Permeability, selectivity and free volume of membrane materials –
new results and ideas
Yu.P. Yampolskii
A.V.Topchiev Institute of Petrochemical Synthesis, Russian Academy of Sciences,
29 Leninsky Pr., 119991 Moscow, Russia, e.mail Yampol@ips.ac.ru
Today, enormous information is accumulated on gas permeation parameters of
polymers – potential materials for gas separating membranes. Thus, the Database
created by us contains gas permeation parameters (gas permeability, diffusivity, etc)
for about 700 glassy homo-polymers and more than 20 gases. Different methods have
been proposed to calculate gas permeability coefficients and other transport properties
on the basis of chemical structure of polymers. Although all this can be useful in an
assessment of the achievements of membrane materials science, it gives little insight
in physical reasons, why one polymer is extremely permeable and another is very
permselective. Other questions that still await answers are: What are the reasons (in a
quantitative way) of a well-known trade-off between permeability and selectivity of
different polymers? Which physical parameters of polymers are responsible for the
effects of chemical structure on permeability and selectivity? In other words, which
parameters should be controlled? Some preliminary answers to those questions will be
given below.
The governing physical concept that explains and quantitatively describes
transport of small molecules in polymers is free volume model. However, more and
more recent evidences indicate that only combined consideration of free volume and
activated diffusion models can give an adequate microscopic description of diffusivity
and describe observed phenomena. It can be added that these results are consistent
with computer simulations reported recently. In this key-note lecture I will summarize
some observations of this kind obtained during last several years by us and several
other groups in USA, Germany, and Italy.
1. Trade-off behavior. It has been shown [1] that very good correlations are
observed between permeability coefficients Pi and Pj of different gases in glassy
polymers. An example of such correlation is shown in Figure 1. Similar correlations
were also observed for the diffusion coefficients Di and Dj. They are direct
consequences of fulfillment of free volume model. It should be noted that free volume
exists in polymeric materials in the form of a size distribution of microcavities.
Strictly speaking, different gases should be sensitive to different parts of this
distribution, so the diffusion coefficient Di of gas Mi will “feel” a different value of Vf
than the diffusion coefficient Dj of gas Mj (see e.g. Park and Paul [2]). However, if we
assume in the first approximation that a single parameter Vf can characterize
permeation of both gases, then the observed linear relationships
logP2 = a + b logP1
(1)
lead directly to the well-known trade-off equation
log α12 = -a – (b-1) log P1
(2)
where a = log A2 – (B2/B1) log A1 and b = B2/B1, while A and B are the parameters of
free volume equation
Pi = Aiexp(Bi/Vf)
(3)
The equation (2) describes the position of the median line in the Robeson diagram [3],
i.e. it explains a steepness of the clouds of the data points in the Robeson diagram
plotted for different gas pairs (see e.g. Figure 2). The parameters a and b well
correlate with cross-sections of diffusant molecules:
b = 0.8(d22/d12) + 0.2
(4)
and
a = -7.15(1 – d22/d12)
(5)
Thus, simple free volume model describes average regularity of permeability
and permselectivity. It can be added that another simple model proposed by Freeman
[4] and based on approximate parameters of transition state theory well describes the
position of “upper bound” in the Robeson diagrams.
2. Free volume: historical perspective and current development. Free
volume as it had been proposed half a century ago [5] was interpreted primarily as an
abstract concept. Empirical methods for its evaluation were developed and often used
[6], sometimes successfully but often giving rather poor correlations. A big
disadvantage of the Bondi’s approach is that a universal equation is employed to
relate van der Waals Vw and occupied Vocc volumes in all the polymers: Vocc = 1.3 Vw.
The Bi parameter in Eq.3 should correlate with molecular dimensions of the diffusing
molecules. However, an application of this equation (with free volume calculated
using the Bondi scheme) gives very poor results in addressing diffusivity of polymers
in respect of different gases. It means that interpretation of free volume as a universal
property of the materials is over-simplified: instead, a concept of accessible free
volume should be considered. This property accounts for those parts of free volume
size distribution that are “seen” by diffusing molecules of significantly different size,
e.g. He and Xe. Several examples of calculation of this parameter can be found in the
recent literature [7,8].
