Chemistry Reference
In-Depth Information
volume hole above which movement of a molecule in the free volume hole is per-
mitted and
ν
f
is the average free volume associated with one molecule, which is
defined as
V
f
N
ν
f
5
(6-68)
where V
f
is the total free volume and N is the number of molecules in the liquid.
Note that
Eq. (6-67)
is scaled with T
1/2
as u, the velocity of the molecule, is
scaled with T
1/2
1
2
mu
2
3
2
k
B
T). This temperature dependency of diffu-
sivity has been shown to be in good agreement with experimental observations,
suggesting that the temperature dependency of the free volume in a liquid is cap-
tured satisfactorily in the theory.
The extension of the free volume theory to concentrated polymer solutions is
not a straightforward task. Much effort, notably by Vrentas and Duda, has been
devoted to such a purpose
[11]
. The following discussion summarizes the key
development of their work.
As shown above, the original free volume theory was developed for the self-
diffusion coefficient D
1
for a one-component liquid or glass. However, the rela-
tion can be readily extended to describe the self-diffusion of a species in a binary
mixture with the penetrating species (small molecules) at low concentrations.
(note that
5
V
1
V
FH
D
1
5
D
01
e
2
γ
(6-69)
In
E
q. (6-69)
, D
01
is the temperature-independent pre-exponential constant.
And V
1
is the critical m
ol
ar free volume above which species 1 is able to move
through the free volume; V
FH
is the free volume available per mole of the mixture.
Unlike small molecule mixtures, displacement of the entire polymer molecule from
one free volume hole to another one is not possible. However, monomers in indi-
vidual polymer molecules can be considered as the basic jumping units. According
to Vrentas and Duda, the number of jumping units in a polymer mixture is
V
FH
mole of jumping unit
V
FH
V
FH
5
g
5
(6-70)
M
1j
1
ω
2
ω
1
=
M
2j
V
FH
is the specific free volume of a mixture with a weight fraction
where
ω
i
of
species i, and with jumping unit molecular weights of M
ij
. Note that Mij for sim-
ple molecules is the entire molecular weight of the component but in the case of
a polymer chain is a small fraction of the total chain molecular weight.
Combining
Eqs. (6-69) and (6-70)
results in an expression for solvent self-
diffusion in a polymer solution as follows:
!
V
1
1ω
2
ξ
V
2
Þ
V
FH
2
2γðω
1
D
1
5
D
01
exp
(6-71)
