Polymer Physics

2. Diluent or plasticizer. The mixture of polymer (labeled with the subscript p)
   and diluent (labeled with the subscript d) can be regarded as a polymer solution
   system. The expansion coefficient of free volume is af. Assuming the
   contributions of each component in the system to the free volume are propor-
   tional to their volume fractionf, one can obtain according to the equivalence
   phenomenon of free volume

afpðTgTgpÞfpþafdðTgTgdÞfd¼ 0 (6.72)

Two contributions compensate to each other, andfpþfd¼1; therefore

Tg¼

afpfpTgpþafdfdTgd

afpfpþafdfd

(6.73)

Ifafpafd,(6.73) can be simplified as

Tg¼fpTgpþfdTgd (6.74)

Replacing the volume fractions above with weight fractions, one obtains the

so-called Wood Equation, which is often used for random copolymer systems

(Wood 1958 ).

IfTgpafpTgdafd,(6.73) can be simplified as

1

Tg

¼

fp

Tgp

þ

fd

Tgd

(6.75)

Replacing the above volume fractions with weight fractions, one obtains the so-called

Fox Equation, which is often used for polymer-diluent mixtures (Fox 1956 ).

3. Random copolymer. Considering the mass fractionsWAandWBproportional to
   the contributions of free volume, according to the equivalence phenomenon of
   free volume,

afAðTgTgAÞWAþafBðTgTgBÞWB¼ 0 (6.76)

AssumingK¼afB/afA,

Tg¼

TgAþðKTgBTgAÞWB

1 þðK 1 ÞWB

(6.77)

This equation is known as Gordon-Taylor equation (Gordon and Taylor 1952 ).

4. Cross-linking. The cross-links restrict the mobility of network chains. Thus,
   Tgrises with the crosslink densityr(crosslink points per gram) (Ueberreiter and
   Kanig 1950 ; Fox and Loshaek 1955 ),

6.3 Glass Transition and Fluid Transition 117