Polymer Mixtures - Polymer Science and Engineering - page 239

molecule. This analysis yields the following expression for Δ S m , the entropy of

mixing N 1 moles of solvent with N 2 moles of polymer.
Δ

S m 52 R ð N 1 ln φ 1 1 N 2 ln φ 2 Þ (5-16)

where the φ i are volume fractions and subscripts 1 and 2 refer to solvent and

polymer, respectively. The polymer consists of r 2 segments, each of which can
displace a single solvent molecule from a lattice site. Thus r 2 is defined as
r 2 5 V 1

M =ρ (5-17)

where M is the molecular weight of the polymer that would have density in the

corresponding amorphous state at the solution temperature and V 1 is the molar
volume of the solvent. The number of lattice sites needed to accommodate this
mixture is (N 1 1 N 2 r 2 )L, where L is Avogadro's constant.
Equation (5-16) is similar to Eq. (5-3) , except that volume fractions have
replaced mole fractions. This difference reflects the fact that the entropy of mix-
ing of polymers is small compared to that of micromolecules because there are
fewer possible arrangements of solvent molecules and polymer segments than
there would be if the segments were not connected to each other.
Equation (5-17) applies also if two polymers are being mixed. In this case the
number of segments r i in the ith component of the mixture is calculated from
r i 5

ρ

M i =ρ ι V r (5-17a)

where V r is now a reference volume equal to the molar volume of the smallest
polymer repeating unit in the mixture. The corresponding volume fraction ϕ i is

X N i r i

φ i 5 N i r i = (5-18)

The entropy gain per unit volume of mixture is much less if two polymers are
mixed than if one of the components is a low-molecular-weight solvent, because
N 1 is much smaller in the former case.
To calculate

Δ

H m (the enthalpy of mixing) the polymer solution is approxi-
mated by a mixture of solvent molecules and polymer segments, and

Δ

H m is
estimated from the number of 1, 2 contacts, as in Section 5.2.1 . The terminol-
ogy is somewhat different in the Flory Huggins theory, however. A site in the
liquid lattice is assumed to have z nearest neighbors and a line of reasoning
similar to that developed above for the solubility parameter model leads to the
expression

Δ H m 5 zw ð N 1 1 N 2 r 2 Þφ 1 φ 2 L (5-19)

for the enthalpy of mixing of N 1 moles of solvent with N 2 moles of polymer.
Here w is the increase in energy when a solvent-polymer contact is formed from
molecules that were originally in contact only with species of like kind.
Now the Flory

Huggins interaction parameter χ (chi) is defined as

X zwL = RT (5-20)

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