**4.4 Spatially heterogeneous electrodes**

Porous electrodes, partially blocked electrodes, microelectrode arrays, electrodes made of composite materials, some modified electrodes and electrodes with adsorbed species are spatially heterogeneous in the electrochemical sense. The simulation of non-Cottrellian electrode responses at such surfaces is challenging both because of the surface variation and

Lange and Doblhofer solved the transport equations by digital simulation techniques with boundary conditions appropriate for the system electrode/membrane-type polymer coating (Lange & Doblhofer, 1987). They have concluded that the current transients follow Cottrell equation, however, the observed "effective" diffusion coefficients are different from the tabular ones. In the 90s an important effort has been devoted to examination of the nature of the diffusion processes of membrane-covered Clark-type oxygen sensors by solving the axially symmetric two-dimensional diffusion equation. Gavaghan and co-workers have presented a numerical solution of 2D equations governing the diffusion of oxygen to a circular disc cathode protected from poisoning by the medium to be measured by a tightly

The current-time behaviour of membrane-covered microdisc clinical sensors was examined with the aim to explain their poor performance when pulsed (Sutton et al., 1996). It has been shown by Sutton and co-workers that the Cottrellian hypothesis is not applicable to this type of sensor and it is not possible to predict this behaviour from an analytical expression, as might be the case for membrane-covered macrodisc sensors and unshielded microdisc

Gmucová and co-workers have shown that changes in kinetic of a redox reaction manifested as a deviation from the Cottrellian behaviour can be utilized in the preparation of ion selective electrodes. The electroactive hydrophobic end of a molecule used for the Langmuir-Blodgett film modification of a working electrode can induce a change in the kinetic of redox reactions. Ion selective properties of the poly(3-pentylmethoxythiophene) Langmuir–Blodgett film modified carbon-fiber microelectrode have been proved using a model system, mixture of copper and dopamine ions. While in case of the typical steadystate voltammetry the electrode remains sensitive to both the copper and dopamine, the kinetic-sensitive properties of voltcoulometry disable the observation of dopamine

Recently, a sensing protocol based on the anomalous non-Cottrellian diffusion towards nanostructured surfaces was reported by Gmucová and co-workers (Gmucová et al., 2011). The potassium ferrocyanide oxidation on a gold disc electrode covered with a system of partially decoupled iron oxides nanoparticle membranes was investigated using the kineticsensitive voltcoulometry. Kinetic changes were induced by the altered electrode surface morphology, i.e., micro-sized superparamagnetic nanoparticle membranes were curved and partially damaged under the influence of the applied magnetic field. Thus, the targeted changes in the non-Cottrellian diffusion towards the working electrode surface resulted in a marked amplification of the measured voltcoulometric signal. Moreover, the observed effect depends on the membrane elasticity and fragility, which may, according to the authors, give rise to the construction of sensors based on the influence of various physical, chemical or biological external agents on the superparamagnetic nanoparticle membrane Young's

Porous electrodes, partially blocked electrodes, microelectrode arrays, electrodes made of composite materials, some modified electrodes and electrodes with adsorbed species are spatially heterogeneous in the electrochemical sense. The simulation of non-Cottrellian electrode responses at such surfaces is challenging both because of the surface variation and

stretched plastic membrane which is permeable to oxygen (Gavaghan et al., 1992).

electrodes.

moduli.

**4.4 Spatially heterogeneous electrodes** 

(Gmucová et al., 2007).

because of the often random distribution of the zones of different electrode activity. The Cottrell equation becomes invalid even if the electrode reaction causes motionof the electrolyte/electrode boundary. Thereby it was modified by Oldham and Raleigh to take account of this effect, as well as to the data published on the inter-diffusion of silver and gold (Oldham & Raleigh, 1971).

