Polymer Physics

This equation is also called Flory-Huggins-Scott equation. If bothr 1 andr 2 are
very large, the common mixing driving forces, i.e. the mixing entropy, will be
extremely small. Therefore, the common non-polar polymers are difficult to mix
with each other. One way to enhance compatibility is to introduce specific
interactions between two species of polymers, such as hydrogen bonding or polar
group interactions. Another way is to introduce highly repulsive comonomer C into
the component B, making the latter a random copolymer BxC 1 x, which modifies
the Flory-Huggins interaction parameter intow¼xwABþ(1x)wACx(1x)wBC.
WhenwBCis very large, the Flory-Huggins interaction parameter can change its
sign to favor the mixing (Kambour et al. 1983 ; ten Brink et al. 1984 ).

8.3.1.3 Shear Flow

For polymer solutions under shear flow, Wolf has considered the elastic energy of
polymer strain resulted from shear-induced deformation (Wolf 1984 ),

DFm

kT

¼N 1 lnf 1 þN 2 lnf 2 þwN 1 f 2 þ

N 1 v 1 þN 2 v 2

RT

Je^0 ^2 g^02 (8.25)

whereJe^0 is the steady shear compliance,is shear viscosity under the shear rateg’,
v 1 andv 2 are the molar volumes of solvent and polymer, respectively.

8.3.1.4 Equilibrium Swelling of Network Polymers

The lightly cross-linked polymer network is known as the gel, which would not
swell into fully stretched chains in a good solvent, due to the conformational
entropy loss associated with polymer deformation, but rather, reaches an equilib-
rium swelling. The mixing free energy provides the driving force for the network
swelling, which can be calculated from the Flory-Huggins equation, as

DFm

kT

¼N 1 lnf 1 þN 2 lnf 2 þwN 1 f 2 (8.26)

The conformational entropy of the chain between the cross-linking points works
against the network swelling, which can be calculated from the classical elastic free
energy of cross-linked network, as

DFe

kT

¼

n 0

2

ðl^21 þl^22 þl^23  3 lnðl 1 l 2 l 3 Þ¼

3 rV 0

2 Mc

ðf 22 =^3  1 þ

1

3

lnf 2 Þ (8.27)

Here,n 0 is the total number of network chains, the deformation ratiol 1 ¼l 2 ¼
l 3 ¼l¼(V/V 0 )1/3¼f 2 1/3, V 0 and V are the volume before and after the
swelling, respectively, andV/V 0 ¼Qis the swelling ratio.n 0 ¼W/Mc¼rV 0 /Mc,

8.3 Developments of Flory-Huggins Theory 157