Semi-integration techniques

Convolution potential sweep voltammetry (CPSV) refers to the mathematical transformation of LSV current—potential data resulting in curves with shapes like conventional polarograms which are suitable for logarithmic analysis. The method was first proposed for the study of electrode kinetics by Imbeaux and Saveant [74] but is equivalent in all respects to a semi-integral technique reported earlier by Oldham [75— 77]. A very readable description of the method has been presented by Bard and Faulkner [21]. 

For bubbly flows most of the early papers either adopted a macroscopic population balance approach with an inherent discrete discretization scheme as described earlier, or rather semi-empirical transport equations for the contact area and/or the particle diameter. Actually, very few consistent source term closures exist for the microscopic population balance formulation. The existing models are usually solved using discrete semi-integral techniques, as will be outlined in the next sub-section. 

The convolution or semi-integral technique offers advantages in the processing of data obtained by CV for diffusion controlled systems. For a reversible redox couple, in an experiment of CV, the convoluted current curves, l(t) vs. E for forward and reverse sweeps are superimposed, returning to zero at a E sufficiently positive fixrm the formal potential of the redox couple 0/R, where Cr(0, t) = 0, as shown in Figure lb. 

Turning now to digital methods of semi-integration, we shall describe a technique by which the semi-integral m may be determined from a set of equally spaced current values, i 0, ii, i2,. . . , . . . , . If A is the time... 

An answer to this lies in the transformation of the linear sweep response into a form which is readily analysable, i.e. the form of a steady-state voltammetric wave. Two independent methods of achieving this goal have been described the convolution technique by Saveant and co-workers11,12, and semi-integration by Oldham13. In this section we describe the convolution technique, and demonstrate the equivalence of the two approaches at the end. 

The mathematically most sophisticated technique involves computing convolution or semi-integrals of the voltammetric data obtained in digital form (Imbeaux and Saveant, 1973 Nadjo et al., 1974 Oldham and Spanier, 1970 Oldham, 1972, 1973). The latter results in the transformation of the LSV wave into a form resembling a polarogram (Fig. 19) which is amenable... 

Both m t) and /(/), which represent the integral in equation 6.7.3, have been used in presentations of this transformation technique clearly the convolutive (20) and semi-integral (21, 22) approaches are equivalent. 

Studies made with this instrumentation on other voltammetrlc techniques such as anodic stripping voltammetry allow one to conclude that the optimization of initial d.c. linear sweep or stripping data leads to optimum performance In the semi-integral, semi-differential and derivative approaches and that, under Instrumental equivalent conditions where d.c. experiments have been optimized with respect to electronic noise and background correction, detection limits are not markedly different within the sub-set of related approaches. Obviously, the resolution and ease of use of a method providing a peak-type readout (semi-differential) are superior to those with sigmoidally shaped read- outs (semi-integral). 

Other procedures applied to the LSV technique Semidifferentiating (deconvolutive) procedure may be considered as a counterpart to semi-integrating (convolutive) transformation. The replacement of Pick s laws by formulations involving semidifferentiation was proposed 25 years ago [122]. Five years later [123] the deconvolutive transformation of recordered (sampled) currents represented by the equation... 

Another way of classifying the integration techniques depends on whether or not the method is explicit, semi-implicit, or implicit. The implicit and semi-implicit methods play an important role in the numerical solution of stiff differential equations. To maintain the continuity of the section, we will first describe the explicit integration techniques in the context of one-step and multistep methods. The concept of stiffness and implicit methods are considered in a separate subsection, which also marks the end of this section. 

Bentley CL, Bond AM, HoUenkamp AF, Mahon PJ, Zhang J (2012) Advantages available in the application of the semi-integral electroanalysis technique for the determination of diffusion coefficients in the highly viscous ionic liquid l-methyl-3-octylimidazolium hexafluorophos-phate. Anal Chem 85(4) 2239-2245. doi 10.1021/ac303042r... 

Whereas it is generally sufficient (at least for the pubhshed methods) to specify the semi-empirical MO technique used in order to define the exact method used for the calculations, ab-initio theory offers far more variations, so that the exact level of the calculation must be specified. The starting point of most ab-initio jobs is an SCF calculation analogous to those discussed above for semi-empirical MO calculations. In ab-initio theory, however, all necessary integrals are calculated correctly, so that the calculations are very much (by a factor of about 1000) more time-consuming than their semi-empirical counterparts. 

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