Polymer Mixtures - Polymer Science and Engineering - page 240

solvent interaction energy per
mole of solvent, divided by RT, which itself has the dimensions of energy. Since
ϕ 1 5

This dimensionless quantity is the polymer

N 2 r 2 ), Eq. (5-19) can be recast to give the enthalpy of forming a
mixture with volume fraction

N 1 /(N 1 1

ϕ 2 of polymer in N 1 moles of solvent as
Δ

H m 5 RT χ N 1 φ 2 (5-21)

V 1 , where V 0 is the molar
volume of the solvent. Then the enthalpy of mixing per unit volume of mixture is
Δ

The total volume V of this solution is ð N 1 1 N 2 r 2 Þ

N 1 φ 2
ð N 1 1 N 2 r 2 Þ V 1 5

χ χφ 1 φ 2

V 1

H m
V 5

RT RT

(5-22)

Huggins value in Eq. (5-22) is now equated to the solubility
parameter expression of Eq. (5-10) , it can be seen that

If the Flory

2

V 1 ðδ 1 2 δ 2 Þ χ 5 = RT (5-23)

Equation (5-23) suffers from the same limitations as the simple solubility
parameter model, because the expression for H m is derived by assuming that

intermolecular forces are only nondirectional van der Waals interactions. Specific
interactions like ionic or hydrogen bonds are implicitly eliminated from the
model. The solubility parameter treatment described to this point cannot take
such interactions into account because each species is assigned a solubility param-
eter that is independent of the nature of the other ingredients in the mixture. The
χ

Δ

parameter, on the other hand, refers to a pair of components and can include
specific interactions even if they are not explicitly mentioned in the basic
Flory

Huggins theory. Solubility parameters are more convenient to use because
they can be assigned a priori to the components of a mixture. values are more

realistic, but have less predictive use because they must be determined by experi-
ments with the actual mixture.
From Eqs. (5-16) and (5-21) the Gibbs free energy change on mixing at tem-
perature T is

χ

Δ

G m 5 Δ

Δ ðχ

N 1 φ 2 1

φ 1 1 φ 2 Þ

H m 2 T S m 5 RT N 1 ln N 2 ln (5-24)

Now, since

T ; P ; N 2 5 T ; P ; N 2 2 T ; P ; N 2 5

G 1 @Δ G m @ G solution

@

@

RTlna 1

@ N 1 N 1 @ N 1

then

T ; P ; N 2 5 μ 1 2

@ G m
@

G 1 5

RTlna 1 (5-25)

N 1

where a 1 is a fictitious concentration called activity mentioned in Section 3.1.4.
Thus, the difference in chemical potential of the solvent in the solution ( μ 1 ) and

G 1 Þ

in the pure state at the same temperature ð (i.e., RTlna 1 ) can be expressed in