Linear polymers may be thought of as very long flexible chains made up of single units called monomers. When placed in a solvent at low dilutions, they may exhibit several different types of morphology. If the interactions between different parts of the chain are primarily repulsive, they tend to be in extended configurations with a large entropy. If, however, the forces are sufficiently attractive, the chains collapse into compact objects with little entropy. The collapse transition between these two states occurs at the theta point, where the energy of attraction balances the entropy difference between the two states. This turns out to be a continuous phase transition, to be described later in Section 9.5. However, even in the swollen, entropy dominated, phase, it turns out that the statistics of very long chains are governed by non-trivial critical exponents. Like the percolation problem, this is a purely geometrical phenomenon, yet, through a mapping to a magnetic system, all the standard results of the renormalization group and scaling may be applied. Before describing this, however, it is important to recall some of the simpler approaches to the problem.

Random walk model

If the problem of a long polymer chain is equivalent to some kind of critical behaviour, we would expect universality to hold, and some of the important quantities to be independent of the microscopic details. This means that we may forget all about polymer chemistry, and regard the monomers as rigid links of length a, like a bicycle chain.