Davies and co-workers have shown that by use of the concept of a ''diffusion domain'' computationally expensive three-dimensional simulations may be reduced to tractable twodimensional equivalents which gives results in excellent agreement with experiment (Davies et al., 2005). Their approach predicts the voltammetric behaviour of electrochemically heterogeneous electrodes, e.g., composites whose different spatial zones display contrasting electrochemical behaviour toward the same redox couple. Four categories of response on spatially heterogeneous electrode have been defined by the authors depending on the blocked and unblocked electrode surface zones dimensions. In the performed analysis of partially blocked electrodes the difference between "macro" and ''micro'' was shown to be critical. The question how to specify whether the dimensions of the electro-active or inert zones of heterogeneous electrodes fall into one category or another one can be answered using the Einstein equation, which indicates that the approximate distance, *δ*, diffused by a species with a diffusion coefficient, *D*, in a time, *t*, is 2*Dt* . The work carried out in the Compton group on methods of fabricating and characterising arrays of nanoelectrodes, including multi-metal nanoparticle arrays for combinatorial electrochemistry, and on numerical simulating and modelling of the electrochemical processes was reviewed in the frontiers article written by Compton (Compton et al., 2008).

An improved sensitivity of voltammetric measurements as a consequence of either electrode or voltammetric cell exposure to low frequency sound was reported by Mikkelsen and Schrøder (Mikkelsen & Schrøder, 1999; Mikkelsen & Schrøder, 2000). According to the authors the longitudinal waves of sound applied during measurements make standing regions with different pressures and densities, which make streaming effects in the boundary layer at least comparable to the conventional stirring. As an alternative explanation of the marked sensitivity enhancement the authors suggested a possible change in the electrical double layer structure. Later, a study of the dopamine redox reactions on the carbon fiber microelectrode by the kinetics-sensitive voltcoulometry (Gmucová et al., 2002) revealed an impressive shift towards the ideal kinetic described by Cottrell equation, achieved by an electrochemical pretreatment of the electrode accompanied by its simultaneous exposure to the low frequency sound.

The diffusion equation including the delay of a concentration flux from the formation of a concentration gradient, called diffusion with memory, was formulated by Aoki and solved under chronoamperometric conditions (Aoki, 2006). A slower decay than predicted by the Cottrell equation was obtained.

A theoretical study of the current–time relationship aimed at the explanation of anomalous response in differential pulse polarography was reported by Lovrić and Zelić. The effect was explained by the adsorption of reactant at the electrode surface (Lovrić & Zelić, 2008). The situation connected with the formation of metal preconcentration at the electrode surface, followed by electrodissolution was modelled by Cutters and Compton. The theory to explore the electrochemical signals in such a case at a microelectrode or ultramicroelectrode arrays was derived (Cutress & Compton, 2009).

A Review of Non-Cottrellian Diffusion Towards Micro- and Nano-Structured Electrodes 13

The most important conclusions, as outlined in (Pajkossy, 1991), are as follows. If a capacitive electrode is of fractal geometry, then the electrode impedance will be of the constant phase element form (i.e., the impedance, *Z*, depends on the frequency, *ω*, as

constant phase element exponent can be established. Assume that a real surface is irregular from the geometrical point of view and that the diffusion-limited current can be measured on it, the surface irregularities can be characterized by a single number, the fractal dimension. The time dependence of the diffusion limited flux to a fractal surface is a power-

The determination of fractal dimension of a realistic surface has been reported by Ocon and co-workers (Ocon et al., 1991). The thin columnar gold electrodeposits (surface roughness factor 50-100) grown on gold wire cathodes by electroreducing hydrous gold oxide layers have been used for this purpose, the fractal dimension has been determined by measuring the diffusion controlled current of the Fe(CN)4-/Fe(CN)3- reaction. Several examples of diffusion controlled electrochemical reactions on irregular metal electrodeposits type of electrodes were described in (Arvia & Salvarezza, 1994). Using the fractal geometry relevant information about the degree of surface disorder and the surface growth mechanism was

obtained and the kinetic of electrochemical reactions at these surfaces was predicted.

an intermediate roughness a more complicated form has been obtained.

Kant has discussed rigorously the anomalous current transient behaviour of self-affine fractal surface in terms of power spectral density of the surface (Kant, 1997). The nonuniversality and dependence of intermediate time behaviour on the strength of fractality of the interface has been reported, the exact result for the low roughness and the asymptotic results for the intermediate and large roughness of self-affine fractal surfaces have been derived. The intermediate time behaviour of the reaction flux for the small roughness interface has been shown to be proportional to *t* -1/2 + const *t* -3/2+*H*, however, for the large roughness interfaces the dependence ~ *t* -1+*H*/2, where *H* is Hurst's exponent, was found. For

Shin and co-workers investigated the diffusion toward self-affine fractal interfaces by using diffusion-limited current transient combined with morphological analysis of the electrode surface (Shin et al., 2002). Here, the current transients from the electrodes with increasing morphological amplitude (roughness factor) were roughly characterised by the two-stage power dependence before temporal outer cut-off of fractality. Moreover, the authors suggested a method to interpret the anomalous current transient from the self-affine fractal electrodes with various amplitudes. This method, describing the anomalous current transient behaviour of self-affine electrodes, includes the determination of the apparent selfsimilar scaling properties of the self-affine fractal structure by the triangulation method.

