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can be treated as a continuum medium. Strictly speaking, the two equations are
not comparable as the penetrating molecules and those made up of the surround-
ing are comparable in size in the transition state theory. Nevertheless, if in the
displacement process the first molecule passes through the shortest pass around
the second one (which is equal to
r), it has been theoretically proven
[16]
that
the self-diffusion coefficient can be written in the following form:
π
k
B
T
π
D
5
(6-65)
r
η
Einstein equation except that the
factor 6 is not in the denominator. The similarity between
Eqs. (6-42) and (6-65)
suggests that a general form of the Stokes
The above equation resembles the Stokes
Einstein equation can be used to cal-
culate the self-diffusion coefficient,
k
B
T
a
D
5
(6-66)
π
r
η
where the factor a in the denominator remains to be determined theoretically or
experimentally for a specific system.
6.6.2
Free Volume Theory
The free volume theory was originally formulated for analyzing the diffusion pro-
cess in simple liquids and glasses made up of molecules that are represented by
hard spheres. The theory was later extended to the study of diffusion of small
molecules in concentrated polymer solutions. In the original work of Cohen and
Turnbull in 1959
[19]
, the authors proposed that diffusion of molecules in a liquid
occurs when they move through the volume that is not occupied by other mole-
cules in the liquid. Such empty voids are termed free volume. The amount of free
volume available in a liquid depends on how much the temperature of the liquid
is above its glass transition temperature. According to the theory, each molecule
in the liquid is confined to a cage formed by the surrounding molecules. Owing
to the natural thermal (local density) fluctuations, holes in the vicinity of the cage
emerge. If a hole is large enough, the molecule in the cage will jump into it. Such
displacement gives rise to the diffusive motion. In a sense, diffusion can be
thought of as a free volume holes redistribution process. Based upon this concept,
Cohen and Turnbull derived the following equation to relate the diffusion coeffi-
cient and the free volume of the liquid:
D
5
ga
ue
2
γν
ν
f
(6-67)
In this equation, g is a geometric factor; a
approximately equals the molecu-
lar diameter; and
is a numerical factor to account for free volume overlaps (a
value usually between 1/2 and 1). Here,
γ
ν
is the minimum required size of a free
