Current transient fractality

Johans et al. derived a model for diffusion-controlled electrodeposition at liquid-liquid interface taking into account the development of diffusion fields in both phases [91]. The current transients exhibited rising portions followed by planar diffusion-controlled decay. These features are very similar to those commonly observed in three-dimensional nucleation of metals onto solid electrodes [173-175]. The authors reduced aqueous ammonium tetrachloropalladate by butylferrocene in DCE. The experimental transients were in good agreement with the theoretical ones. The nucleation rate was considered to depend exponentially on the applied potential and a one-electron step was found to be rate determining. The results were taken to confirm the absence of preferential nucleation sites at the liquid-liquid interface. Other nucleation work at the liquid-liquid interface has described the formation of two-dimensional metallic films with rather interesting fractal shapes [176]. 

As mentioned in potentiostatic current transient method, when the fractal dimension is determined by using diffusion-limited electrochemical technique, the diffusion layer length acts as a yardstick length.122 In the case of cyclic voltammetry, it was... 

Since diffusing species move randomly in all directions, the diffusing species may sense the self-affine fractal surface and the self-similar fractal surface in quite different ways. Nevertheless a little attention has been paid to diffusion towards self-affine fractal electrodes. Only a few researchers have realized this problem Borosy et al.148 reported that diffusion towards self-affine fractal surface leads to the conventional Cottrell relation rather than the generalized Cottrell relation, and Kant149,150 discussed the anomalous current transient behavior of the self-affine fractal surface in terms of power spectral density of the surface. 

In order to examine the current response to the imposition of the potential step on the self-affine fractal interface, the current transients were calculated theoretically by random walk simulation.153 The simulation cell was taken as the square area bottom boundary which is replaced by one of the self-affine fractal profiles in Figure 7. The details of the simulation condition were described in their publication.43... 

Figures 8a and 8b present the simulated current transients obtained from the self-affine fractal interfaces of r/ = 0.1 0.3 0.5 and r] = 1.0 2.0 4.0, respectively, embedded by the Euclidean two-dimensional space. It is well known that the current-time relation during the current transient experiment is expressed as the generalized Cottrell equation of Eqs. (16) and (24).154 So, the power exponent -a should have the value of - 0.75 for all the above self-affine fractal interfaces.

However, the simulated current transients in Figure 8 never exhibited the expected power exponent of -0.75 with the exception of the original self-affine fractal interface ( = 1) the... 

H. -C. Shin et al., A study on the simulated diffusion-limited current transient of a self-affine fractal electrode based upon the scaling property, J. Electroanal. Chem., 531 p. 101, Copyright 2002, with permission from Elsevier Science. 

Figure 8. Simulated current transients obtained from the self-affine fractal profiles h(x) of various morphological amplitudes rj of (a) 0.1, 0.3, and 0.5 (b) 1.0, 2.0, and 4.0 in h(x) = 7]fws(x). Reprinted from H.-C. Shin et al., A study on the simulated diffusion-limited current transient of a self-affine fractal electrode based upon the scaling property, J. Electroanal. Chem., 531, p. 101, Copyright 2002, with permission from Elsevier Science.

Fractal Dimensions of the Profiles h(x) at Various Morphological Amplitudes rj in h(x) = 77/wsCv) Determined by the Current Transient Technique (2nd Column) and the Triangulation Method (3rd Column). Here, /ws(-v) Means the Weierstrass Function with a Self-Affine Fractal Dimension t/Fsa = 1.5 ... 

From the above results, it is noted that the self-similar scaling property investigated by the triangulation method can be effectively utilized to analyze the diffusion towards the self-affine fractal interface. This is the first attempt to relate the power dependence of the current transient obtained from the self-affine fractal curve to the self-similar scaling properties of the curve. 

In practical application, it was reported that the platinum particles dispersed in highly porous carbonized polyacrylonitrile (PAN) microcellular foam used as fuel-cell electrocatalyst160 have the partially active property. The fractal dimension of the platinum particles was determined to be smaller than 2.0 by using the potentiostatic current transient technique in oxygen-saturated solutions, and it was considered to be a reaction dimension, indicating that not all of the platinum particle surface sites are accessible to the incoming oxygen molecules. 

Go, J.-Y., and Pyun, S.-L 2004. A study on lithium transport through fi-actal l.p.sCoOj film electrode by analysis of current transient based upon fractal theory. Electrochimica Acta 49, 2551-2562. 

Jung, K.-N., and Pyun, S.-I. 2006b. The cell-impedance-controlled lithium transport through LiMn2O4 film electrode with fractal surface by analyses of ac-impedance spectra, potentiostatic current transient and linear sweep voltammograms. Electrochimica Acta 51, 4649 658. 

The ion-exchanged polypyrrole shows increased porosity and increased fractal dimension. This is seen by analyzing the current transients during the oxidation of solution species (such as hydroquinone) at this polymer electrode. The current transient shows a r" dependence as mentioned above. This is shown in Fig. 6. 

Lee, J.-W. Pyun, S.-I. (2005). A Study on the Potentiostatic Current Transient and Linear Sweep Voltammogram Simulated from Fractal Intercalation Electrode Diffusion... 

Serna, C. Molina, A. (1999) General Solutions for the 1/1 Response for Reversible Processes in the Presence of Product in a Multipotential Step Experiment at Planar and Spherical Electrodes whose Areas Increase with any Power of Time.. Electroanal. Chem. Vol.466, No.l, (May 1999), pp. 8-14, ISSN 1572-6657 Shin, H.-Ch. Pyxm, S.-I. Go, J.-Y. (2002). A Study on the Simulated Diffusion-Limited Current Transient of a Self-Affine Fractal Electrode Based upon the Scaling Property. J. Electroanal. Chem. Vol.531, No.2, (August 2002), p>p. 101-109, ISSN 1572-6657... 

K. N. Jung and S. I. Pyun, Electrochim. Acta, 52, 2009 (2007). Theoretical Approach to Cell-Impedance-Controlled Lithium Transport through Lij Mu204 Film Electrode with Partially Inactive Fractal Surface by Analyses of Potentiostatic Current Transient and Linear Sweep Voltammogram. 

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