Polymer Physics

the integrated total 1/N 2 !. 1/2 is the symmetric factor for the double estimation
starting from either one of two chain ends. Introducing the Stirling’s approxima-
tion, whenA>>1,

lnA!AlnAA (8.14)

We then obtain

Sm

k

¼lnO¼N 1 lnð

N 1

N

ÞN 2 lnð

N 2

N

ÞN 2 ðr 1 ÞþN 2 lnð

q

2

ÞþN 2 ðr 2 Þlnðq 1 Þ

(8.15)

For the purely disordered bulk phase of polymers,N 1 ¼0,N¼rN 2 , we have

Sm

k

(^)
N 1 ¼ 0
¼N 2 ðr 1 ÞþN 2 lnð
qr
2
ÞþN 2 ðr 2 Þlnðq 1 Þ (8.16)
The mixing entropy of polymer solutions is the difference of entropy between
the disordered bulk phase and the disordered solution phase, as given by
DSm
k

¼

Sm

k



Sm

k

N 1 ¼ 0

¼N 1 lnð

N 1

N

ÞN 2 lnð

rN 2

N

Þ¼N 1 lnf 1 N 2 lnf 2 (8.17)

wheref 1 andf 2 are the volume fractions of small solvent molecules and polymers,
respectively.

8.2.4 Calculation of Mixing Heat and Free Energy

According to the mean-field assumption, the mixing heat

DUm¼BP 12 (8.18)

Here,P 12 is the number of pairs between solvent site 1 and monomer site 2, the
mixing interactions in each pairB¼B 12 (B 11 þB 22 )/2. Each chain approxi-
mately holds (q2)rcoordination sites, and each coordination site contains the
probability of solvent occupationf 1 , thus forN 2 chains,

P 12 ¼N 2 ðq 2 Þrf 1 (8.19)

One can define

w

ðq 2 ÞB

kT

(8.20)

8.2 Flory-Huggins Lattice Theory of Polymer Solutions 155