During the last decades, several experimental methods for investigation of free
volume, so-called probe methods, were proposed and widely used. A list of those
methods is given in Table 1. The probes that are used for sampling free volume in
polymers differ by the size and shape, whereas the methods are based on different
principles of observation of behavior of the probes in polymers. The smallest probe is
used in the positron annihilation lifetime spectroscopy (PALS): in this method
information on free volume is provided by lifetimes of o-positronium atoms (o-Ps),
o
that is, hydrogen-like electron-positron pair (e-e+) having the size of about 1 A . Much
larger molecules serve as probes in the methods of electrochromic, photochromic and
o
spin probes: their size can be as large as 10-20 A . In these methods, the relation
between the dimensions of the probes and microcavities, where they reside, determine
their behavior in the polymers. For instance, in inverse gas chromatography, vapor
retention characteristics and thermodynamic parameters deduced from them depend
on relative size of solute molecules and microcavities in polymers. In contrast to the
PALS and 129Xe/NMR techniques, all other probe methods employ series of probes
with varying sizes: by observing the behavior of smaller and larger probes
conclusions on the hole size of a polymer can be made.
As now various probe methods have been applied for investigation of free
volume in certain polymers, a question can be asked about reliability of the results of
these methods. Some answers to this question are presented in Figure 3 and Table 2.
Figure 3 shows size distribution of free volume elements in several glassy polymers
according to the PALS and IGC methods [9]. It is seen that relative positions of the
curves in both plots are very similar: the largest free volume elements are present in
the most permeable polymers – PTMSP and amorphous Teflon AF2400. A wider
comparison of the results of probing free volume is made in Table 2. It can be noted
that observations based on different methods are in reasonable agreement. These
results are also consistent with the predictions of computer modeling of the same
polymers [8, 11].
3. Relationship of the transport parameters to free volume. As the mean
size of free volume elements (FVE) is found one can calculate the fractional free
volume provided the concentration N of FVEs is known. The problem of
determination of this value is not simple. Recently it was shown that rather good
estimate of N in different polymers can be obtained on the basis of analysis of
activation energies of diffusion using approximate Meares equation [12]. The N
values found are consistent with the estimations using PALS [9]. Thus, it was shown
that the length of diffusion jump in glassy polymers virtually coincides with the
average distance between adjacent FVEs. These results enabled us to explain several
well documented but still not comprehensible observations such as dependence of
activation energy of diffusion or diffusion selectivity on free volume.
This and several other studies indicate that the quantitative description of gas
permeation in glassy polymers should account for the parameters of the processes that
take place in “the walls” of FVEs. Thus, extremely important is the role of interchain
interactions and small scale movements in densely packed areas surrounding FVEs.
Good illustration of this thesis was given by Faupel et al [13], who showed that
deviations from the free volume model of diffusion can be well accounted by
cohesion energy density. According to this work, the activation energy of diffusion
can be attributed to energy required to separate the polymer chains in the saddle point
of a hopping event. So correlations of diffusivity with free volume found via the
PALS method can be markedly improved by taking into account the parameters of the
processes that take place in the walls of FVEs.
Concluding remarks. Thus, today prerequisites are created for a transfer from
phenomenological approaches relating chemical structure of polymers and their
transport properties to physically more consistent models that account for not only
free volume interpreted as integral parameter of the material but also its size
distribution, topology, connectivity, as well as the parameters of the processes that
proceed in more densely packed regions of polymers, which surround FVEs. It is
likely that a decisive role in development of such models will be played by computer
modeling, the field that is making now a very fast progress. However, the methods of
molecular dynamics still deeply need validation of the computational results. This
validation depends on experimental determination of solubility and diffusion
coefficients and on the results of different probe methods for characterization of free
volume. Hence, so far experimental studies of membrane properties of polymers and
their free volume are still called for. Future will show how long this situation will last.
References.
1. A. Alentiev, Yu. Yampolskii, J. Membr. Sci., 165, 201 (2000).
2. J. Y. Park, D. R. Paul, J. Membr. Sci., 125, 23 (1997).
3. L.M.Robeson, J.Membr. Sci., 62, 165 (1991).
4. B. D. Freeman, Macromolecules, 32, 375 (1999).
5. Ya. I. Frenkel, Kinetic theory of liquids, Moscow-Leningrad, 1945.
6. A.Bondi, Physical properties of molecular crystals, liquids, and gases, Wiley, New
York, 1968.
