Polymer Physics

One can see that whenr¼1, this equation can be reduced to the Flory-Huggins-
Scott equation for binary polymer blends. The lattice fluid theory can predict both
UCST and LCST (lower critical solution temperature) types of phase diagrams for
polymer blends, with further considerations of specific interactions (Sanchez and
Balazs 1989 ), see more introductions about LCST in Sect.9.1.

8.3.3 Dilute Solutions

Apparently, the random-mixing approximation in the classical lattice statistical
theory is not applicable to the dilute solutions of polymers. In 1950, Flory and
Krigbaum treated the suspended polymer coils in dilute solutions as rigid spheres
with an effective excluded volumeu(Flory and Krigbaum 1950 ). The combinatorial
entropy depends upon the total number of waysOto putN 2 spheres in a big volume
V(supposed in the unit ofu). The number of ways to put the first sphere isV,toput
the second sphere isVu, to put the thirdV 2 u, and so on so forth, then the total
number of ways

O¼

NY 2  1

i¼ 0

ðViuÞ (8.37)

The mixing heat can be neglected in dilute solutions, and then the mixing free
energy

DFm¼TDSm¼kTlnO¼kT½N 2 lnVþ

Xn^2 ^1

i¼ 0

lnð 1 iu=Vފ

kT½N 2 lnV

u

V

nX 2  1

i¼ 0

iŠ¼N 2 kTðlnV

N 2 u

2 V

Þð 8 : 38 Þ

The above simplified process omitted the higher order expansions of the logarithmic
term under the dilute condition ofiu/V<<1. Therefore, the osmotic pressure of
solvent is

P¼ðm 1 m^01 Þ=v 1 ¼Na

@DFm

@N 1

=v 1 ¼Na

@DFm

@V



@V

@N 1

=v 1 (8.39)

whereNais the Avgadro constant, andv 1 is the molar volume of the solvent. Since
∂V/∂N 1 ¼v 1 /Na, one obtains

P¼@DFm=@VkT½N 2 =Vþðu= 2 ÞðN 2 =VÞ^2 Š (8.40)

160 8 Statistical Thermodynamics of Polymer Solutions