Excluded Volume in a Chain Molecule

Consider a hypothetical process in which the volume excluded by each of the two spheres increases from 0 to Vg. The space available to the other sphere decreases from the volume V of the system to V - v. Therefore, the configurational entropy of the sphere changes by 

When the system has N identical spheres in the volume of V, there are N /2 pairs of excluded-volume interaction. Then, the change in the total free energy due to the excluded volume is AA/k T = (N /2)vJV. The change per sphere, (AA/k T)/N, is proportional to the density N/V. At low concentrations, the excluded volume is negligible. As the concentration increases and Nv approaches V, the effect becomes stronger. The same effect appears in the van der Waals equation of state for a real gas The correction to the volume is equal to the excluded volume. 

2 Excluded Volume in a Chain Molecule In the polymer solution, the excluded volume does not disappear even in the low concentration limit. Connectivity of monomers makes the sitnation different from that in the suspension of 

Suppose a polymer chain consisting of N spheres of diameter b (pearl-necklace model see Fig. 1.34). We consider the dilute solution limit in which each chain is isolated from the other chains in the solution. When the chain dimension is R, these N spheres are contained in a cube of volume close to R, but no other spheres 

The excluded volume is not limited to a pair of monomers on the same chain (intrachain interaction). It exists equally for a pair of monomers on different chains (interchain interaction). At higher concentrations, the interchain interaction is the dominant part of the excluded volume effect. 

---

Figure 1.34. Excluded volume in a chain molecule. The two white beads cannot overlap each other.

---