**3. Electrochemical cells with bulk resistance**

Mathematical modeling of kinetics and mass-transfer in electrochemical events and related electroanalytical experiments, generally consists of dealing with various physico-chemical parameters, as well as complicated mathematical problems, even in their simplest statement.

An analysis of the transient response in potential controlled experiments is a standard procedure which can yield information about many electrochemical processes and several kinetic parameters. However, a resistance in series (i.e., solution resitance, electrode coating resistance, sample resitance in solid state electrochemistry) can have a serious effect on electrochemical measurements. Thus, the presence of migration leads to essential deviations from the Cottrellian behaviour. Electrochemical systems that exhibit bulk ohmic resistances cannot be characterized accurately using the Cottrell equation. Electrochemical experiments in solution without added supporting electrolyte, i.e., without suppressed migration, became possible with the progress of microelectrodes. The expressions for current vs. time responses to applied voltage steps across the whole system, and corresponding concentration profiles within the cell or membrane were derived by Nahir and Buck and compared with experimental results (Nahir & Buck, 1992). Voltammetry in solutions of low ionic strength has been reviewed by Ciszkowska and Stojek (Ciszkowska & Stojek, 1999). A mathematical model of migration and diffusion coupled with a fast preceding reaction at a

A Review of Non-Cottrellian Diffusion Towards Micro- and Nano-Structured Electrodes 7

solution resistance on the reversible charge transfer at an arbitrary rough electrode was studied and the significant deviation from the classical Cottrellian behavior was explained as arising from the resistivity of the solution and geometric irregularity of the interface. In the short time domain it was found to be dependent primarily on the resistance of the electrolytic solution and the real area of the surface. Results obtained for various electrode roughness models were reported. In the absence of the surface roughness, the current crossover to classical Cottrell response as the diffusion length exceeds the diffusion-ohmic length, but in the presence of roughness, there is formation of anomalous intermediate region followed by classical Cottrell region. Later, the theoretical results elucidating the influence of an uncompensated solution resistance on the anomalous Warburg's impedance in case of rough surfaces has been published by the same authors (Srivastav & Kant, 2011).

Modified electrodes include electrodes where the surface was deliberately altered to impart functionality distinct from the base electrode. During last decades a large number of different strategies for physical and chemical electrode modification have been developed, aimed at the enhancement in the detection of species under interest. Particularly in biosciences and environmental sciences such electrodes became of great importance. One of the issues raised in the research of redox processes taking place at modified electrodes has

Historically, liquid and solid electrochemistry grew apart and developed separately for a long time. Appearance of novel materials and methods of thin films preparation lead to massive development of chemically modified electrodes (Alkire et al., 2009). Such electrodes represent relatively modern approach to electrode systems with thin film of a selected chemical bonded or coated onto the electrode surface. A wide spectrum of their possible applications turned the spotlight of electrochemical research towards the design of electrochemical devices for applications in sensing, energy conversion and storage, molecular electronics etc. Only several examples of possible electrode coatings are mentioned in this chapter, all of them in close contact with the study of the electron transfer

Marked deviations from Cottrellian behaviour were encountered in the theoretical study (Thompson et al., 2006) describing the diffusion of charge over the surface of a microsphere resting on an electrode at a point, in the limit of reversible electrode kinetics. A realistic physical problem of truncated spheres on the electrode surface was modelled in the above mentioned work, and the effect of truncation angle on chronoamperometry and voltammetry was explored. It has been shown that the most Cottrell-like behaviour is observed for the case of a hemispherical particle resting on the surface, but only at short times is the diffusion approximated well by a planar diffusion model. Concurrently, Thompson and Compton have developed a model for the voltammetric response due to surface charge injection at a single point on the surface of a microsphere on whose surface the electro-active material is confined. The cyclic voltammetric response of such system was investigated, the Fickian diffusion constrained on spherical surfaces showed strong deviations from the responses expected for planar diffusion. The Butler–Volmer condition

been the analysis of changes in the diffusion towards their altered surfaces.

