**1. Introduction**

The past few decades have seen a massive and continued interest in studying electrochemical processes at artificially structured electrodes. As is well known, the rate of redox reactions taking place at an electrode depends on both the mass transport towards the electrode surface and kinetics of electron transfer at the electrode surface. Three modes of mass transport can be considered in electrochemical cells: diffusion, migration and convection. The diffusional mass transport is the movement of molecules along a concentration gradient, from an area of high concentration to an area of low concentration. The migrational mass transport is observed only in the case of ions and occurs in the presence of a potential gradient. Convectional mass transport occurs in flowing solutions at rotating disk electrodes or at the dropping mercury electrode.

In 1902 Cottrell derived his landmark equation describing the diffusion current, *I*, flowing to a planar, uniformly accessible and smooth electrode of surface area, *A*, large enough not to be seriously affected by the edge effect, in contact with a semi-infinite layer of electrolyte solution containing a uniform concentration, *cO*, of reagent reacting reversibily and being present as a minor component with an excess supporting electrolyte under unstirred conditions, during the potential-step experiment (Cottrell, 1902)

$$I = nFAcc\_O \sqrt{\frac{D}{\pi t}} \,\tag{1}$$

where *n* is the number of electrons entering the redox reaction, *F* is the Faraday constant, *D* is the diffusion coefficient, and *t* is time.

It has long been known that the geometry, surface structure and choice of substrate material of an electrode have profound effects on the electrochemical response obtained. It is also understood that the electrochemical response of an electrode is strongly dependent on its size, and that the mass transport in electrochemical cell is affected by the electrode surface roughness which is generally irregular in both the atomic and geometric scales. Moreover, the instant rapid development in nanotechnology stimulates novel approaches in the preparation of artificially structured electrodes. This review seeks to condense information on the reasons giving rise or contributing to the non-Cottrellian diffusion towards microand nano structured electrodes.

A Review of Non-Cottrellian Diffusion Towards Micro- and Nano-Structured Electrodes 5

were reported by Molina and co-workers (Molina et al., 2011b). Here, the evolution of the mass transport from linear (high sizes) to radial (microelectrodes) was characterized, and the conditions required to attain a stationary state were discussed. The use of differential pulse voltammetry at spherical electrodes and microelectrodes for the study of the kinetic of charge transfer processes was analyzed and an analytical solution was presented by Molina and co-workers (Molina et al., 2010b). The repored expressions are valid for any value of the electrode radius, the heterogeneous rate constant and the transfer coefficient. The anomalous shape of differential pulse voltammetry curves for quasi-reversible processes with small values of the transfer coefficient was reported, too. Moreover, general working curves were given for the determination of kinetic parameters from the position and height of differential pulse voltammetry peak. Sophisticated methods based on graphic programming units have been used by Cuttress and Compton to facilitate digital electrochemical simulation of processes at elliptical discs, square, rectangular, and

microband electrodes (Cuttress & Compton, 2010a; Cuttress & Compton, 2010b).

(spheres, disks, bands, and cylinders) have been deduced (Molina et al., 2011a).

**3. Electrochemical cells with bulk resistance** 

A general, explicit analytical solution for any multipotential waveform valid for an electrochemically reversible system at an electrode of any geometry is continually in the centre of interest. This problem has been solved many times (e.g., Aoki et al., 1986; Cope & Tallman, 1991; Molina et al., 1995; Serna & Molina, 1999). A general theory for an arbitrary potential sweep voltammetry on an arbitrary topography (fractal or nonfractal) of an electrode operating under diffusion-limited or reversible charge-transfer conditions was developed by Kant (Kant, 2010). This theory provides a possibility to make clear various anomalies in measured electrochemical responses. Recently, analytical explicit expressions applicable to the transient *I-E* response of a reversible charge transfer reaction when both species are initially present in the solution at microelectrodes of different geometries

