A triangular model of fractal growth with application to adsorptive spin-coating of polymers

Abstract

Over the last 40 years, applied mathematicians and physicists have proposed a number of mathematical models that produce structures exhibiting a fractal dimension. This work has coincided with the discovery that objects with fractal dimension are relatively common in the natural and human-produced worlds. One particularly successful model of fractal growth is the diffusion limited aggregation (DLA) model, a model as notable for its simplicity as for its complex and varied behavior. It has been modified and used to simulate fractal growth processes in numerous experimental and empirical contexts. In this work, we present an alternative fractal growth model that is based on a growing mass that bonds to particles in a surrounding medium and then exerts a force on them in an iterative process of growth and contraction. The resulting structure is a spreading triangular network rather than an aggregate of spheres, and the model is conceptually straightforward. To the best of our knowledge, this model is unique and differs in its dynamics and behavior from the DLA model and related particle aggregation models. We explore the behavior of the model, demonstrate the range of model output, and show that model output can have a variable fractal dimension between 1.5 and 1.83 that depends on model parameters. We also apply the model to simulating the development of polymer thin films prepared using spin-coating which also exhibit variable fractal dimensions. We demonstrate how the model can be adjusted to different dewetting conditions as well as how it can be used to simulate the modification of the polymer morphology under solvent annealing.