The lattice model of polymer solutions

In the last note, we looked at the lattice model of small-molecule solutions. We now turn to polymers, using the same machinery. The lattice theory of polymer solutions is known as Flory-Huggins theory. In many ways, you will find this theory similar to the smallsolute case, except that the statistics are a little more complicated given that a polymer is a connected entity. Figure 1 shows a lattice model of a polymer. Each filled circle represents a chain segment, a piece of the polymer that is about the same size as a solvent molecule. A chain segment is not necessarily a monomer. In fact, for many polymeric materials, a chain segment will be smaller than a monomer. A polymer is taken to be made up of νs chain segments (on average). The value of νs can be estimated as the ratio of the molar volumes of the polymer and solvent. As in the small-molecule case, the solvent is assumed to have a coordination number z. Recall again that the flat illustration of figure 1 is only an aid to visualization and that, in general, the lattice will be three-dimensional, and may have a non-cubic geometry (e.g. it could be a hexagonal close-packed lattice). Neglecting the ends of a chain, which, for long polymers, represent a tiny minority of segments, a linear polymer segment makes z− 2 contacts with lattice sites to which the segment is not bonded. Another key assumption in the present note is that all the monomers are the same. There is therefore no energetic difference between placing one monomer or another in contact with the solvent. That being the case, there are only two significant possibilities to consider: Either the polymer is well solvated, i.e. it assumes a more-or-less random conformation in the solvent, or it doesn’t dissolve at all. In the next lecture, we will talk about a class of polymers for which this is not true, namely proteins.