Chemistry Reference
In-Depth Information
less the tendency for miscibility of the system. Essentially, by mixing two macro-
molecules that individually have many degrees of conformational freedom, any con
gu-
rational entropy associated with this freedomwill be reduced upon mixing and interaction
of the two components. Consequently, we would expect that all other factors being equal
the miscibility of amorphous binary systems involving at least one polymer would be
reduced relative to those for two small molecules and that a mixture of two polymers
would have a strong tendency to lose less free energy and possibly phase separate. Indeed,
many amorphous polymer blends exhibit signi
cant immiscibility for this reason.
To further examine the thermodynamics of mixing in amorphous solids and the
possible relationships with the structural features of the amorphous state, for example,
molar volume or density, it will be useful to consider the volumetric changes that can
occur with the mixing of components having molar volumes of
V
A
and
V
B
. First, recall
that the volume per molecule or mole of any component X at any temperature is
determined by the volume occupied by the mass of a molecule,
V
occup
, plus any free
volume
V
free
that exists between the molecules, as determined by the degree and nature of
the molecular packing, as expressed in Equation 1.25:
V
X
V
occup
V
free
:
(1.25)
Consequently, when two molecules A and B are mixed, the net volume change will have
contributions from both their occupied and free volumes in proportion to the volume
fraction of each of the individual components; as temperature is changed, these
volumetric changes will be strongly affected by changes in
V
free
. For an ideal solution,
where
H
mix
is equal to zero, we would expect that the volume of the mixture,
V
AB
,
would be the sum of the weighted average of the molar volume of each component:
Δ
V
AB
n
A
V
A
n
B
V
B
;
(1.26)
where
V
A
and
V
B
are themolar volumes and
n
A
and
n
B
are themole fractions of components
A and B, respectively. This means that for ideal mixing, the occupied and free volumes of
each component will be additive in a manner weighted exactly by each component
s mole
fraction. From the thermodynamic analysis described above, it would be expected that
miscible mixtures of amorphous solids, where
'
H
mix
is not equal to zero, will exhibit
nonideal mixing so that there would be an excess molar volume change,
V
excess
, that can be
either positive or negative depending on the nature of the intermolecular interactions giving
rise to an enthalpy change. In such a case
V
AB
can be written as
Δ
V
AB
n
A
V
A
n
B
V
B
V
excess
;
(1.27)
where very strong interactions between A and B lead to a negative excess volume, that is, a
reduction in overall volume, as in the well-known case of the mixing of ethanol and water,
while stronger interactions between A and A and B and B than between A and B lead to a
positive excess molar volume or an increase in overall molar volume.
From Equation 1.27 and our previous discussion of factors that might in
uence the
glass transition temperature of any amorphous solid, it would appear that the various
