Polymer Physics

However, the conformation statistics in Flory’s treatment gives the conformational
free energy, rather than the conformational entropy adapted in the Gibbs-DiMarzio
theory. In addition,Wwas calculated with respect to the fully ordered state; therefore,
lnW¼0 simply implies the return to the fully ordered state, rather than frozen in a
disordered state. Furthermore,Ecreflects the static semi-flexibility, while the glass
transition should be related with the dynamic semi-flexibility of polymer chains.
Therefore, fundamental assumptions of the Gibbs-DiMarzio thermodynamic theory
are misleading.
A more proper theoretical consideration to interpret glass transition starts from the
dynamic point of view. Fox and Flory supposed that the motion of polymer chains is
realized via chain monomers entering the void sites of free volume, and the free
volume contains a relatively large thermal expansion coefficient above the glass
transition temperature; and thus they explained phenomenologically the slope change
of the volume-temperature curve atTg(Flory and Fox 1951 ). The free volume is

Vf¼<V>V 0 (6.54)

whereis the average occupied volume of molecules in the liquid,V 0 is the
van der Waals volume of molecules. WhenTTg, the free volume will be kept
as constant as the free volume of the glass stateVg, and polymers are in the glass
state as (Fig.6.17)

Vf¼Vg (6.55)

WhenT>Tg, with respect toTg, one has

Vf¼VgþðTTgÞ

ð

dV

dT



dV 0

dT

Þ (6.56)

AroundTg, sinceVf<<V, approximately one hasVV 0. From the thermal
expansion coefficient (6.49), one obtains

a 0 ¼

1

V 0

dV 0

dT P

j (6.57)

Therefore, the expansion coefficient of the free volume

af

1

V

ð

dV

dT



dV 0

dT

ÞjP (6.58)

Taking into the expression forVf, one defines the fraction of free volume

f

Vf

V

(6.59)

fg

Vfg

V

(6.60)

6.3 Glass Transition and Fluid Transition 113