Biomedical Engineering Reference
In-Depth Information
Substitute this relation into Fick's first law, where the effective diffusion coefficient is used:
DK
L
p
J
=
(
C
−
C
)
(5.44)
A
B
D
measures the diffusivity of the solute in the membrane. The activity coefficient measures
the solute-solvent interaction, which will be discussed in more detail in the subsequent
section. For the sake of discussion, in the following sections, the effective diffusion coef-
ficient is defined as the product of the partition coefficient and the diffusivity of the solute,
which accounts for the combined effect of partition (
K
p
) and diffusion (
D
) on the flux.
Free Volume
In the nonporous description of membrane, diffusion is a result of the random movement
of molecules through the free volume in the membrane. Free volume is defined as the
difference between the actual volume of the liquid and the minimum volume, which it
would occupy, of its molecules, which were packed firmly in contact with each other [36].
It could also be interpreted as the volume difference between temperature
T
and absolute
zero. Molecules in the liquid at temperature
T
are moving at the average thermal veloc-
average thermal veloc-
ity but, most of the time, is confined to a cage bounded by their immediate neighbors.
Fluctuation in molecular arrangement opens up a hole in the cage, which is big enough
to permit a considerable displacement of the molecule contained in the cage. Net move-
ment of the displaced molecule could result only if another molecule jumps into the hole
before the displaced molecule returns to its original position. Diffusion occurs as a result
of redistribution of the free volume within the liquid. This description of diffusion by
Cohen and Turnbuli [37] was originally to describe self-diffusion of liquid with impurities
(solute). Because another molecule has to fill the void left by the displaced molecule, the
free volume (difference between the cage volume and volume of the molecule) has to be
large enough to permit another molecule to jump in after displacement.
In a solvent-solute binary system, solute diffusion coefficient is therefore proportional
to the probability of finding a free volume, which is greater than its own volume. The total
probability of finding a hole of volume exceeding
v
* is [37]
are moving at the average thermal veloc-
moving at the average thermal veloc-
the average thermal veloc-
P
(
v
*) = exp[-
κ
v
*/
v
f
]
(5.45)
where
k
is a factor to correct for overlap of free volume (0.5 ≤
κ
≤ 1 [38]), and
v
f
is the aver-
age free volume.
κ
is introduced because the total free volume is not the summation of
individual free volume:
∑
∑
κ
N v
i
i
V
N
v
=
f
=
i
(5.46)
f
N
i
i
Diffusion coefficient of solute in a liquid at infinite dilution is given by
D
→
a
*
u
exp(-
γ
v
*/
v
f
)
(5.47)
a
* is the diameter of the cage or the rough diameter of the molecule [37].
