Polymer models

One of the most transparent²⁵ applications of animals is to the modelling of polymers in solution. A polymer is a large molecule that is made up of many monomers connected together by chemical bonds; depending on how the monomers are connected to each other polymers can have very complicated topologies.

Polymers in dilute solution undergo a phase transition known as the coil-globule transition (see figure 20). At low temperatures (or in a poor solvent) the attractive interactions between monomers pull the polymer into a dense ball-like configuration or ``globule''. At high temperatures (or in a good solvent) the interactions are mediated by the solvent molecules, and the typical configurations are open coils. At a specific temperature, the $\theta$-point, the polymers undergo a phase transition (much like when water boils into vapour).

These phases (coil and globule) are characterised by the asymptotic behaviour of the average size²⁶ of a polymer containing $n$ monomers. If we consider the radius of gyration, $R_g(n)$, of a polymer (which is the average distance of a monomer from the polymer's centre of mass), then the average radius of gyration²⁷ of polymers with $n$ monomers is expected to behave as

$$\langle R_g \rangle_n \sim A n^{\nu} n \rightarrow \infty$$ (9)

where the critical exponent, $\nu$, is expected to be the same for all (mathematical or real) linear polymers. The value of $\nu$ changes between the phases; for linear polymers in three dimensions the best numerical estimates of $\nu$ are:

• in the swollen phase: $\nu \approx 0.588$,
• in the collapsed phase: $\nu = 1/3$.

It should be noted that if the polymer behaved like a random walk then one would have $\nu=1/2$.

Figure 20: In the collapsed phase the polymer forms dense ball-like ``globules'', while in its swollen phase it forms more open coils.

Consider a linear polymer (each monomer is connected to only two others). In a real polymer the monomers occupy positions in continuous space ( $\mathbb{R}^3$), and the bonds between them are constrained to have only certain angles (depending upon the nature of the monomers). We can simplify this situation by embedding the polymer into discrete space ( $\mathbb{Z}^3$), requiring that the monomers exist at integer coordinates only a single lattice spacing apart. Since the monomers cannot occupy the same space, the polymer embedded into discrete space is in fact a self-avoiding walk. To account for interactions between monomers, one can study the number of nearest-neighbour contacts (the number of points at which the walk passes within one lattice spacing of itself); by favouring or disfavouring these contacts one can study attractive and repulsive interactions -- see figure 21. Lattice trees and bond animals can be used to model polymers with more complicated topologies.

The self-avoiding walk model was introduced by Orr [35] to explore the geometric properties of linear polymers in a good solvent. At first glance it appears as though this model is far too simple to have any hope of modelling such a complex situation, however the phenomenon of universality tells us many quantities are not dependent on the specific details of the system, rather they are determined only by its universality class. The universality class is determined by the very general properties of the system, such as its dimension, and not by the very specific details (such as the type of lattice). All systems (real or mathematical) within the same universality class share the same dominant asymptotic behaviour close to a phase transition -- and so any member of the universality class can be used to determine this behaviour. For example, the critical exponent, $\nu$, in any given phase is expected to be the same for all linear polymers in three dimensions, regardless of precise details of the system -- $\nu$ for SAWs is exactly the same as $\nu$ for almost any ²⁸ linear polymer! On the other hand, other quantities such as the exact location of the $\theta$-point are not universal, and can only be determined by experiment.

Figure: Embedding a polymer into discrete space ( $\mathbb{Z}^3$) gives an interacting self-avoiding walk.