Flory terms and the N-body interaction terms

Flory's model is sometimes referred to as an early attempt at mean field calculations. Other authors claim it is not. Actually, what seems most clear from the example we offer here is that Flory's model works from the angle of the free volume that is similar to the $V-Nb$ term of the Van der Waals equation. Presuming $V » Nb$, the FE description for the Van der Waals equation (62) simplifies to

$$\Delta F=-Nk_B T\ln \left( V \right)+Nk_B T\left( {\frac{Nb}... ...frac{aN^2}{V}+\;\mbox{terms}\;\mbox{independent}\;\mbox{of}\;V$$ (84)

which after solving $P = - \partial F/ \partial V$, the simplified Van der Waals equation becomes

$$\left\{ {P+\left( {a-k_B Tb} \right)\mathop {\left( {\frac{N}{V}} \right)}\nolimits^2 } \right\}V=Nk_B T$$ (85)

and we can see that the analogy with the $\Theta$ point is very strong: $\psi_1 ( 1 - \Theta / T ) (N/V)^2$. So the term $b$ of (85) refers to the free volume correction ($\psi _1 )$ and the $a$ term refers to the heat of mixing ($\kappa _1 )$. It now should be almost clear that the entropy of mixing used in Flory's model is analogous to the free volume correction ($b)$ used in the Van der Waals model of a real gas.

In this respect, Flory's model is part of the tradition of mean field models in as much as the Van der Waals equation is to be taken to be a mean field model. It should be clear that whereas Flory's original theory had some clear weakness there were a number of good points about the approach that are worth imitating. There are some clear features of mean field theory in Flory's model. Perhaps the only complaint that can be raised is no ``mean field'' was specifically specified, but over all, it fits in the traditions of models like the Van der Waals equation which can also be solved conceptually without specifically obtaining the precise parameters of a mean field. From this brief expose, we hope the reader has gained some appreciation for how far Flory managed to get on this problem, as well as where he ran into troubles. We also hope the reader might be inspired to refrain from reckless approximation strategies and conscientiously write down equations.