Polymer Physics

wherefis the charge fraction of monomers on the polymer chain. This equation is
known as Flory-Rehner-Donnan equation. With the increase off, the equation will
give two solutions of swelling ratios under a specificw, indicating a sudden volume
transition. This phenomenon is called volume phase transition, and has been
verified experimentally (Tanaka et al. 1980 ). It can be applied as the environmen-
tally responsive smart gels.

8.3.2 Compressible Fluids

The classical lattice statistical model only considers the mixtures of two incom-
pressible fluids. Flory recognized that the line of the actual mixing interaction
parameter versus the reciprocal temperature does not have zero intercept, i.e.

w¼Aþ

B

T

(8.34)

Here, the termAincludes the contribution of interaction entropy (Flory 1970 ).
Such a contribution can be understood from the concept of compressible free volume
in the fluids. When two fluids are mixed with each other, part of molecules of one
species enters the free volume of another species, and then the total volume is not a
simple addition of the two individual components. Yamakawa made an approximate
estimation from the expansion theory (Yamakawa 1971 ). Prigogine attributed this
contribution to a combinatorial contribution of molecular geometry and a
non-combinatorial contribution of molecular structures, and proposed an equation-
of-state theory (Prigogine1957b). Flory, Orwell and Vrij further considered the
contribution of free volume, and employed separate parameters to describe the
hard-core volume and surface contacts of chain units (Flory et al. 1964 ;Flory
1965 ; Orwall and Flory 1967 ). This work makes the equation of state fit better to
the experimental results, and derives the so-called Flory-Orwell-Vrij equation of state
for pure polymers, as given by

P^0 V^0

T^0

¼

V^01 =^3

V^01 =^3  1



1

V^0 T^0

(8.35)

where P’¼P/P, V’¼V/V, T’¼T/T, and P,V and T are adjustable.
Sanchez and Lacombe supposed that in a binary polymer blend, free volume
occupiedN 0 lattice sites, and the bulk polymer densityrN/(NþN 0 ), where
N¼SNiriandriwas the chain length ofith fraction, then they developed the lattice
fluid theory to calculate Helmholtz free energy (Sanchez and Lacombe 1974 ;
Sanchez 1978 ), as given by

DFm

NkT

¼

f 1

r 1

lnðrf 1 Þþ

f 2

r 2

lnðrf 2 Þþð 1 rÞlnð 1 rÞ=rþwrf 1 f 2 (8.36)

8.3 Developments of Flory-Huggins Theory 159