Chemistry Reference
In-Depth Information
terms of
χ
by differentiating
Eq. (5-24)
with respect to N
1
. Here, a
1
and
χ
are
related by the following equation:
φ
2
1
χφ
1
r
2
2
2
lna
1
5
ln
φ
1
1
1
(5-26)
2
G
1
can be obtained from measurements of any of several
thermodynamic properties of the polymer solution [e.g., osmotic pressure as
shown in Eq. (3-15)]. It can be shown then that the second virial coefficient
(Eq. 3-24) is given by
Experimentally,
μ
1
2
2
V
1
A
2
5
ð
0
:
5
2
χÞ=ρ
(5-27)
ρ
where
0 in theta
mixtures (Section 3.14) where the polymer is insoluble, the condition for compati-
bility is
is the polymer density at the particular temperature. Since A
2
5
χ
,
0.5. When the mixture is produced from two polymers A and B,
Eq. (12-24) can be recast in the form
Δ
G
m
5
RTV
½χ
AB
ϕ
A
ð
1
2
φ
A
Þ
1
ðφ
A
=
V
A
Þ
ln
φ
A
(5-28)
1
ðφ
B
=
V
B
Þ
ln
ð
1
2
φ
A
Þ
where V is the total volume of the mixture, V
i
is the molar volume of species i,
and
χ
AB
is the interaction parameter for the two polymeric species. Since
V
i
5
M
i
/
ρ
i
this is also equivalent to
Δ
G
m
5
RTV
½χ
AB
ϕ
A
ð
1
2
φ
A
Þ
1
ðφ
A
ρ
A
=
M
A
Þ
ln
φ
A
(5-29)
1
ðφ
B
ρ
B
=
M
B
Þ
ln
ð
1
2
φ
B
Þ
The logarithmic terms are negative because the
ϕ
i
are less than one. Therefore,
Δ
G
m
is less negative and the mixture is less stable the higher the molecular weights
of the components. In fact, mixtures of high polymers are indicated to be always
incompatible unless
0. This situation will occur only when the enthalpy of
mixing is less than or equal to zero, i.e., when there are some specific interactions
(not of the van der Waals type) between the components of the mixture.
The Flory
χ
AB
#
Huggins theory predicts that the solubility of polymers will be
inversely related to their molecular sizes. Compatibility of polymers with other
materials is certainly affected by the molecular weight of the macromolecules.
Higher molecular weight materials are generally less soluble in solvents. The
influence of molecular weight on the stability of other mixtures is more complex.
Higher molecular weight species are generally more difficult to disperse, espe-
cially if they are minor components of mixtures in which the major species are
lower molecular weight, less viscous substances. If they can be dispersed ade-
quately, however, their diffusion rates and consequent rates of segregation will be
correspondingly less and the dispersion may appear to be stable as a result.
The Flory
Huggins model differs from the regular solution model in the inclu-
sion of a nonideal entropy term due to the difference in the sizes of molecules of
different kinds and replacement of the enthalpy term in solubility parameters by