A general transport phenomenon in the intercalation electrode with a fractal surface under the constraint of diffusion mixed with interfacial charge transfer has been modelled by using the kinetic Monte Carlo method based upon random walk approach (Lee & Pyun, 2005). Go and Pyun (Go & Pyun, 2007) reviewed anomalous diffusion towards and from fractal interface. They have explained both the diffusion-controlled and non-diffusion-controlled transfer processes. For the diffusion coupled with facile charge-transfer reaction the

< 1). However, no unique relation between fractal dimension *Df* and

. This equation provides a possibility for the experimental

= (*Df* - 1)/2) between fractal

 *Z* i

with 0 <

dimension and the exponent

determination of the fractal dimension.

law function of time, and there is a unique relation (

#### **5. Fractal concepts**

A possible cause of the deviation of measured signals from the ideal Cottrellian one is of geometric origin. The irregular (rough, porous or partially active) electrode geometry can and does cause current density inhomogeneities which in turn yield deviations from ideal behaviour. Kinetic processes at non-idealised, irregular surfaces often show nonconventional behaviour, and fractals offer an efficient way to handle irregularity in general terms. Rough and partially active electrodes are frequently modelled using fractal concepts; their surface roughness of limited length scales irregularities is often characterized as selfaffine fractal. Fractal geometry is an efficient tool for characterizing irregular surfaces in very general terms. An introduction to the methods of fractal analysis can be found in the work (Le Mehaute & Crepy, 1983). Electrochemistry at fractal interfaces has been reviewed by Pajkossy (Pajkossy, 1991). Diffusion-limited processes on such interfaces show anomalous behavior of the reaction flux.

Pajkossy and co-workers have published an interesting series of papers devoted to the electrochemistry on fractal surfaces (Nyikos & Pajkossy, 1986; Pajkossy & Nyikos, 1989a; Pajkossy & Nyikos, 1989b; Nyikos et al., 1990; Borosy et al., 1991). Diffusion to rough surfaces plays an important role in diverse fields, e.g., in catalysis, enzyme kinetics, fluorescence quenching and spin relaxation. Nyikos and Pajkossy have shown that, as a consequence of fractal electrode surface, the diffusion current is dependent on time as <sup>21</sup> *Df ti* , where *Df* is the fractal dimension (Nyikos & Pajkossy, 1986). For a smooth, twodimensional interface ( <sup>2</sup> *Df* ) the Cottrell behaviour <sup>21</sup> *ti* is obtained. In electrochemical terms this corresponds to a generalized Cottrell equation (or Warburg impedance) and can be used to describe the frequency dispersion caused by surface roughness effects. Later, the verification of the predicted behaviour for fractal surfaces with *Df* > 2 (rough interface), and *Df* < 2 (partially blocked surface or active islands on inactive support) was reported (Pajkossy & Nyikos, 1989a). The fractal decay kinetics has been shown to be valid for both contiguous and non-contiguous surfaces, rough or partially active surfaces. Using computer simulation, a mathematical model, and direct experiments on well defined fractal electrodes the fractal decay law has been confirmed for different surfaces. According to the authors, this fractal diffusion model has a feature which deserves some emphasis: this being its generality. It is based on a very general assumption, i.e., selfsimilarity of the irregular interface, and nothing specific concerning the electrode material, diffusing substance, etc. is assumed. Based on the generalized Cottrell equation, the calculation and experimental verification of linear sweep and cyclic voltammograms on fractal electrodes have been performed (Pajkossy & Nyikos, 1989). The generalized model has been shown to be valid for non-linear potential sweeps as well. Its experimental verification on an electrode with a well defined fractal geometry *Df* = 1.585 was presented for a rotating disc electrode of fractal surface (Nyikos et al., 1990). The fractal approximation has been shown to be useful for describing the geometrical aspects of diffusion processes at realistic rough or irregular-interfaces (Borosy et al., 1991). The authors have concluded that diffusion towards a self-affine fractal surface with much smaller vertical irregularity than horizontal irregularity leads to the conventional Cottrell relation between current and time of the Euclidean object, not the generalised Cottrell relation including fractal dimension.

