Modelling Gas Separation in Porous Membranes
91
permeability polymers like poly(trimethylsilylpropyne), as has been proved by computer
modelling, low activation energy of diffusion, negative activation energy of permeation,
solubility controlled permeation in this and similar polyacetylenes
[10,25]
(see also
Chapters 2 and 3 of this book). Although polymeric membranes have often been viewed
as non - porous, in the modelling framework discussed here it is convenient to consider
them nonetheless as porous. Glassy polymers have pores
that can be considered as
‘ frozen ’ over short times scales, as demonstrated in Figure 5.3 a, while rubbery polymers
have dynamic fl uctuating pores (or more correctly free volume elements) that move,
shrink, expand and disappear, as illustrated in Figure 5.3 b [14] .
5.6
Transition State Theory (TST)
The diffusion of molecules within porous networks similar to that of microporous silica
and non - porous glassy polymers can be modelled within the framework of the so - called
transition state theory [17,22,28,33,45] . A gas molecule bounces around in a reactant
cavity eventually bouncing towards the transition state by which it transports through to
the product cavity and therefore successfully
makes a diffusive jump, as demonstrated in
Figure 5.4 a. Within glassy polymers, see Figure 5.4 b, the transition state is a dynamical
section that becomes available through polymer chain motions. Within microporous
silica, see Figure 5.4 c, the transition state is a permanent pathway for the transport of the
gas molecule. The transition state theory offers a method to express the rate of diffusion
D (or diffusivity) within these porous networks in the following way:
Microporous glass
Carbon nanotubes
Zeolite
Carbon layers
Polymer
Silica
Figure 5.2 Porous structure within various types of membranes [3,22,37] . Microporous
glass fi gure from [22] , reprinted with permission of John Wiley & Sons, Inc. Silica fi gure from
[3] , reprinted with permission of Wiley - VCH Verlag GmbH & Co. KGaA. Carbon nanotubes
fi gure reprinted with permission from Science, Aligned multiwalled carbon nanotube
membranes, by B. J. Hinds, N. Chopra, T. Rantell, R. Andrews, V. Gavalas and L. G. Bachas,
303, 62 – 65. Copyright (2004) American Association for the Advancement of Science.
92
Membrane Gas Separation
D
=
the probability that the molecule will travel towards a ttransition
the probability that the molecule will pas
g
ρ
( )
×
ss through
the transition
the velocity of the molecule
E
ρ
( )
×
through the transition
the jump length from the react
u
( )
×
aant cavity to the product cavity
.
λ
( )
This formula
D
u
=
ρ ρ λ
g
E
provides some insight into the major factors contributing to the separation of particular
molecules. If the transition state has the form of a narrow constriction then the smaller
molecules are more likely to pass through and therefore have a higher rate of diffusion
than their larger counterparts. On the other hand, if the transition state is wide enough for
both molecules
to freely pass through, then the velocity at which they travel may be the
a) Glassy polymer
b) Rubbery polymer
Figure 5.3 Computer simulations performed by Greenfi eld and Theodorou [14] for free
volume clusters before and after 10
7
Monte Carlo steps within (a) glassy polymer and
(b) rubbery polymer. Reprinted with permission from Macromolecules, Geometric analysis
of diffusion pathways in glassy and melt atactic polypropylene by M. L. Greenfi eld and
D. N. Theodorou, 26, 5461 – 5472. Copyright (1993) American Chemical Society
Modelling Gas Separation in Porous Membranes
93
dominant factor in determining the rate of diffusion. Further, within glassy polymers the
rate of diffusion could be dominated by the rate of polymer chain movements in the walls
of free volume elements or closed pores which occasionally provide a transition pathway
for the molecules.
product
cavity
jump length (
λ
)
jump length
λ
d
n
d
p
reactant
cavity
(a)
(b)
(c)
neck
(transition state)
hole size
Figure 5.4 Transition State Theory for diffusion in condensed media. (a) General
representation of the transition state theory. (b) Diffusive jump in glassy polymer [17] .
Reprinted from Journal of Membrane Science, 73 , E. Smit, M. H. V. Mulder, C. A. Smolders,
H. Karrenbeld, J. van Eerden and D. Feil, Modelling of the diffusion of carbon dioxide in
polyimide matrices by computer simulation, 247 – 257, Copyright (1992), with permission
from Elsevier. (c) Diffusive jump in microporous silica, reprinted with permission from
AIChE, Theory of gas diffusion and permeation in inorganic molecular - sieve membranes by
A. B. Shelekhin, A. G. Dixon and Y. H. Ma, 41, 58 – 67, Copyright (1995) AIChE
