Electron Transfer through a Modified Electrode with a Fractal Structure: Cyclic Voltammetry and Chronoamperometry Responses C. P. Andrieux* ,† and P. Audebert* ,‡ Laboratoire d’Electrochimie Mole ´ culaire, UniVersite ´ Denis Diderot, 2 Pl. Jussieu, 75005 Paris, France, and Laboratoire de Photophysique et Photochimie Supramole ´ culaires et Macromole ´ culaires, Ecole Normale Supe ´ rieure de Cachan, 61 AV. du P t Wilson, 94235 Cachan, France ReceiVed: April 25, 2000; In Final Form: October 2, 2000 The problem of charge transfer by electron hopping through a volumic-modified electrode with a fractal structure was addressed, both in the case of chronoamperometry and linear sweep voltammetry. A behavior intermediate between that seen in classical diffusion and that of a thin layer was found. The currents in chronoamperometry and the peak currents in cyclic voltammetry both depend linearly on a bilogarithmic scale with, respectively, the time and the scan rate. The slopes of the curves in both cases depend on the fractal dimension, provided the spectral dimension remains constant. Plots are given in the case where the spectral dimension is equal to 4/3. Introduction The field of modified electrodes, although active for ∼20 years, continues to attract the attention of several scientists because of its numerous applications, particularly in the field of sensors. 1 Another field of renewed interest is establishing the relation between the electrochemical response of a volumic- modified electrode and its nanostructure, with the goal of gaining insight into the electrode material substructure. For example, the spatial distribution of the electroactive sites on the electrode material is expected to strongly influence the electrochemical response through the electron-transfer kinetics. In addition, close examination of the electrochemical response of a functionalized polymer is expected to give precious information about its structure. This approach has been introduced and used by us with functionalized conducting polymers 2-4 and recently with several sol-gel polymers, 5-10 for which the knowledge of the nanostructure is of great importance and often difficult to determine by other methods. Analysis of the current recorded from a modified electrode with a homogeneous repartition of the redox sites has long been addressed by Andrieux and Save ´ant, who found that the electron transfer was diffusive-like, with an apparent diffusion coefficient D ) kΔx 2 C°, 11 where C° represents the concentration of redox sites, ∆x is the average distance between two sites, and k is the isotopic rate constant inside the material (which is usually much different from its solution value). Further effort was made by Save ´ant to determine the role of k according to the polymer structure, 12 as well as the role of migration when it could occur. 13 However, no further effort was made to calculate the electro- chemical response of modified electrodes with a particular, nonisotropic distribution of the electroactive sites. Here we address the problem of both a linear concentration variation inside a homogeneous electrode as well as the problem of electron transport in a fractal electrode; that is, an electrode with a fractal distribution of the redox sites. Although the problem of diffusion toward an electrode with a fractal surface was addressed relatively early by several groups, 14-18 the problem of electron transfer in three-dimen- sional fractal electrodes (2 < df < 3) has not yet been addressed, probably because no examples of such electrodes were known. However, it is well known that inorganic silica xerogels and aerogels very often display such fractal structures, and the electrochemical behavior of a modified electrode prepared from a ferrocene functionalized xerogel exhibits quite anomalous behavior. These results will be discussed in light of the theoretical treatment presented next. Results and Discussion The problem of anomalous diffusion has been addressed by Alexander and Orbach, 19 who postulated that the number of sites visited was proportional to t d s /2 , where d s is introduced as the spectral dimension. On a fractal object, the number of accessible sites is proportional to r d f, where d f is the fractal dimension and r represents the distance to an arbitrary origin. Therefore, the diffusion coefficient D is no longer independent of time, but depends on time according to the following law: D ∝〈r 2 (t)〉/t ∝ 1/t (1 - d s /d f ) ∝ 1/r 2(d f /d s - 1) ,(d s /d f being <1). We will use θ to represent the value 2(d f /d s -1). The problem that we address is the one of a modified electrode containing electroactive sites able to undergo fast electron exchange, with the charge-transfer kinetics obeying the Andrieux-Save ´ant-Laviron hypotheses and with a fractal instead of an isotropic distribution of the accessible sites. Therefore, we continue to postulate the invariance of the electron exchange rate constant k throughout the polymer, which constitutes a limiting hypothesis for the applicability of the model, as will be discussed later. The charge-transfer kinetics can now obey the equation ∂C/∂t ) D(δ/x) θ ∂ 2 C/∂x 2 , where the D of the isotropic case has been changed into D(δ/x) θ , accounting for the fractal distribution. Here, δ represents the short distance cutoff length of the fractal structure, below which the system is again isotropic. This case also supposes that the electron exchange is not affected by the geometry of the † Laboratoire d’Electrochimie Mole ´culaire. ‡ Laboratoire de Photophysique et Photochimie Supramole ´culaires et Macromole ´culaires. 444 J. Phys. Chem. B 2001, 105, 444-448 10.1021/jp001565k CCC: $20.00 © 2001 American Chemical Society Published on Web 12/20/2000