Fractal Analysis Methods for Solid Alkane Monolayer Domains at SiO 2 /Air Interfaces

Abstract

A systematic evaluation of various fractal analysis methods is essential for studying morphologies of finite and noisy experimental patterns such as domains of long chain alkanes at SiO2/air interfaces. The derivation of trustworthy fractal dimensions crucially relies on the definition of confidence intervals for the assumed scaling range. We demonstrate that the determination of the intervals can be improved largely by comparing the scaling behavior of different morphological measures (area, boundary, curvature). We show that the combination of area and boundary data from coarse-grained structures obtained with the box-counting method reveals clear confidence limits and thus credible morphological data. This also holds for the Minkowski density method. It also reveals the confidence range. Its main drawback, the larger swing-in period at the lower cutoff compared to the box-counting method, is compensated by more details on the scaling behavior of area, boundary, and curvature. The sandbox method is less recommendable. It essentially delivers the same data as box-counting, but it is more susceptible to finite size effects at the lower cutoff. It is found that the domain morphology depends on the surface coverage of alkanes. The individual domains at low surface coverage have a fractal dimension of ≈ whereas at coverages well above ≈the scaling dimension is 2 with a large margin of uncertainty at ≈coverage. This change in morphology is attributed to a crossover from a growth regime dominated by diffusion-limited aggregation of individual domains to a regime where the growth is increasingly affected by annealing and the interaction of solid growth fronts which approach each other and thus compete for the alkane supply.