Polymer Physics

gauche-conformation with energyEcraised, the partition function for each step of
walks is thus defined as

zc 1 þðq 2 Þexp½Ec=ðkTފ (8.53)

which can replaceq1in(8.52). Here, the collinear connection has only one choice
due to the ground state, while the non-collinear connection can haveq2 choices.
Therefore, one can get the partition function with the conformation contributions of
semi-flexible chains as

lnO¼N 1 ln

N 1

N

N 2 ln

N 2

N

N 2 ðr 1 ÞþN 2 ln

q

2

þN 2 ðr 2 Þlnzc (8.54)

The conformational states of semi-flexible chains can be represented by the
disorder parameterf, defined as

f¼

ðq 2 Þexp½Ec=ðkTފ

1 þðq 2 Þexp½Ec=ðkTފ

(8.55)

In fact,fis the statistical average fraction of non-collinear connections along the
chain. fis close to one whenT!1, corresponding to the flexible chain.
fapproaches zero when the temperature is low enough, corresponding to the rigid
chain. In the latter case,zcapproachese, which makes the conformation contribution
Zconfin the total partition function close to one, implying the spontaneous return to
the ground state (the fully ordered state). Flory suggested that this case corresponds
to crystallization driven by the chain semi-flexibility. This spontaneous ordering
process has been verified by computer simulations. But in practical concentrated
solutions, semi-flexible polymer chains are often frozen into a disordered state at low
temperatures, similar to the glass transition induced by the random stacking of rigid
rods. In fact, for a better description of polymer crystallization, one may need to
further consider the parallel packing interactions between crystalline polymer
chains, as will be further introduced in Sect.10.2.

Question Sets

1. Why can the lattice model calculate the mixing free energy of polymer
   solutions?


2. What is the purpose of mean-field assumption?


3. Why are two species of non-polar polymer chains not easy to mix with each
   other?


4. Why is the shear flow easy to induce phase separation in the multi-component
   polymer systems?


5. How to make use of equilibrium swelling to measure the average molecular
   weight of network polymers in the cross-linked polymer systems?


6. How to make use of volume phase transition of polyelectrolyte gels to design the
   smart gel system?

164 8 Statistical Thermodynamics of Polymer Solutions