Chemistry Reference
In-Depth Information
Based upon the Onsager analysis, Darken proposed the following equation to
calculate the mutual diffusion coefficient D for metallic alloys
[9]
:
1
@
ln
γ
x
B
D
A
1
x
A
D
B
Þ
D
5 ð
1
(6-10)
@
ln x
where x
A
and x
B
are the mole fractions of species A and B, and D
A
and D
B
are
the corresponding self-diffusion coefficients of species A and B. Owing to the
Gibbs
Duhem relation, the thermodynamic correction term in
Eq. (6-10)
is the
same for both species A and B as shown in the following equation:
@
ln
ln x
A
5
@
γ
A
ln
ln x
B
5
@
γ
B
ln
γ
(6-11)
@
@
@
ln x
where
is the activity coefficient. Obviously, the Darken equation allows one
to estimate the mutual diffusion coefficient based upon the values of self-
diffusion coefficients obtained from either experiment or theoretical calculation.
It is worth noting that direct measurement of the mutual diffusion coefficient
is very difficult. The Darken equation has been extended to polymer mixtures
by Hartley
γ
and Crank
[12]
. However,
solution
theory
such
as
the
Flory
Huggins theory (Section 5.2.2) should be used to estimate the thermody-
namic correction term.
Although the Darken equation holds for the entire range of concentrations,
calculation of self-diffusion coefficients at high concentrations of polymer
molecules is not trivial. This is because chain dynamics are affected by entan-
glements, especially for high-molecular-weight systems. However, for concen-
trated polymer solutions, there is no need to estimate the self-diffusion
coefficient of entangled chains surrounded by solvent molecules as its magni-
tude is several orders of magnitude lower than that of the solvent. As a result,
ignoring the self-diffusion coefficient term of the polymer molecules in
Eq. (6-10)
would not introduce significant errors to the estimation of the
mutual diffusion coefficient. It is worth noting that the self-diffusion coefficient
of solvent can be estimated using the free volume theory that will be discussed
in
Section 6.6
.
6.5
Self-Diffusion of Polymer Chains in Dilute Polymer
Solutions
The key concept involved in understanding the self-diffusion of polymer chains
in the liquid state is Brownian motion. The two well-known theories of poly-
mer dynamics (i.e., the Rouse and Zimm models) are formulated based on the
dynamics of Brownian particles. In particular, the models are used to obtain
self-diffusion of unentangled polymer chains in solutions and melts. In this sec-
tion, the random walk theory as the basis to yield the self-diffusion coefficient
