Fractal Electrodes Research Articles

Diffusion toward Fractal Interfaces: Potentiostatic, Galvanostatic, and Linear Sweep Voltammetric Techniques

Diffusion to fractal electrodes has been investigated by three different techniques for the oxidation of ferrocyanide in a gel electrolyte: the current response to a potential step, the potential response to a current step, and the current response to a linearly increasing potential. The current density vs. time relationship under potentiostatic control, the transition timevs. current density under galvanostatic control, and the peak current density vs. sweep rate in a voltammetric experiment all followed power laws similar to those describing the experiments carried out with a planar electrode. The usual −0.5 exponent, however, became in the case of a fractal electrode of dimension . The experimental results were analyzed and described by solving a generalized diffusion equation involving a fractional derivative operator.

Electrocatalytic hydrogenation of conjugated enones on nickel boride, nickel, and Raney nickel electrodes

The selectivity of the electrocatalytic hydrogenation (ECH) of conjugated enones to the corresponding carbonyl compounds has been investigated in aqueous methanol under constant current at nickel boride, fractal nickel, and Raney nickel electrodes made of pressed powders. The influence of various parameters (water percentage, pH, concentration of substrate, and current density) on the selectivity of the carbon–carbon double bond hydrogenation was studied with cyclohex-2-en-1-one at nickel boride electrodes. Under given electrolysis conditions, fractal nickel electrodes were found to give the highest selectivity. The selectivity of the ECH of a variety of conjugated enones at fractal nickel electrodes was determined under electrolysis conditions that gave the best selectivity with cyclohexenone at nickel boride electrodes. Keywords: electrocatalytic hydrogenation, α,β-unsaturated ketones, selective hydrogenation, nickel based cathodes.

Linear and non-linear behavior of fractal and irregular electrodes

We describe a simple way to compute the response of an irregular resistive interface to a Laplacian field in d = 2. It permits to find the linear response of electrodes with an arbitrary geometry from the image only of the electrode. It also allows to compute the non-linear response of self similar electrodes. This method applies in principle to arbitrary irregular geometry in d = 2 and it permits to predict generally that the slope of the Tafel plot is divided by the fractal dimension.

A comparison of nickel boride and Raney nickel electrode activity in the electrocatalytic Hydrogenation of Phenanthrene

The electrocatalytic activity of nickel boride in the electrocatalytic hydrogenation (ECH) of phenanthrene in ethylene glycol–water at 80 °C has been compared to that of Raney nickel and fractal nickel. The intrinsic activity of the electrode material (real electrode activity) is the same for nickel boride and Raney nickel electrodes and is lower for fractal nickel electrodes. The apparent electrode activity of nickel boride pressed powder electrodes (Ni2B electrodes) is less than that of codeposited Raney nickel (RaNi) electrodes and pressed powder fractal nickel/Raney nickel (Ni/RaNi = 50/50 to 0/100) electrodes. The apparent activity of Ni2B electrodes is improved by adding sodium chloride to the powder and dissolving it after pressing (Ni2B–NaCl electrodes). The Ni2B–NaCl electrodes have the same apparent activity as codeposited RaNi and pressed powder Ni/RaNi (20/80 to 0/100) electrodes. The apparent and real electrode activity of Ni/RaNi electrodes increases with the RaNi content up to a 20/80 ratio. The Tafel and alternating current (ac) impedance parameters were determined for the hydrogen evolution reaction (HER) in 1 M aqueous sodium hydroxide at 25 °C at nickel boride and at codeposited RaNi electrodes. The intrinsic electrocatalytic activity for HER, expressed by the ratio of the exchange current density over the roughness factor (I0/R), is similar for Ni2B, Ni2–NaCl, and codeposited RaNi electrodes. Surface characterization of Ni2B and Ni2B–NaCl electrodes was carried out by BET, ac impedance, scanning electron microscopy, and mercury porosimetry. No direct relation between the apparent electrode activity in ECH and the surface measured by BET and ac impedance was found. The ac impedance measurements were also carried out in the presence of sodium trans-cinnamate.

SCALING IN THE FREQUENCY-DEPENDENT ADMITTANCE OF ELECTRODEPOSITED FRACTAL ELECTRODES

We describe measurements of the complex admittance of the interface between a class of fractal electrodes and electrolytic solutions over the frequency range 100 Hz to 40 kHz. The electrodes are grown by the diffusion-limited electrodeposition of copper from aqueous solution and have fractal dimensions comparable to those obtained for computer simulations of the diffusion-limited aggregation model, Df = 2.5. The admittance measured at various stages of the electrodeposit's growth displays constant phase angle (CPA) scaling with frequency. This behavior appears to have its origins in the disordered geometry of the system, although our measurements do not agree with the scaling predictions of theories for fractal blocking electrodes.

