Flory's polymer swelling model

The Flory model for the polymer-solvent interactions originally addressed polymer swelling in which the root-mean-square (rms) end-to-end separation distance was seen to increase quite dramatically compared to the Gaussian polymer chain (GPC) model. This effect occurs because there are strong attractive interactions between the polymer and the solvent and therefore, the solvent quickly occupies much of the GPC volume coordinating with the polymer. The effect is so pronounced, that the rms end-to-end separation distance is seen to increase exponentially as $\sqrt {\left\langle {r^2} \right\rangle } =\xi^{2/3}N^{3/5}b$; substantially faster than the GPC ( $r_o =\sqrt {\left\langle {r^2} \right\rangle } =(\xi N)^{1/2}b$).

The swelling phenomena involves two major thermodynamic contributions, one is an elastic response from the polymer chain to its structure being forced from its ideal equilibrium configuration $r_o$ to a value $\alpha r_o$. This is $F_{el}(\alpha)$. The other contribution is due to the overall attractive (or repulsive) interaction between the polymer with itself and the solvent that is carrying it. In good solvent, the polymer has more interest in interacting with the solvent and exhibits comparatively little self interaction. However, in poor solvent, the polymer tends to favor interacting with itself and only allows partial interaction with the solvent. This latter contribution is known as the internal interactions: $F_{int}(\alpha)$.

When the polymer swells or contract as $r = \alpha r_o$, the distribution function in (3) must also adjust to reflect this changed distribution by an equivalent amount. Thus

$$r_{ij}^\prime =\alpha r_{ij} \;\mbox{and}\;\vartheta _{ij}^\prime =\frac{\vartheta _{ij} }{\alpha ^\delta }$$ (7)

where substitution of these expressions into (3) shows that (3) is invariant under these transformations.

We now proceed to run through the steps in a similar fashion as Flory's model, to solve for the free energy contribution to the swelling. We describe first $F_{el}$ followed by $F_{int}$. After showing Flory's solution, we turn to the Van der Waals equations to view this analogy with respect to this mean field approximation. We then show how to use Flory's strategy combined with our understanding of the mean field approximation used in the Van der Waals gas equation to find a reasonable physical model for both polymer swelling and the formation of globular structures (effectively polymer contraction or the opposite of swelling). Finally, in the last section we return to Flory's solution and look more closely at how it relates to the mean field approximations relative to the Van der Waals approximation.