A possible cause of the deviation of measured signals from the ideal Cottrellian one is of geometric origin. The irregular (rough, porous or partially active) electrode geometry can and does cause current density inhomogeneities which in turn yield deviations from ideal behaviour. Kinetic processes at non-idealised, irregular surfaces often show nonconventional behaviour, and fractals offer an efficient way to handle irregularity in general terms. Rough and partially active electrodes are frequently modelled using fractal concepts; their surface roughness of limited length scales irregularities is often characterized as selfaffine fractal. Fractal geometry is an efficient tool for characterizing irregular surfaces in very general terms. An introduction to the methods of fractal analysis can be found in the work (Le Mehaute & Crepy, 1983). Electrochemistry at fractal interfaces has been reviewed by Pajkossy (Pajkossy, 1991). Diffusion-limited processes on such interfaces show

Pajkossy and co-workers have published an interesting series of papers devoted to the electrochemistry on fractal surfaces (Nyikos & Pajkossy, 1986; Pajkossy & Nyikos, 1989a; Pajkossy & Nyikos, 1989b; Nyikos et al., 1990; Borosy et al., 1991). Diffusion to rough surfaces plays an important role in diverse fields, e.g., in catalysis, enzyme kinetics, fluorescence quenching and spin relaxation. Nyikos and Pajkossy have shown that, as a consequence of fractal electrode surface, the diffusion current is dependent on time as <sup>21</sup> *Df ti* , where *Df* is the fractal dimension (Nyikos & Pajkossy, 1986). For a smooth, twodimensional interface ( <sup>2</sup> *Df* ) the Cottrell behaviour <sup>21</sup> *ti* is obtained. In electrochemical terms this corresponds to a generalized Cottrell equation (or Warburg impedance) and can be used to describe the frequency dispersion caused by surface roughness effects. Later, the verification of the predicted behaviour for fractal surfaces with *Df* > 2 (rough interface), and *Df* < 2 (partially blocked surface or active islands on inactive support) was reported (Pajkossy & Nyikos, 1989a). The fractal decay kinetics has been shown to be valid for both contiguous and non-contiguous surfaces, rough or partially active surfaces. Using computer simulation, a mathematical model, and direct experiments on well defined fractal electrodes the fractal decay law has been confirmed for different surfaces. According to the authors, this fractal diffusion model has a feature which deserves some emphasis: this being its generality. It is based on a very general assumption, i.e., selfsimilarity of the irregular interface, and nothing specific concerning the electrode material, diffusing substance, etc. is assumed. Based on the generalized Cottrell equation, the calculation and experimental verification of linear sweep and cyclic voltammograms on fractal electrodes have been performed (Pajkossy & Nyikos, 1989). The generalized model has been shown to be valid for non-linear potential sweeps as well. Its experimental verification on an electrode with a well defined fractal geometry *Df* = 1.585 was presented for a rotating disc electrode of fractal surface (Nyikos et al., 1990). The fractal approximation has been shown to be useful for describing the geometrical aspects of diffusion processes at realistic rough or irregular-interfaces (Borosy et al., 1991). The authors have concluded that diffusion towards a self-affine fractal surface with much smaller vertical irregularity than horizontal irregularity leads to the conventional Cottrell relation between current and time of the Euclidean object, not the generalised Cottrell relation including fractal dimension.

**5. Fractal concepts** 

anomalous behavior of the reaction flux.

The most important conclusions, as outlined in (Pajkossy, 1991), are as follows. If a capacitive electrode is of fractal geometry, then the electrode impedance will be of the constant phase element form (i.e., the impedance, *Z*, depends on the frequency, *ω*, as *Z* i with 0 < < 1). However, no unique relation between fractal dimension *Df* and constant phase element exponent can be established. Assume that a real surface is irregular from the geometrical point of view and that the diffusion-limited current can be measured on it, the surface irregularities can be characterized by a single number, the fractal dimension. The time dependence of the diffusion limited flux to a fractal surface is a powerlaw function of time, and there is a unique relation ( = (*Df* - 1)/2) between fractal dimension and the exponent . This equation provides a possibility for the experimental determination of the fractal dimension.