Investigations of the negative plate of lead/acid cells 3. Model calculations of the impedance of self-similar porous electrodes

In this paper a model for the calculation of the admittance (reciprocal impedance) of a porous metal electrode is derived, under anodic constant-current conditions, during precipitation of an insoluble salt at the metal/electrolyte interface. An example of the discharge, at low current density, of the negative plate in a lead/sulphuric acid battery during which lead sulphate is formed. The discussion is based on the assumption that porosity has a self-similar fractal character. The impedance of fractal electrodes will be discussed in terms of geometrical and physical parameters.

Equivalent-circuit, scaling, random-walk simulation, and an experimental study of self-similar fractal electrodes and interfaces.

A comprehensive study of the macroscopic transport parameters of self-similar interfaces is presented. The iteration of a simplified equivalent leads to the prediction of a simple mathematical expression for the impedance of fractal electrodes in d=2 and 3 dimensions. The same value is predicted by scaling arguments and verified by extended numerical simulations in d=2. Experiments on model electrodes confirm the theoretical prediction. We introduce the approximate concept of an information fractal. It gives a very simple access to the theory and a description of the regions of the fractal surface which are really active for the transport. The same result should apply to transport across fractal membranes and to certain Eley-Rideal heterogeneous catalysis processes.

Impedance studies of porous lanthanum-phosphate-bonded nickel electrodes in concentrated sodium hydroxide solution

The hydrogen evolution reaction (HER) was investigated on a lanthanum-phosphate-bonded nickel (LPBN) powder electrode in 30 wt.% NaOH at 70°C using ac impedance and steady-state polarization techniques. Circuits containing one or two constant-phase elements (CPEs) in parallel with a resistance and corresponding to fractal and porous electrode models were tested in order to interpret the ac impedance data. The experimental impedance spectra were well described by the porous electrode model and the circuit containing two CPEs. The results obtained from the ac impedance and steady-state measurements allowed the mechanism and kinetics of the HER to be evaluated. Comparison of these parameters with those obtained on the polycrystalline nickel electrode in 1 M alkaline solution at 25°C indicates that an increase in activity is principally due to an increase in the real surface area.

Active surface and adaptability of fractal membranes and electrodes

We study the properties of a Laplacian potential around an irregular object of finite surface resistance. This can describe the electrical potential in an irregular electrochemical cell as well as the concentration in a problem of diffusion towards an irregular membrane of finite permeability. We show that using a simple fractal generator one can approximately predict the localization of the active zones of a deterministic fractal electrode of zero resistance. When the surface resistance r s is finite there exists a crossover length L c : In pores of sizes smaller than L c the current is homogeneously distributed. In pores of sizes larger than L c the same behavior as in the case r s =0 is observed, namely the current concentrates at the entrance of the pore. From this consideration one can predict the active surface localization in the case of finite r s . We then introduce a coarse-graining procedure which maps the problem of non-null r s into that of r s =0. This permits us to obtain the dependence of the admittance and of the active surface on r s . Finally, we show that the fractal geometry can be the most efficient for a membrane or electrode that has to work under very variable conditions

Study of electrode activities towards the hydrogen evolution reaction by a.c. impedance spectroscopy

Abstract Electrical equivalent models were tested for the description of the hydrogen evolution reaction on various polycrystalline and rough electrodes in alkaline solution. It was found that the model containing the constant phase element best describes the experimental impedance curves, but in some cases the fractal or porous electrode models had to be used. Determination of the surface roughness allowed us to estimate the real electrode activity. It was found that, except for the electrodes containing noble metals, the main contribution to the activity arises from the large surface area.

Local probe microscopies and their applications to semiconducting materials

The aim of this work is two-fold: -to describe the principles of scanning tunneling (STM) and scanning force (SFM) microscopies; -to summarize the most important results obtained so far on atom-resolved surfaces of Si or GaAs and from the study of photocatalysts like TiO 2 (single crystals, powders, fractal electrodes, transparent ceramics).

Diffusion-limited processes on fractal surfaces generated as a set of scaled Walsh functions

The Monte Carlo technique was used for modelling mass transport to fractal electrode surfaces under conditions of a potentiostatic pulse and linear scan voltammetry. Fractal surfaces of different fractal dimension D were constructed using the Walsh-Hadamard transform. The effects of surface geometry on the form of the current response and long-time cut-off are discussed with respect to the validity of diagnostic procedures used for the electrode kinetics on smooth electrodes.

The Monte Carlo simulation of the voltammetry at fractal surfaces

The single-sweep voltammetry of a reversible electrode reaction proceeding on a fractal metallic electrode was simulated by the Monte Carlo technique. The diffusion of oxidized and reduced forms was represented by two-dimensional movements in a plane perpendicular to the electrode surface. This model yields the theoretical current-voltage curve for a planar electrode and higher currents are observed at fractal surfaces. The simulations describe not only the macroscopic behaviour, but at the same time the stochastic properties of the electrode reactions. Schottky's theorem was confirmed for the planar electrode and the slope of Schottky's plot is almost doubled on a fractal surface. The distribution of reaction sites (or the penetration depth) was evaluated and can be used for more complex reaction schemes.

