Polymer Physics

Therefore, the partition function of polymer solutions can be expressed as

Z¼ð

N

N 1

ÞN^1 ð

N

N 2

ÞN^2 ð

q

2

ÞN^2 zcðr^2 ÞN^2 eðr^1 ÞN^2 zpðr^1 ÞN^2 zmrN^2 (10.8)

where

zc 1 þðq 2 Þexp½Ec=ðkTފ

zmexp½ðq 2 Þ

N 1

N



B

kT

Š

zpexpf

q 2

2

½ 1 

2 ðr 1 ÞN 2

qN

Š

EP

kT

g

At the right-hand side of (10.8), the first five terms come from Flory’s semi-
flexibility treatment (8.55), the sixth term comes from the mean-field estimation for
the pair interactions of parallel bonds (10.7), and the last term comes from the
mean-field estimation for the mixing interactions between the chain units and the
solvent molecules (8.21). According to the Boltzmann’s relationF¼kTlnZ,
the free energy of the solution system can be obtained as

F

kT

¼N 1 ln

N 1

N

þN 2 ln

N 2

N

N 2 ln

qzcr^2

2 er^1

þN 2 ðr 1 Þ

q 2

2

½ 1 

2 N 2 ðr 1 Þ

qN

Š

Ep

kT

þ

N 1 N 2 rðq 2 ÞB

NkT

(10.9)

In the practical systems, the mixing free energy change is estimated with the
reference to the amorphous bulk phase of the polymer, so

DFm¼FmFmjN 1 ¼ 0 (10.10)

From (10.9), one can obtain the expression of the mixing free energy consistent
with the Flory-Huggins equation, as given by

DFm¼DUmTDSm¼kTðN 1 lnf 1 þN 2 lnf 2 þwN 1 f 2 Þ (10.11)

wheref 1 ,f 2 are the volume fractions, and

w¼

ðq 2 ÞBþð 1 

2

q

Þð 1 

1

r

Þ

2

Ep

kT

(10.12)

194 10 Polymer Crystallization