Polymer Physics

He directly obtained

Pij¼

Nvoidvoid

Nvoid

¼ 1 

rjþi

N

(8.8)

Also in 1942, Huggins considered the doubly vacant case in the total amount of
coordination pairsqN/2 (Huggins 1942 ), as

1

2

qN¼NvoidvoidþNvoidoccupyþNoccupyoccupy (8.9)

The part of double occupation pairs should be subtracted, because they contain
two consecutively connected monomers along the chain, and do not belong to
statistical events. The subtracted part is

Moccupyoccupy¼qðrjþiÞ

r 2

qr

(8.10)

Therefore, he adopted

Pij¼

Nvoidvoid

Nvoid

¼

Nvoidvoid

qN= 2 Moccupyoccupy

¼

qðNrjiÞ

qN 2 ðr 2 ÞðrjþiÞ=r

(8.11)

This term looks like the probability of vacant sites on the surface of each site,
and thus has the name of “the surface fraction”. Equation (8.8) has been called “the
volume fraction”. Huggins consideration is more sophisticated, although for a large
coordination number it does not result in a significant difference from Flory’s
consideration.
The total ways in the arrangement of the solution systemOis equal to the ways
to arrangeN 2 chains, and there is only one way to arrange solvent molecules in the
available single sites as they are identical. Therefore, we have

O¼

1

2 N^2 N 2!

P

N 2  1

j¼ 0

Wjþ 1 (8.12)

Inserting (8.5), we obtain

O¼

qN^2 ðq 1 ÞN^2 ðr^2 Þ

2 N^2 NN^2 ðr^1 ÞN 2!

 P

N 2  1

j¼ 0

ðNrjÞ!

½Nrðjþ 1 ފ!

¼

qN^2 ðq 1 ÞN^2 ðr^2 Þ

2 N^2 NN^2 ðr^1 ÞN 2!



N!

ðNrN 2 Þ!

; ð 8 : 13 Þ

Here, 1/N 2! is the symmetric factor resulted from double estimation of identical
chains: 1 for the first chain, 1/2 for the second chain,...1/N 2 for the last chain, then

154 8 Statistical Thermodynamics of Polymer Solutions