Random-walk simulation of the response of irregular or fractal interfaces and membranes.

A method of simulating the response of an irregular interface is investigated. It is based on an exact mapping between the Laplace equation and the steady-state diffusion equation with mixed boundary conditions. Simulations in two dimensions show that diffusion-limited aggregation (DLA) and other self-similar fractal electrodes exhibit the so-called constant-phase-angle (CPA) behavior. In the case of DLA electrodes the CPA exponent is found by this method to be close to the inverse of the fractal dimension. We show that this is directly related to the fact that in d=2 the admittance is proportional to the overall size of the self-similar electrode.

Electrochemistry at fractal interfaces: the coupling of ac and dc behaviour at irregular electrodes

Impedance models pertaining to the ideally polarizable (blocking) fractal interface and those describing diffusion-limited charge transfer across fractal surfaces are reviewed. In the blocking case no general prediction can be made: although the non-trivial frequency dispersion often exhibits constant phase angle behaviour, the frequency exponent, α, is not uniquely related to the fractal dimension of the interface. In contrast, the generalized diffusion impedance (or Contrell response) is universal, owing to the fact that the time-dependent “yardstick”—the diffusion length—measures the fractal dimension proper. After discussing some consequences of these models, a new conjecture is presented: the ac behaviour of the blocking interface and the Tafel slope of the dc polarization curve measured in the presence of an electroactive material are interrelated at fractal electrode surfaces. Three examples are given where the modified Tafell slope of the irregular electrode is in fact the original value—which is characteristic to the electrode reaction and which is observed at smooth, planar interfaces—multiplied by the frequency exponent, α, of the blocking impedance.

Scaling-law analysis to describe the impedance behavior of fractal electrodes.

In order to explain the frequency dispersion of the impedance spectra of irregular surfaces and the frequent occurrence of the constant-phase element (CPE), several authors have proposed a fractal description for the capacitive (blocking) interface. These models, however, lead to different conclusions. In this paper it is shown how analysis of the scaling laws of electrochemical cells can be applied to predict the dependence of the CPE exponent on the Hausdorff dimension. Our general treatment reproduces the results of about a dozen published models. Furthermore, four other, new fractal models are presented, and it is shown that analysis of the scaling laws yields the same result as independent numerical or analytical methods.

Tafel current at fractal electrodes: Connection with admittance spectra

Various types of fractal electrode/electrolyte interfaces are investigated in the presence of an electrode reaction, the rate of which is described by the Butler-Volmer equation in a wide range of overpotentials. Applying scaling analysis, an interesting connection between the Tafel current at a fractal electrode, both in the limit of high and small overpotentials on the one hand and the admittance behaviour of the same electrode on the other, reveals itself. In particular, from Tafel plots in the limit of high overpotentials the effective transfer coefficients are seen to be modified by a geometrical factor α which is equal to the exponent appearing in the description of the CPE frequency dependence found in admittance measurements. Furthermore, at small overpotentials the current is shown to contain information on the exchange current density in a way which applies to all electrode geometries which give rise to CPE behaviour, i.e. not specifically to fractal electrodes.

On skewed ARC plots of impedance of electrodes with an irreversible electrode process

By a careful analysis of the form of the current implied by the boundary conditions satisfied by the solution of Poisson's equation, we generalize previous results and show that irreversible redox processes at a self similarly rough (fractal) electrode give rise to a skewed arc plot in the complex impedance-plane. We also discuss the form of scaling relations obeyed by the current at the electrode to ensure the validity of Ohm's law in the solution.

Fractal electrodes and constant phase angle response

Previous papers on fractal electrodes by Wang and Pajkossy and Nyikos are discussed.

Diffusion to fractal surfaces—II. Verification of theory

Decay of the diffusion controlled current of particles diffusing from an initially homogeneous medium to a completely absorbing fractal boundary was previously shown to exhibit t −α time-dependence instead of the conventional t − 1 2 one with the exponent α being determined by the fractal dimension, D f, of the interface as α =( D f−1)/2. 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. We verify the predicted behaviour for fractal surfaces with D f>2 (rough interface), and D f<2 (partially blocked surface or active islands on inactive support). In addition, the fractal decay kinetics is shown to be valid for both contiguous and non-contiguous surfaces. Computer simulation, a mathematical model, and direct experiments on well defined fractal electrodes are the tools for verifying the fractal decay law for the different surfaces. The predicted power law behaviour is observed, and the predicted α( D f) relationship was seen to prevail in each case.