Thermodynamic Relations for Dilute Polymer Solutions

Thermodynamic Relations for Dilute Polymer Solutions.—Let us consider the total number of ways in which identical polymer molecules of excluded volume u may be distributed over a volume V of solution. The solution will be assumed to be so dilute that n u is much less than F, thus allowing ample room for all molecules without forced overlapping of their molecular domains. The number of locations available for the center of gravity of the first polymer molecule introduced into the solution will be proportional to V, The first molecule effectively excludes a volume u consequently the number of locations available to the second molecule will be proportional to V—u. The volume available to the third molecule will be V—2Uj and so forth. Provided the final solution is sufficiently dilute to justify assumption of independent volume exclusion on the part of the individual molecules, the total excluded volume will remain additive in the number of polymer molecules, and the total number of arrangements for polymer molecules can be written 

On the other hand, if the solution becomes too concentrated the same volume element may often be subject to exclusion by two (or more) different molecules hence the total volume excluded is somewhat less than V — iu Eq. (66) takes account only of binary encounters. Ternary and higher encounters, which become increasingly prominent with increase in concentration, are neglected. Eq. (66) is limited accordingly to low concentrations. 

The total partition function of the system is represented by Q of Eq. (66) hence 

Since iu/V is always much less than unity in the sufficiently dilute solution, the logarithmic terms of the summation may be expanded in series with neglect of higher terms to give 

The first term in Eq. (68) represents the ideal free energy of mixing term in dilute solutions, as will be apparent below. If the entire volume were available to atl of the molecules, which would be an acceptable assumption if either 7 were very large or u were very small, it would be the only term, for then Q = Const. The second term in 

---

Parameter occurring in thermodynamic relations for dilute polymer solutions (Chaps. XII and XIV). 

---