Impact of Fractal Geometry on Permittivity and Related Quantities

Abstract

A new physical model of relative permittivity and derived quantities of fractal structures, contrary to the current concept, predicts their dependence on the fractal dimension (D) and on the extent (R) of the considered structure. The scaling of considered quantities predicted by our theory is  R D-d , where d is dimension of Euclidean space. Hence, fractality, a feature often found in physical, chemical and biological systems, influences also basic properties of such systems that, so far, were believed to be material and not structure related. The theory has been experimentally verified by electrochemical measurements of capacitance of adsorbed layers of nonionic surfactant Triton-X-100 and linoleic acid that have fractal structure. The experimental results, substantiating theoretical predictions, are presented and the influence of relevant parameters is discussed. IntroductionThe dielectric features of material, relative permittivity, susceptibility, polarizibility and related properties such as capacitance play a significant role in various chemical, physical and biological systems, and on scales ranging from microscopic to macroscopic. Among other things capacitance is link to the knowledge of the fluctuations of several physical quantities, e.g. voltage and electromagnetic field fluctuations, 1-6 dipole moment, 5-7 pH and charge, 8 and also to polarizibility and dielectric dispersion of colloidal and polyelectrolite systems [5][6][7] . Hence, it is crucial for understanding of biological systems that include ionic channels and cell membranes, [9][10][11] but also other systems such as adsorbed and spread films [12][13][14] . On the other hand it has been shown that many of these systems are fractal structures or exhibit fractal behavior, 15 and can be described with effective fractal dimension D. Therefore it was worthwhile to examine the possible influence of fractal geometry on fundamental and derived electric properties. In that context we examine the relative permittivity and specific capacitance of fractal structures and their dependence on geometrical features.The notion of these quantities in classical (non-fractal) system implies that they are constant and characteristics of the material of the considered system be it physical, chemical or biological.Starting from considerations of fundamental material properties we have developed a general theoretical model that predicts dependence of relative permittivity and consequently Cs of fractal structures (e.g. adsorbed layer, molecular or particle aggregates, porous material etc.) on structure size and fractal dimension. These theoretical predictions have been experimentally verified by electrochemical measurements of capacitance of adsorbed layers possessing fractal structure, in 3 conjunction with recently introduced method for determination of fractal dimension of such layers 16 . This specific testbed was chosen because structural and dynamic properties of adsorbed molecular films are of both fundamental and applied interest in diverse areas. These ...

The simple electrochemical method for detection of 3d molecular reorientation in adsorbed layers of organic substances

“…After that time, the rate of change of relative coverage and consequently that of the capacitive current, is governed only by the diffusion flux, and complies with Eqs. (8) and (9). Hence, we suppose that the inflection point on the dI c /dt curve and subsequent steep decrease of the rate of change corresponds to reorientation (erection) of molecules initially adsorbed flat.…”

“…Recently, it has been shown that these processes can be detected by the corresponding changes of the fractal dimension, D, of the adsorbed layer [5,6]. Moreover, the fractal dimension is also sensitive to the changes in hydrodynamic conditions and influences the basic material properties [7,8]. To determine the fractal dimension of an adsorbed layer, a new method was developed employing the scaling of a hanging mercury drop electrode [6].…”

Hydrodynamic influence on the fractal morphology of the linoleic acid adsorbed layer at the mercury/electrolyte interface

“…The same relation can be derived using the recent finding [25] that the specific capacitance, Cs, of any fractal structure scales as Cs ~ R D-d . Now, the total capacitance is given by Ct = ACs, and as A ~ R 2 , taking into account Eq.2, we again get Eq.…”

“…It has been demonstrated that in the case of the linoleic acid (LA) adsorption at the mercury electrode, subtle changes in the adsorption process resulting in structural changes of the adsorbed layer are reflected in, and can be observed through, the change in the corresponding fractal dimension [14]. Moreover, recently it was shown that geometrical features, such as fractality also reflect itself in basic material properties such as permittivity, polarization and specific 4 capacity [25]. This indicates that fractal analysis could provide more subtle information of considered system than usually recognized.…”

Interfacial adsorption with molecular reorientation

“…The situation is somewhat different in case of fractal layer where these structural changes are directly reflected in the corresponding change of a fractal dimension of the adsorbed layer [28,29]. Moreover, fractal dimension is also sensitive to the hydrodynamic conditions during the adsorption and influences some basic material properties [30,31]. In the case of adsorption on the mercury drop electrode, the method of scaling of the hanging mercury drop electrode [29] provides means for determination of the fractal dimension and consequently detection/characterization of different structural transformations.…”