The determination of fractal dimension of a realistic surface has been reported by Ocon and co-workers (Ocon et al., 1991). The thin columnar gold electrodeposits (surface roughness factor 50-100) grown on gold wire cathodes by electroreducing hydrous gold oxide layers have been used for this purpose, the fractal dimension has been determined by measuring the diffusion controlled current of the Fe(CN)4-/Fe(CN)3- reaction. Several examples of diffusion controlled electrochemical reactions on irregular metal electrodeposits type of electrodes were described in (Arvia & Salvarezza, 1994). Using the fractal geometry relevant information about the degree of surface disorder and the surface growth mechanism was obtained and the kinetic of electrochemical reactions at these surfaces was predicted.

Kant has discussed rigorously the anomalous current transient behaviour of self-affine fractal surface in terms of power spectral density of the surface (Kant, 1997). The nonuniversality and dependence of intermediate time behaviour on the strength of fractality of the interface has been reported, the exact result for the low roughness and the asymptotic results for the intermediate and large roughness of self-affine fractal surfaces have been derived. The intermediate time behaviour of the reaction flux for the small roughness interface has been shown to be proportional to *t* -1/2 + const *t* -3/2+*H*, however, for the large roughness interfaces the dependence ~ *t* -1+*H*/2, where *H* is Hurst's exponent, was found. For an intermediate roughness a more complicated form has been obtained.

Shin and co-workers investigated the diffusion toward self-affine fractal interfaces by using diffusion-limited current transient combined with morphological analysis of the electrode surface (Shin et al., 2002). Here, the current transients from the electrodes with increasing morphological amplitude (roughness factor) were roughly characterised by the two-stage power dependence before temporal outer cut-off of fractality. Moreover, the authors suggested a method to interpret the anomalous current transient from the self-affine fractal electrodes with various amplitudes. This method, describing the anomalous current transient behaviour of self-affine electrodes, includes the determination of the apparent selfsimilar scaling properties of the self-affine fractal structure by the triangulation method.

A general transport phenomenon in the intercalation electrode with a fractal surface under the constraint of diffusion mixed with interfacial charge transfer has been modelled by using the kinetic Monte Carlo method based upon random walk approach (Lee & Pyun, 2005). Go and Pyun (Go & Pyun, 2007) reviewed anomalous diffusion towards and from fractal interface. They have explained both the diffusion-controlled and non-diffusion-controlled transfer processes. For the diffusion coupled with facile charge-transfer reaction the

A Review of Non-Cottrellian Diffusion Towards Micro- and Nano-Structured Electrodes 15

knowledge on the kinetic of charge transfer during the studied redox reaction has lead to a significant number of theoretical, computational, phenomenological and, last but not least, experimental studies. Based on them one can conclude: nowadays, an un-usual behaviour is

This work was supported by the ASFEU project Centre for Applied Research of Nanoparticles, Activity 4.2, ITMS code 26240220011, supported by the Research & Development Operational Programme funded by the ERDF and by Slovak grant agency

Alkire, R.C.; Kolb, D.M.; Lipkowsky & J. Ross, P.N. (Eds.). (2009). *Advances in Electrochemical* 

Verlag GmbH & Co. KGaA, ISBN 978-3-527-31420-1, Weinheim Germany Aoki, K. & Osteryoung, J. (1981) Diffusion-Controlled Current at the Stationary Finite Disk

Aoki, K. & Osteryoung, J. (1984) Formulation of the Diffusion-Controlled Current at Very

Aoki, K.; Tokuda, K.; Matsuda, H. & Osteryoung, J. (1986) Reversible Square-Wave

Aoki, K. (1991) Nernst Equation Complicated by Electric Random Percolation at Conducting

Aoki, K. (2006) Diffusion-Controlled Current with Memory. *J. Electroanal. Chem.* Vol.592,

Arvia, A.J. & Salvarezza, R.C. (1994). Progress in the Knowledge of Irregular Solid Electrode

Belding, S.R.; Campbell, F.W.; Dickinson, E.J.F. & R.G. Compton, R.G. (2010) Nanoparticle-

