Polymer Physics

and obtains

fðT>TgÞ¼fgþðTTgÞaf (6.61)

The free volume is related to the polymer viscosityaccording to Doolittle
empirical equation (Doolittle 1951 ), as given by

¼Aexpð

BV 0

Vf

ÞAexpð

B

f

Þ (6.62)

whereAandBare constants,B¼0.51. With respect toTg,

ln½

ðTÞ

ðTgÞ

Š¼Bð

1

f



1

fg

Þ¼B½

1

fgþðTTgÞaf



1

fg

Š (6.63)

Merging with the right-hand two terms, one obtains

log½

ðTÞ

ðTgÞ

Š¼

B

2 : 303 fg

TTg

fg

af

þTTg

(6.64)

In another more general form,

log½

ðTÞ

ðTgÞ

Š¼C 1

TTg

C 2 þTTg

(6.65)

This is the well-knownWLF equation. In the temperature range fromTgto
Tgþ 100 C, most polymers obey the WLF equation. The averaged constants
C 1 ¼17.44 and C 2 ¼51.6. Normally B1, one may obtain fg¼0.025.
fgappears to be independent of the molecular structures, thus the glass transition
is also referred as an equal-free-volume transition verified by the experiments. It
can be proved that WLF equation is actually a reflection of VFT-type liquids.

Fig. 6.17 Illustration of the
separation of polymer
volume-temperature curve
into the van der Waals
volume and the free volume

114 6 Polymer Deformation