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

### **2. Electrode geometry**

Cottrell equation, derived for a planar electrode, can be applied to electrodes of other simple geometries, provided that the temporal and spatial conditions are such that the semi-infinite diffusion to the surface of the electrode is approximately planar. However, in both the research and application spheres various electrode geometries are applied depending on the problem or task to be solved. Most electrodes are impaired by an ''edge effect'' of some sort and therefore do not exhibit uniform accessibility towards diffusing solutes. Only the well defined electrode geometry allows the data collected at the working electrode to be reliably interpreted. The diffusion limited phenomena at a wide variety of different electrode geometries have been frequently studied by several research teams. Aoki and Osteryoung have derived the rigorous expressions for diffusion-controlled currents at a stationary finite disk electrode through use of the Wiener-Hopf technique (Aoki & Osteryoung, 1981). The chronoamperometric curve they have obtained varies smoothly from a curve represented by the Cottrell equation and can be expressed as the Cottrell term multiplied by a power series in the parameter *Dt r* , where *r* is the electrode radius. Later, a theoretical basis for understanding the microelectrodes with size comparable with the thickness of the diffusion layer, providing a general solution for the relation between current and potential in the case of a reversible reaction was given by the same authors (Aoki & Osteryoung, 1984). A userfriendly version of the equations for describing diffusion-controlled current at a disk electrode resulting from any potential perturbation was derived by Mahon and Oldham (Mahon & Oldham, 2005). Myland and Oldahm have proposed a method that permits the derivation of Cottrell's equation without explicitly solving Fick's second law (Myland & Oldham, 2004). The procedure, based on combining two techniques – the Green's Function technique and the Method of Images, has been shown to successfully treat several electrochemical situations. Being dependent on strict geometric conditions being met, it may provide a vehicle for a novel approach to electrochemical simulation involving diffusion in nonstandard geometries. In the same year Oldham reported an exact method used to find the diffusion-controlled faradaic current for certain electrode geometries that incorporate edges and vertices, which is based on Green's equation (Oldham, 2004). Gmucová and coworkers described the real electrochemical response of neurotransmitter dopamine on a carbon fiber microelectrode as a power function, i.e., *t* (Gmucová et al., 2004). That power function expanded to the polynomial terms can be, in conformity with (Aoki & Osteryoung, 1981; Mahon & Oldham, 2005), regarded as a Cottrell term, multiplied by a series of polynomial terms used to involve corrections to the Cottrell equation.

The variation of the diffusion layer thicknesses at planar, cylindrical, and spherical electrodes of any size was quantified from explicit equations for the cases of normal pulse voltammetry, staircase voltammetry, and linear sweep voltammetry by Molina and coworkers (Molina et al., 2010a). Important limiting behaviours for the linear sweep voltammetry current-potential curves were reported in all the geometries considered. These results are of special physical relevance in the case of disk and band electrodes which possess non-uniform current densities since general analytical solutions were derived for the above-mentioned geometries for the first time. Explicit analytical expressions for diffusion layer thickness of disk and band electrodes of any size under transient conditions

