Department of Mathematics and Statistics, Mount Holyoke College, South Hadley, Massachusetts, United States of America.
Department of Chemistry, Mount Holyoke College, South Hadley, Massachusetts, United States of America.
PLoS One. 2024 Feb 23;19(2):e0298916. doi: 10.1371/journal.pone.0298916. eCollection 2024.
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.
在过去的 40 年中，应用数学家和物理学家提出了许多数学模型，这些模型产生的结构具有分形维数。这项工作恰逢发现分形维数的物体在自然和人为产生的世界中相对常见。分形生长的一个特别成功的模型是扩散限制聚集（DLA）模型，该模型以其简单性和复杂多样的行为而著称。它已经被修改并用于模拟众多实验和经验背景下的分形生长过程。在这项工作中，我们提出了一种替代的分形生长模型，该模型基于生长的质量与周围介质中的颗粒结合，然后在生长和收缩的迭代过程中对其施加力。由此产生的结构是一个扩展的三角网络，而不是球体的聚集，并且该模型在概念上很简单。据我们所知，这种模型是独特的，其动力学和行为与 DLA 模型和相关的粒子聚集模型不同。我们探索了模型的行为，展示了模型输出的范围，并表明模型输出的分形维数可以在 1.5 到 1.83 之间变化，具体取决于模型参数。我们还将模型应用于模拟使用旋涂法制备的聚合物薄膜的发展，这些薄膜也表现出可变的分形维数。我们演示了如何调整模型以适应不同的去湿条件，以及如何使用它来模拟溶剂退火下聚合物形态的修改。
