The structure of macromolecules in dilute solution

In order to discuss the behavior of single chains in solution, an appropriate geometric description must be chosen. Because only global dimensions are being considered, it is convenient to adopt the model of a chain molecule that consists of m subchains that are themselves long enough to exhibit the asymptotic behavior of a random coil. Each subchain is characterized by an end-to-end vector, f, and the total end-to-end vector, R, is the vector sum of the subchain vectors (Equation 2.26). It is also necessary to specify the location of they enter of mass of each subchain, R. The intersubunit vectors, R i = Ric Ri, are important quantities in the description of the global conformational state of a real-chain molecule in dilute solution. 

Now consider a pair of subchains in solution. The Gibbs energy of the solution is a function of the distance between the two subchains, R, averaged over all conformations of the pair of subunits and of the associated 

The interaction of any two subchains can be characterized by an integral called the excluded volume for a pair of subchains  

If the energy of interaction of two mers is more favorable than the interaction of a solvent molecule and a mer, the potential of mean force for weak overlap of two subchains will be negative. The value of the excluded volume will depend on the temperature through the denominator in the exponential term of Equation 5.3. It is possible to have positive excluded volumes in one range of temperature and negative excluded volumes in a different range. The intersubunit excluded volume vanishes at a particular temperature. Because the potential of mean force depends on both the polymer and the solvent, it is even possible to observe more than one temperature at which = 0. The existence of a compensation point where the intersubunit 

---