# Gels

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Final Project for AP225 Fall 2011, written by Hyerim Hwang

## Definition

A gel can be thought to be liquid which behaves like solids. It has cross-linked network so that it can have a stick and hard structure but it has rubber-like properties at the same time because the segments which connects the network are flexible. There are many different gels with different classes of subunit and bonding. We may classify them under two large groups, chemical gels and physical gels. The bonds of chemical gels are covalent and those of physical gels are physical interactions. To make chemical gels, units should be polymerized to link themselves by chemical bonds. Epoxy resin is an example of chemical gels, which is formed from a short polymer with reactive groups on both ends and a hardener. Physical gels are joined together by physical bonds so these bonds can be easily broken by heat. In this way, physical gels are known as thermoreversible gels.

Figure 1. Gels.
Figure 2. Percolation model.As more bonds are added, clusters of points are formed and then they joined to form a cluster which spans the entire system.

## Gelation

One goes from liquid to a gel with adding bonds. If we create more bonds in a continuous way, the properties of the liquid change discontinuously, that is when some bonds has made, the liquid changes abruptly from a liquid to solid-like material, gel. There are many theoretical models to describe the gelation. The one of them is the percolation model. By this theory, an array of points can be joined by bonds. As more bonds are added, clusters of points are formed, until at some stage we form a cluster that spans the entire lattice. But there are still the questions we need to ask; the fraction of bonds that need to be made in order to obtain an infinite cluster, how the average size varies with the fraction bonds, the proportion of bonds to the infinite lattice as a function of fraction of bonds, and elasticity of the network.

## Percolation Theory

Percolation theory is the study of the connectivity of networks. If you take a piece of paper and punch small holes in it at random positions, it will remain connected if the density of holes is small. If you punch many holes that most of the paper is punched away, the paper will fall apart into small clusters. You may think of it as a phase transition in percolation, where the paper first falls apart. The computation exercise develops class libraries for creating percolation networks and breadth-first search algorithms for finding their clusters. The scaling exercise introduces scaling and critical phenomena methods for studying the phase transtion for percolation.

## References

1. Israelachvili, Jacob N. (2011). Intermolecular and Surface Forces. Academic Press. ISBN 9780123919274.

2. Jones, Richard A. N. (2002). Soft Condensed Matter. University Press. ISBN 9780198505891.

3. Last, B. J., Thouless, D. J., 1971. "Percolation Theory and Electrical Conductivity." Phys. Rev. Lett. 27, 1719–1721.