Bieniasz, L.K. (2011). Extension of the Adaptive Huber Method for Solving Integral

Borosy, A. P.; Nyikos, L. & Pajkossy, T. (1991). Diffusion to Fractal Surfaces-V. Quasi-

Buck, R.P.; Mahir, T.M.; Mäckel, R. & Liess, H.-D. (1992) Unusual, Non-Cottrell Behavior of

*Science and Engineering, Volume 11, Chemically modified electrodes,* WILEY-VCH

Electrode. *J. Electroanal. Chem.* Vol.122, No.1, (May 1981), pp. 19-35, ISSN 1572-6657

Small Stationary Disk Electrodes. *J. Electroanal. Chem.* Vol.160, No.1-2 (January

Voltammograms Independence of Electrode Geometry. *J. Electroanal. Chem.* Vol.207,

Polymer-Coated Electrodes. *J. Electroanal. Chem.* Vol.310, No.1-2, (July 1991), pp. 1–

Surfaces. *Electrochimica Acta* Vol.39, No.11-12, (August 1994), pp. 1481–1494, ISSN

Modified Electrodes. *Physical Chemistry Chemical Physics* Vol.12, No.37, (October

Equations Occurring in Electroanalysis, onto Kernel Function Representing Fractional Diffusion. *Electroanalysis,* Vol.23, No.6, (June 2011), pp. 1506-1511, ISSN

Random Interfaces. *Electrochimica Acta.* Vol.36, No.1, (1991), pp. 163–165, ISSN

Ionic Transport in Thin Cells and in Films. *J. Electrochem. Soc.* Vol.139, No.6, (June

the Cottrellian one.

**8. References** 

**7. Acknowledgment** 

VEGA contract No.: 2/0093/10.

1984), pp. 335–339, ISSN 1572-6657

12, ISSN 1572-6657

0013-4686

1521-4109

0013-4686

No.1-2, (July 1986), pp. 335–339, ISSN 1572-6657

No.1, (July 2006), pp. 31-36, ISSN 1572-6657

2010), pp. 11208–11221, ISSN 1463-9084

1992), pp. 1611–1618, ISSN 0013-4651

electrochemical responses at fractal interface were treated with the help of the analytical solutions to the generalised diffusion equation. In order to provide a guideline in analysing anomalous diffusion coupled with sluggish charge-transfer reaction at fractal interface, i.e., non-diffusion-controlled transfer process across fractal interface, this review covered the recent results concerned to the effect of surface roughness on non-diffusion-controlled transfer process within the intercalation electrodes. It has been shown, that the numerical analysis of diffusion towards and from fractal interface can be used as a powerful tool to elucidate the transport phenomena of mass (ion for electrolyte and atom for intercalation electrode) across fractal interface whatever controls the overall transfer process.

A theoretical method based on limited scale power law form of the interfacial roughness power spectrum and the solution of diffusion equation under the diffusion-limited boundary conditions on rough interfaces was developed by Kant and Jha (Kant & Jha, 2007). The results were compared with experimentally obtained currents for nano- and microscales of roughness and are applicable for all time scales and roughness factors. Moreover, this work unravels the connection between the anomalous intermediate power law regime exponent and the morphological parameters of limited scales of fractality.

Kinetic response of surfaces defined by finite fractals has been addressed in the context of interaction of finite time independent fractals with a time-dependent diffusion field by a novel approach of Cantor Transform that provides simple closed form solutions and smooth transitions to asymptotic limits (Nair & Alam, 2010). In order to enable automatic simulation of electrochemical transient experiments performed under conditions of anomalous diffusion in the framework of the formalism of integral equations, the adaptive Huber method has been extended onto integral transformation kernel representing fractional diffusion (Bieniasz, 2011).

The fractal dimension can be simply estimated using the kinetics-sensitive voltcoulometry introduced by Thurzo and co-workers (Thurzo et al., 1999). On the basis of the multipoint analysis principles the transient charge is sampled at three different events in the interval between subsequent excitation pulses and the sampled values are combined according the appropriate filtering scheme. The third sampling event chosen at the end of measuring period and slow potential scans make the observation of non-Cottrellian responses easier. The parameter that enters the power-law time dependence of the transient charge, as well as the fractal dimension can be simply determined from two voltcoulograms obtained for two distinct sets of sampling events (Gmucová et al., 2002).
