Flory theory of chain expansion

The Flory theory of chain expansion results in an explicit expression for the expansion factor. The key concept is that the energy of interaction of all the pairs of subchains depends on the distribution of subchains relahve to the center of mass. The theory starts with a consideration of the probability distribution for the radius of gyration of a long-chain molecule. The exact probability distribution for the radius of gyration of a phantom chain is complicated, but a good approximation was chosen by Flory  

To consider the effect of intersubunit interactions, Flory defined an energy for the chain as a function of the radius of gyration, averaged over all global conformations with the same radius of gyration, W(Rg)- The probability distribution for the radius of gyration of a real chain is then postulated to be  

The energy fxmction, W(Rg), was calculated by assuming that the local energy of interaction is proportional to the square of the local subunit density. The subunit density was obtained by assuming that the subchains were distributed relative to the center of mass in a Gaussian distribution. This model is called the smoothed-density model and will be used many times in this chapter. The number density of subunits at a location S relative to the center of mass in a macromolecule of overall radius of gyration Rq is given by  

The potential is then proportional to the integral of the square of the subunit density. The result can be expressed in the form  

The chain-expansion factor can be calculated by evaluating the mean-squared radius of gyration using Equation 5.5. Flory obtained an equation that can be written as  

---