**4. Modified electrodes** 

kinetic on them.

**4.1 Micro- and nanoparticle modified electrodes** 

microelectrode was developed by Jaworski and co-workers (Jaworski et al., 1999). Myland and Oldham have shown that on macroelectrodes the Cottrellian dependence can be preserved even when supporting electrolyte is absent. The limiting current, however, was shown to depart in magnitude from the Cottrellian prediction by a factor (greater or less than unity) that depends on the charge numbers of the salt's ions and that of the electroproduct (Myland & Oldham, 1999). A generalized theory of the steady-state voltammetric response of a microelectrode in the absence of supporting electrolyte and for any values of diffusion coefficients of the substrate and the product of an electrode process was presented by Hyk and Stojek (Hyk & Stojek, 2002).

The influence of supporting electrolyte on the drugs detection was studied and data obtained using cyclic voltammetry, steady-state voltammetry and voltcoulometry on the same analyte were compared to each other by Orlický and co-workers. Under unsupported conditions different detection limits of the above mentioned methods were observed. Some species were easily observed by the kinetics-sensitive voltcoulometry even for concentrations near or under the sensitivity limit of voltammetric methods (Orlický et al., 2003). Thus, systems obeing deviations from Cottrell behaviour should find their application in sensorics. Later, it has been revealed that the dopamine diffusion current towards a carbon fiber microelectrode fulfills, within experimental errors and for concentration similar to those in a rat striatum, the behaviour theoretically predicted by the Cottrell equation. Nevertheless, under unsupported or weakly supported conditions non-Cottrellian responses were observed. Moreover, markedly non-Cottrellian responses were observed for dopamine concentrations lower or higher than the physiological ones in the rat striatum. It has been also shown, that the non-Cottrellian behaviour of diffusion current involves the nonlinearity of the dopamine calibration curve obtained by kinetics-sensitive voltcoulometry, while voltammetric calibration curve remains linear (Gmucová et al., 2004). Similarly, Caban and co-workers analysed the contribution of migration to the transport of polyoxometallates in the gels by methods of different sensitivity to migration (Caban et al., 2006).

Mathematical models of the ion transport regarded as the superposition of diffusion and migration in a potential field were analyzed by Hasanov and Hasanoglu (Hasanov & Hasanoglu, 2008). Based on the Nernst-Planck equation the authors have derived explicit analytical formulae for the concentration of the reduced species and the current response in the case of pure diffusive as well as diffusion–migration model, for various concentrations at initial conditions. The proposed approach can predict an influence of ionic diffusivities, valences, and initial and boundary concentrations to the behaviour of non-Cottrellian current response. In addition to these, the analytical formulae obtained can also be used for numerical and digital simulation methods for Nernst-Planck equations. The mathematical model of the nonlinear ion transport problem, which includes both the diffusion and migration, was solved by the same authors (Hasanov & Hasanoglu, 2009). They proposed a numerical iteration algorithm for solving the nonlocal identification problem related to nonlinear ion transport. The presented computational results are consistent with experimental results obtained on real systems.

The quantitative understanding of generalized Cottrellian response of moderately supported electrolytic solution at rough electrode/electrolyte interface was enabled with the Srivastav's and Kant's work (Srivastav & Kant, 2010). Here, the effect of the uncompensated solution resistance on the reversible charge transfer at an arbitrary rough electrode was studied and the significant deviation from the classical Cottrellian behavior was explained as arising from the resistivity of the solution and geometric irregularity of the interface. In the short time domain it was found to be dependent primarily on the resistance of the electrolytic solution and the real area of the surface. Results obtained for various electrode roughness models were reported. In the absence of the surface roughness, the current crossover to classical Cottrell response as the diffusion length exceeds the diffusion-ohmic length, but in the presence of roughness, there is formation of anomalous intermediate region followed by classical Cottrell region. Later, the theoretical results elucidating the influence of an uncompensated solution resistance on the anomalous Warburg's impedance in case of rough surfaces has been published by the same authors (Srivastav & Kant, 2011).
