The fractal dimension of the percolation by invasion cluster at the percolation threshold is 91/48=1.89
Author: Alexis Monnerot-Dumaine; Source: Wikipedia.org

Since 7th this month I have become a PhD candidate, after graduated from Master in June (In China it is possible and prevailing one graduate from a Master and go on PhD under a different supervisor). My new project will be quite theoretically tuned which I am not familiar with at all. I have to make a plan for the initial literature review, which could be a project itself, by setting a logical framework of questions to answer.

The core key word of my project is physical gelation. Gelation is the process of gelling, from a sol to a gel. Eggs gel when heating for example. But in contrast to eggs, physical gelation is reversible. After randomly reading several papers of this field I can make the following statement:

Chemical (irreversible) gelation has been described with multiple models/theories from the classical Flory-Stockmayer to the most recent ones. All these theories in fact belong to percolation models differing in minor but decisive aspects, while a universal scaling relationship is observed in all these theories and therefore a fractal/self-similar structure in gelling is implied. Whereas the development of the gelation theories agree quite well now with experiments regarding chemical gelation, physcial (reversible) gelation, because of significantly different nature of crosslinking, is still questionable while these theories are applied.

This understanding of the current state of study is obvious not embodied enough to ask my own question. I need to answer, by reviewing papers and referring to books, the following questions:

The ‘What’ questions: What’s percolation? What’s fractal? What’s self-similarity? What’s the fractal structure in gel?

The ‘How’ questions: How many different percolation theories are there (major ones)? How these theories coincidentally come to a scaling law? How physical gel differ from chemical gel THEORETICALLY, so that it deviates from the prediction of current theories? How does the fractal structure grow when gelling, as predicted by a percolation theory? How different gelling mechanisms differ in geometric growth of fractal structural, so that agreements are met with different percolation theories?

One general questions: Can I observed the fractal structure by microscope?

To answer these questions I decide to start from experiment, as all science practice do. After reading a large number of experimental works I should be clear what phenomena are actually observed (maybe of several classes) and decide the most agreeing theories to learn in order to save time. By comparing the experimental results and the prediction of the theories in the literature I may find some problems existing in there. I guess problems may lie in the complex nature of physical crosslinking, which differs largely among different macromolecular systems. So I may design and synthesize a specific model macromolecule for experiment that can hopefully answer the currently unsolved question from a new and valuable aspect. Finally it will very interesting if I can figure out a method to directly watch the growth of fractal structure during gelling by microscope and compared with the pictures of those pure mathematical simulation, or in other words add another paint to the gallery of naturally occurring fractal, among many others such as ferns, mountains, cauliflower, and aveoli of the lungs. However this relies on comprehensive understanding of fractal geometry and is too early to say now.

Run percolation in you browser and watch the growth of fractal patterns