Membranes - Introduction to Carbon Capture and Sequestration - page 339

ν

1 LL

a

wz z

= exp − UkT /

wB

4 kT L

B y

In this calculation, we have assumed that the concentration in the pores
is suffi ciently low that in our expression for the permeance the transport dif-
fusion coeffi cient can be approximated with the self-diffusion coeffi cient.
Let us now use this result to see how changes in the pore geometry
of our material will infl uence the permeation. Figure 7.6.3 illustrates
some of the changes we can make in the geometry. We can change
the size of our cavity or the diameter of the window, and we assume
that these changes do not infl uence the energies. Volume changes infl u-
ence the entropy of our molecules. For example, if we increase the vol-
ume of the cavity, the entropy of our adsorbed molecules increases and
hence the Henry coeffi cient increases. For the diffusion coeffi cient, an
increase in the entropy of a molecule in the cavity corresponds to a
decrease in the free energy. This decrease, however, increases the bar-
rier to diffusion and we see a decrease in the diffusion coeffi cient. The
opposite will happen if we decrease the volume of the cavity; the Henry
coeffi cient will decrease and the diffusion coeffi cient will increase.
Let us now focus on changing the geometry of the window. As the
window region contributes little to the Henry coeffi cient, changing the
window volume will predominantly affect the diffusion coeffi cient. Making
the window diameter smaller will increase the free energy and hence we
will see a decrease of the diffusion coeffi cient. Similarly, increasing the
window diameter will increase the diffusion coeffi cient.
The results are summarized in Figure 7.6.4 . We see that only
changes in the window diameter will infl uence the permeation, whereas
for our model changes in the cavity volume will cause a change in the
Henry coeffi cient that is compensated by a change in the diffusion coef-
fi cient in an opposite direction. We can use these results to compute the
permeability of our materials.
At this point, it is important to note that we have created a very sim-
ple model based on many assumptions. For example, it is diffi cult to
envision how we would change the volume of a cavity without changing
the energy in the cavity at the same time. Therefore, in our model we see
an exact compensation, which is unlikely in real materials. Despite these
limitations, our model does illustrate why some changes in the material
have signifi cant effects on the diffusion coeffi cients and Henry coeffi -
cients, but not on the permeability of the material.