Chemistry Reference
In-Depth Information
It is obvious that the prediction of the molecular weight dependence of D
G
using the Zimm model is in good agreement with the experiment for polymer
chains in
M
2
1/2
).
θ
solvents (i.e., D
G
~
EXAMPLE 6-5
Consider a polymer chain consisting of 100 segments. Each segment can be modeled as a
Brownian particle as described in Example 6-4 and is subjected to the same conditions.
What is the self-diffusion coefficient of the center of mass of the chain?
Solution
Assume that at 298.15 K, water is a
θ
solvent for the polymer. According to the Zimm
model,
10
223
k
B
T
η
s
p
b
50:196
1
:
38
3
33310
29
52:69310
211
m
2
3
298
:
15
D
G
50:196
10
23
p
s
100
The above result shows that, owing to the connectivity of the beads, the self-diffusion
coefficient is roughly half that of a single particle with the same size of a segment and sur-
rounded by the same solvent.
6.6
Self-Diffusion of Solvent in Polymers
The key concept involved in the mechanistic understanding of the self-diffusion
of small molecules in polymers, either above or below their glass transition tem-
peratures, is the free volume theory. The original free volume theory was devel-
oped for small molecule mixtures and was established based upon the transition
state theory and experimental data on the viscosity of glass-forming liquids. In
the following two sections, the transition state theory will be briefly reviewed to
show the correlation between the diffusion coefficient and viscosity followed by
the introduction of the free volume theory for self-diffusion of simple liquids
(hard sphere particles). Finally, the application of the free volume theory to the
diffusion of small molecules in polymers will be discussed.
6.6.1
Transition State Theory—Diffusion in Simple Liquids
In the previous sections, the self-diffusion of Brownian particles in a solution has
been demonstrated to be inversely proportional to the viscosity of the solvent. The
transition state theory was essentially developed to uncover the connection between
the self-diffusion coefficient and viscosity of the liquids as summarized below
[18]
.
Consider two layers of molecules in a liquid, as shown in
Fig. 6.9
, at a dis-
tance
λ
1
apart. Suppose that one molecule in a layer slides past the other under
the influence of an applied shear force f in the positive x direction. According to
the definition of viscosity,