7. I.Ronova, E.Rozhkov, A.Alentiev, Yu.Yampolskii, Macromolecular Theory and
Simulations (accepted).
8. D.Hofmann, M.Entrialgo-Castano, A.Lerbret, M.Heuchel, Yu.Yampolskii,
Macromolecules (submitted).
9. A. Alentiev, V.Shantarovich, T.C.Merkel, V.Bondar, B.D.Freeman,
Yu.Yampolskii, Macromolecules, 35, 9513 (2002).
10. G.Golemme, J. B. Nagy, A. Fonseca, C.Algieri, Yu.Yampolskii, Polymer
(accepted).
11. D.Hofmann, M.Heuchel, Yu.Yampolskii, V.Khotimskii, V.Shantarovich,
Macromolecules, 35, 2129 (2002).
12. A. Alentiev, Yu. Yampolskii, J. Membr. Sci., 206, 291 (2002).
13. C.Nagel, K.Guenther-Schade, D.Fritsch, T.Strunskus, F.Faupel, Macromolecules,
35, 2071 (2002).
Table 1. Methods for probing free volume in polymers
Method
Probe
Size
Information
o
Positron annihi- o-Ps
Size, size distribution,
1.06 A
lation lifetime
concentration of FVE,
spectroscopy
dependence of size of FVE on
(PALS)
temperature and pressure
o
Hydrocarbons
Inverse gas
Temperature averaged mean
>5 A
chromatography
size of FVE
(>C3)
(IGC)
129
129
Xe-NMR
Xe
About
Size of FVE and its temperature
o
dependence
4A
o
Spin probe
Stable free radicals
Information on the part of size
3
>100
A
method
distribution of free volume that
corresponds to larger holes;
Photochromic
Stilbene, azo120o
temperature dependence of
probe method
benzene derivatives
600 A 3
larger holes size
o
Electrochromic Azo-dyes
3
>800 A
probe method
o
Table 2. Radii ( A ) of free volume elements in polymers measured by different
methods [9].
129
Polymer
PALS
Xe-NMR
IGC
PTMSP
6.8
AF2400
6.0
7.8
6.4
AF1600
4.9
6.7
5.8
PTFE
4.2
5.7
PVTMS
4.3
5.3
PPO
3.4
2.9
3.4
PC
2.9
2.5
PTMSP – poly(trimethylsilyl propyne), AF2400 and AF1600 - copolymers of 2,2bis(trifluoromethyl)-4,5-difluoro-1,3-dioxole and tetrafluoroethylene (TFE content 13
and 35%, respectively), PVTMS – poly(vinyltrimethyl silane), PPO – poly(phenyelen
oxide), PC – polycarbonate.
logP(N2), Barrer
4
3
α =10
α =15
2
1
0
-1
α =1 α =5
-2
-3
-3
-2
-1
0
1
2
3
4
logP(O2), Barrer
logα (O2/N2)
Figure 1. Correlation of the permeability coefficients of O2 and N2 in glassy polymers;
dashed lines are iso-selectivity lines.
1.4
1.2
1
0.8
0.6
0.4
0.2
0
B
A
-3
-2
-1
0
1
2
3
4
logP(O2), Barrer
Figure 2. Permeability/selectivity diagram for O2/N2 pair: A is the line defined by Eq.
(2), B is the upper bound line of Robeson.
0
-0.1
-0.2
AF 2400
- f (R)
-0.3
-0.4
-0.5
AF 1600
-0.6
PVTMS
-0.7
PTMSP
-0.8
0
500
1000
1500
2000
Vf, Е 3
10
?H m [kJ/mol]
5
0
-5
AF 2400
-10
-15
-20
PVTMS
AF 1600
-25
0
200
400
600
800
Vc [cm3/mol]
Figure 3. Estimation of the size of free volume elements in glassy polymers. Top:
Probability density function f(R) in polymers with different size of free volume
elements according to PALS; Bottom: Excess enthalpy of mixing ∆Hm as a function
of critical volume of solutes Vc according to IGC.