Flory's hypothesis

In the previous chapter we have treated a model which takes into account the short range excluded volume effect. The long range excluded volume effect  however has not been taken into account, in concreto: two monomers whose indices along the chain differ by more than 4 units may happen to occupy the same position, something that in a real system is impossible. As we shall see in the next chapter Eq. (1.46) is a direct consequence of this fact. In fact numerous computer simulations have shown that in reality 

 

$$bN^{\nu },$$ R_g = (2.1)

$$\nu$$ = 0.588. (2.2)

One interesting point about $\nu$ is that it is a universal coefficient , i.e. that it is independent of the details of the chain and depends only on very few parameters, like for example the dimension of the chain.

Calculation of the coefficient $\nu$ is an extremely hard problem. In the case of a chain in vacuum like in Eq. (2.2) one may intuitively understand why $\nu >0.5$. Consider building the chain by consecutively adding monomers. At every step there are on average more monomers in the back than in front of the last monomer. Therefore the chain can gain entropy  by going out, and being larger than a chain in which the new monomer does not feel its predecessors.

For chains in a solvent the story may be different. In a good solvent  the situation is like in vacuum. In a bad solvent however apart from the excluded volume, which is nothing but the short range van der Waals interaction , two monomers may feel an effective attraction at distances just outside the van der Waals radius. In case this attraction is strong enough it may cause the chain to shrink. Of course there is a whole range between good and bad solvents, and at some point both effects cancel and Eq. (1.46) holds true. A solvent having this property is called a $\Theta$-solvent .

Now we come to Flory's argument . In a polymer melt, every monomer is isotropically surrounded by other monomers, and there is no way to decide whether the surrounding monomers belong to the same chain as the monomer at hand or to a different one. Consequently there will be no preferred direction and the polymer melt will act as a $\Theta$-solvent.

Flory's hypothesis has gotten some credibility from experiments and computer simulations. Here we shall adopt the hypothesis and use it to calculate the intramolecular distribution function defined below. Using the integral equation theory of simple liquids we shall then calculate the complete radial distribution function.