Neopolarogram
Encyclopedia
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
. The implementation of fractional derivative calculations by means of numerical methods is straight forward. The G1- (Grünwald–Letnikov derivative) and the RL0-algorithms (Riemann–Liouville integral) are recursive methods to implement a numerical calculation of fractional differintegrals. Yet differintegrals are faster to compute in discrete fourier space using FFT
.
in acetonitrile
at 100mV/s, the reference electrode
is 0.1M Ag+/Ag in acetonitrile (+0.04V vs. Fc). (Note to the admin: None of this is original research, it can all be found in books and papers cited in this article!)
as long as the electrochemical reaction is diffusion controlled, the planar diffusion approximation can be applied to the electrode geometry and ohmic drop distortion is minimal. The FWHM of the curve is approximately 100 mV for a system that behaves in the described manor. The maximum is found at the value of the formal potential, this is quivalent to the 1.5th order semiderivative hitting the abszissa 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). Plotting the semiderivatives produced at different scanrates gives a family of curves
that are linearly related by the scanrate quotient in an ideal system.
of an electrochemical cell in cyclic voltammetry
. 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
while the original data is gauss-sigmoid convoluted. This enables the operator to optimize parameters necessary for positive feedback compensation
in an easy manor. If ohmic drop distortion is present the two sigmoid 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 have adverse effects on data quality.
or toluene
, where feedback oszillations are a frequent problem.
Polarography
Polarography is a subclass of voltammetry where the working electrode is a dropping mercury electrode or a static mercury drop electrode ., useful for its wide cathodic range and renewable surface...
or cyclic voltammograms
Cyclic voltammetry
Cyclic voltammetry or CV is a type of potentiodynamic electrochemical measurement. In a cyclic voltammetry experiment the working electrode potential is ramped linearly versus time like linear sweep voltammetry. Cyclic voltammetry takes the experiment a step further than linear sweep voltammetry...
that in effect deconvolute diffusion and electrochemical kinetics. This is achieved by analog or digital implementations of fractional calculus
Fractional calculus
Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator.and the integration operator J...
. The implementation of fractional derivative calculations by means of numerical methods is straight forward. The G1- (Grünwald–Letnikov derivative) and the RL0-algorithms (Riemann–Liouville integral) are recursive methods to implement a numerical calculation of fractional differintegrals. Yet differintegrals are faster to compute in discrete fourier space using FFT
Fast Fourier transform
A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...
.
Applications
The graphs below show the behaviour of fractional derivatives calculated by different algorithms for ferroceneFerrocene
Ferrocene is an organometallic compound with the formula Fe2. It is the prototypical metallocene, a type of organometallic chemical compound consisting of two cyclopentadienyl rings bound on opposite sides of a central metal atom. Such organometallic compounds are also known as sandwich compounds...
in acetonitrile
Acetonitrile
Acetonitrile is the chemical compound with formula . This colourless liquid is the simplest organic nitrile. It is produced mainly as a byproduct of acrylonitrile manufacture...
at 100mV/s, the reference electrode
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 concentrations of each participants of the redox reaction.There are many ways reference...
is 0.1M Ag+/Ag in acetonitrile (+0.04V vs. Fc). (Note to the admin: None of this is original research, it can all be found in books and papers cited in this article!)
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 abscissaAbscissa
In mathematics, abscissa refers to that element of an ordered pair which is plotted on the horizontal axis of a two-dimensional Cartesian coordinate system, as opposed to the ordinate...
as long as the electrochemical reaction is diffusion controlled, the planar diffusion approximation can be applied to the electrode geometry and ohmic drop distortion is minimal. The FWHM of the curve is approximately 100 mV for a system that behaves in the described manor. The maximum is found at the value of the formal potential, this is quivalent to the 1.5th order semiderivative hitting the abszissa 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). Plotting the semiderivatives produced at different scanrates gives a family of curves
Family of curves
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 influence the shape of the curve in a way that is more complicated than a simple linear transformation...
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 dropOhm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...
of an electrochemical cell in cyclic voltammetry
Cyclic voltammetry
Cyclic voltammetry or CV is a type of potentiodynamic electrochemical measurement. In a cyclic voltammetry experiment the working electrode potential is ramped linearly versus time like linear sweep voltammetry. Cyclic voltammetry takes the experiment a step further than linear sweep voltammetry...
. 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
Sigmoid means resembling the lower-case Greek letter sigma or the Latin letter S. Specific uses include:* Sigmoid function, a mathematical function* Sigmoid colon, part of the large intestine or colon...
while the original data is gauss-sigmoid convoluted. This enables the operator to optimize parameters necessary for positive feedback compensation
Positive feedback
Positive feedback is a process in which the effects of a small disturbance on a system include an increase in the magnitude of the perturbation. That is, A produces more of B which in turn produces more of A. In contrast, a system that responds to a perturbation in a way that reduces its effect is...
in an easy manor. If ohmic drop distortion is present the two sigmoid 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 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. This is very useful when cyclic voltammograms are recorded in high resistivity solvents like tetrahydrofuranTetrahydrofuran
Tetrahydrofuran is a colorless, water-miscible organic liquid with low viscosity at standard temperature and pressure. This heterocyclic compound has the chemical formula 4O. As one of the most polar ethers with a wide liquid range, it is a useful solvent. Its main use, however, is as a precursor...
or toluene
Toluene
Toluene, formerly known as toluol, is a clear, water-insoluble liquid with the typical smell of paint thinners. It is a mono-substituted benzene derivative, i.e., one in which a single hydrogen atom from the benzene molecule has been replaced by a univalent group, in this case CH3.It is an aromatic...
, where feedback oszillations are a frequent problem.