Monolayer adsorption on fractal surfaces: A simple two-dimensional simulation

Abstract

We have simulated the adsorption of homologous series of ‘‘molecules’’ on deterministic fractal curves in the whole range 1≤D<2. Adsorption differs from a regular Hausdorf dimension determination in the sense that in the latter process the molecules are put with their center on the curve in order to cover it, whereas in the former process the molecules have to reach the curve, coming from one side only, and are not allowed to cross it. Three series of molecules were used: (i) circles of increasing diameter, r; (ii) rectangles of increasing height, t; (iii) rectangles of high aspect ratio (almost lines) and increasing length, l. The fractal exponents Dr, Dt, and Dl were derived from ln Nm vs ln r, ln t, and ln l plots, respectively, Nm being the number of molecules necessary to reach ‘‘monolayer’’ coverage. With linear molecules, we found Dl=1 in whole range 1≤D<2. With the circular and rectangular molecules, two regimes were observed. In the range 1≤D<1.5, Dr=D and Dt=D−1. These results are consistent with each other and suggest that, in the range 1≤D<1.5, the ln Nm vs ln l surface is a plane of equation ln Nm=α ln l+β ln t+γ, with α=−1 and β=1−D. This provides the basis for predicting the size dependence of Nm for molecules of any aspect ratio. An entirely different regime was observed in the range 1.5<D<2. Instead of increasing with D, Dr and Dt were found to decrease according to Dr=3−D and Dt=2−D, respectively. This can be assigned to the generation of pore necks as the fractal dimension and the local concavity of the curve increases, and can be rationalized in terms of the rules governing the intersection of fractals. The model has been extended to adsorption on fractal surfaces. The prediction is that the maximum fractal dimension which could be measured without error is D=2.5, with molecules with constant aspect ratio. Using molecules of different shape would lead to considerable errors. However, this conclusion only applies to the particular class of fractals that we have considered, i.e., those for which the fractal character is restricted to their boundary. Fractal particles in which the fractal character stems from a particular internal morphology (i.e., from the distribution of mass, like in sponges or aggregates) may lead to an entirely different behavior.