, where feedback oscillations are a frequent problem.

References

Electroanalytical chemistry