Chemistry Reference
In-Depth Information
V
i
where
is t
he
specific hole free volume of component i required for a diffusive
V
1j
V
2j
.
Quantification of the specific free volume, V
FH
, is the most critical step in the
implementation of the free volume concept in the description of the mass trans-
port in polymers. As a first approximation, one might assume that the total
volume of a liquid is composed of two parts, the occupied volume and the free
volume. The specific occupied volume of a liquid is generally defined as the spe-
cific volume of the equilibrium liquid at 0 K (i.e.,
step and
ξ 5
V
0
). Hence, the specific free
volume of a species as a function of temperature is given by
ð
0
Þ
V
FH
5
V
ð
T
Þ 2
V
0
ð
0
Þ
(6-72)
V
ð
T
Þ
where
is the specific volume of an equilibrium liquid at a given temperature T.
V
0
Obviously,
is not directly measurable. It can only be estimated by applying
group contribution methods based upon the knowledge of the chemical composition
of the mixture. Although
Eq. (6-72)
can be used to approximate the total free volume
in a liquid, one question still exists as to whether this free volume is the same as that
defined in simple liquids. As experimental facts show, the continuous redistribution of
the total free volume is not always possible in polymer systems
[11]
. This is because a
portion of the free volume is trapped between the polymer segments in such topologi-
cal restrictions that redistribution of such free volume may not be as easy as in simple
liquids. To account for this situation, free volume can be divided into two categories.
One portion of the free volume, denoted as the interstitial free volume, requires large
redistribution energy and thus does not facilitate mass transport through the mixture.
The remaining free volume, which facilitates molecular transport, is termed the hole
free volume and can be redistributed freely. In this regard, Vrentas and Duda devel-
oped a relationship between the hole free volume and the well-defined volumetric
characteristics of the pure components in the solution as follows:
V
FH
5ω
1
K
11
ð
ð
0
Þ
K
21
2
T
g1
1
T
Þ 1ω
2
K
12
ð
K
22
2
T
g2
1
T
Þ
(6-73)
Here, K
11
and K
21
are the free volume parameters for the solvent while K
12
and
K
22
are the free volume parameters for the polymer. T
gi
and
ω
i
are the glass tran-
sition temperature and mass fraction of the component, respectively; T is the solu-
tion temperature.
In the original free volume theory, a molecular jump does not involve any
activation energy but is solely related to the probability of locating a sufficiently
large free volume hole. This is because hard spheres do not have attractive inter-
actions. However, a jumping unit must overcome the attractive forces from
nearby molecules prior to the jump. Therefore, Cohen and Turnbull modified the
pre-exponential term in
Eq. (6-71)
to
E
RT
D
01
5
D
0
exp
2
(6-74)