Cottrell equation, derived for a planar electrode, can be applied to electrodes of other simple geometries, provided that the temporal and spatial conditions are such that the semi-infinite diffusion to the surface of the electrode is approximately planar. However, in both the research and application spheres various electrode geometries are applied depending on the problem or task to be solved. Most electrodes are impaired by an ''edge effect'' of some sort and therefore do not exhibit uniform accessibility towards diffusing solutes. Only the well defined electrode geometry allows the data collected at the working electrode to be reliably interpreted. The diffusion limited phenomena at a wide variety of different electrode geometries have been frequently studied by several research teams. Aoki and Osteryoung have derived the rigorous expressions for diffusion-controlled currents at a stationary finite disk electrode through use of the Wiener-Hopf technique (Aoki & Osteryoung, 1981). The chronoamperometric curve they have obtained varies smoothly from a curve represented by the Cottrell equation and can be expressed as the Cottrell term multiplied by a power series in the parameter *Dt r* , where *r* is the electrode radius. Later, a theoretical basis for understanding the microelectrodes with size comparable with the thickness of the diffusion layer, providing a general solution for the relation between current and potential in the case of a reversible reaction was given by the same authors (Aoki & Osteryoung, 1984). A userfriendly version of the equations for describing diffusion-controlled current at a disk electrode resulting from any potential perturbation was derived by Mahon and Oldham (Mahon & Oldham, 2005). Myland and Oldahm have proposed a method that permits the derivation of Cottrell's equation without explicitly solving Fick's second law (Myland & Oldham, 2004). The procedure, based on combining two techniques – the Green's Function technique and the Method of Images, has been shown to successfully treat several electrochemical situations. Being dependent on strict geometric conditions being met, it may provide a vehicle for a novel approach to electrochemical simulation involving diffusion in nonstandard geometries. In the same year Oldham reported an exact method used to find the diffusion-controlled faradaic current for certain electrode geometries that incorporate edges and vertices, which is based on Green's equation (Oldham, 2004). Gmucová and coworkers described the real electrochemical response of neurotransmitter dopamine on a

carbon fiber microelectrode as a power function, i.e., *t*

power function expanded to the polynomial terms can be, in conformity with (Aoki & Osteryoung, 1981; Mahon & Oldham, 2005), regarded as a Cottrell term, multiplied by a

The variation of the diffusion layer thicknesses at planar, cylindrical, and spherical electrodes of any size was quantified from explicit equations for the cases of normal pulse voltammetry, staircase voltammetry, and linear sweep voltammetry by Molina and coworkers (Molina et al., 2010a). Important limiting behaviours for the linear sweep voltammetry current-potential curves were reported in all the geometries considered. These results are of special physical relevance in the case of disk and band electrodes which possess non-uniform current densities since general analytical solutions were derived for the above-mentioned geometries for the first time. Explicit analytical expressions for diffusion layer thickness of disk and band electrodes of any size under transient conditions

series of polynomial terms used to involve corrections to the Cottrell equation.

(Gmucová et al., 2004). That

**2. Electrode geometry** 

were reported by Molina and co-workers (Molina et al., 2011b). Here, the evolution of the mass transport from linear (high sizes) to radial (microelectrodes) was characterized, and the conditions required to attain a stationary state were discussed. The use of differential pulse voltammetry at spherical electrodes and microelectrodes for the study of the kinetic of charge transfer processes was analyzed and an analytical solution was presented by Molina and co-workers (Molina et al., 2010b). The repored expressions are valid for any value of the electrode radius, the heterogeneous rate constant and the transfer coefficient. The anomalous shape of differential pulse voltammetry curves for quasi-reversible processes with small values of the transfer coefficient was reported, too. Moreover, general working curves were given for the determination of kinetic parameters from the position and height of differential pulse voltammetry peak. Sophisticated methods based on graphic programming units have been used by Cuttress and Compton to facilitate digital electrochemical simulation of processes at elliptical discs, square, rectangular, and microband electrodes (Cuttress & Compton, 2010a; Cuttress & Compton, 2010b).

A general, explicit analytical solution for any multipotential waveform valid for an electrochemically reversible system at an electrode of any geometry is continually in the centre of interest. This problem has been solved many times (e.g., Aoki et al., 1986; Cope & Tallman, 1991; Molina et al., 1995; Serna & Molina, 1999). A general theory for an arbitrary potential sweep voltammetry on an arbitrary topography (fractal or nonfractal) of an electrode operating under diffusion-limited or reversible charge-transfer conditions was developed by Kant (Kant, 2010). This theory provides a possibility to make clear various anomalies in measured electrochemical responses. Recently, analytical explicit expressions applicable to the transient *I-E* response of a reversible charge transfer reaction when both species are initially present in the solution at microelectrodes of different geometries (spheres, disks, bands, and cylinders) have been deduced (Molina et al., 2011a).
