Journal of Nuclear and Radiochemical Sciences, Vol. 2, No. 1-2, pp. R23-R26, 2001

Reviews
Positron Annihilation Study of Polymer Membrane Materials
V. P. Shantarovich*
Institute of Chemical Physics Russian Academy of Sciences, Moscow 117334, Kosygin st.4, Russia
Received: October 1, 2001; In Final Form: November 16, 2001
Free volume effects are important for positronium formation, pick-off annihilation and chemical reactions. In
order to turn positron annihilation lifetime (PAL) measurements into effective tool for quantitative investigationsof molecular solids, polymers for example, the relation between the intensity of the long-lived positronium
component of lifetime distribution of positron annihilation and concentration of defects (elementary free volumes
EFV) has to be derived. This enables not only calculations of the effective size of elementary free volumes but
also estimations of the specific free volume. Meanwhile, the studies of EFV in polymer systems are of greatinterest, because effective sizes and concentrations of EFV determine many transport, mechanical and other properties of polymers. We demonstrate here an approach to the studies of these problems on example of big group ofglassy polymers distinguished by a wide range of gas permeability P (1−104 Barrer) and selectivity. They are used
therefore as good gas-separating membrane materials.

1. Introduction
Positron annihilation lifetime (PAL) spectroscopy provides
an effective approach for studying elementary free volumes(EFV) in the solid structure. With respect to polymers, thismethod is based, in particular, on the so-called “free volume
model”, suggested by Brandt. 1 This model assumes that
positronium atom (Ps) tends to localize in the free volume
elements of polymer structure due to its repulsion (exchangeinteraction) from surrounding molecules. The correlation
between Ps annihilation characteristics and properties of thematerials was studied in many laboratories and a lot of interesting results were obtained. For example, Tao2 and Eldrup3
derived a simple semi-empirical equation for the dependenceof o-Ps lifetime on the FVE effective size. In addition to wellknown program PATFIT, describing PAL distribution in termsof several components, Gregory 4 and Jean 5 applied the
CONTIN computing program to obtain continuous lifetimeand size-distribution of EFV in polymers. We are not going todiscuss here the whole spectrum of the studies and refer theinterested reader to the book6 and to some other recent publications of more or less general character.7, 8 We shall touch here
one problem: how Ps atom finds itself in the EFV and, correspondingly, what kinetic equations are to be solved in order tofind a dependence between the intensity of the long-lived(positronium) component and EFV concentration. On thebasis of our approach, we discuss our PAL results on positronannihilation in glassy polymer membrane materials.
2. The Problem of Ps Trapping
The simplest way to tackle a problem of trapping is toconsider corresponding kinetic equations. For the simplest
case of the traps of one kinddPf /dt = − (λ +ν3) Pf ,dPt /dt = ν3Pf − γ3 Pt ,o−PsfPf (0) = 3Q/4,
Pt (0) = 0,(1)

where Pi is the probability to find a free or trapped (long-lived)o-Ps, λo−Ps
stands for free (non-trapped) ortho-positroniumf*E-mail: shant@center.chph.ras.ru. FAX: +7-095-1378318.

annihilation rate, and ν3 and γ 3 are the trapping and annihilation rates of o-Ps in elementary free volume.
The lifetime distribution of annihilation radiation is given
then by the following equation
Y (t) = F (t) + I3γ3 exp (−γ3t),(2)where F(t) involves short-lived components. Long-lived
component I3 is related to ortho-positronium in EFV.
I 3 = (3Q/4) ν 3 ( λo−Ps+ ν 3 − γ 3).
f(3)

Eq 2 shows a typical S-like dependence of I3 on the Ps trapping rate ν 3 into elementary free volumes. Q is fraction of
positrons which form Ps in a substance. Subscript “3” comesfrom the number of the components in the lifetime distributionof positron annihilation. The longest one has a number “3”.
Obviously, γ3 is dependent on the mobility of free (just formedin the bulk) positronium atom and on the concentration ofelementary free volumes N3:

ν3 = 4π D o−Ps
N3R3 .
f(4)

D o−Psis a diffusion coefficient of free (non-localized) positrof
nium, and R 3 is an effective radius of the elementary freevolume. Radius R3 can be determined from o-Ps lifetime τ3 =1/γ3 using a well-known equation

τ3 (ns) = (1/2)[1− (R3/R0)+(1/2π)(Sin (2πR3/R0)]−1, (5)suggested by Tao and Eldrup. Parameter R0=R3+∆R, where R3
is already mentioned effective EFV radius and ∆R = 1.66 Å.
The two limiting situations can be considered for the eq 3:− γ 3, then I 3=(3/4) QDo−Ps
R 3 N 3,
1- if one assumes that ν t < λo−Psff
and fractional free volume Vf 3 is roughly proportional to the
product of I3 and the volume of micro-cavity vf 3 :Vf 3 = CI 3vf 3 .

(6)

A similar empirical equation had been suggested by Jean et al.9
They called parameter C an empirical constant.

© 2001 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 12/28/2001

Shantarovich

R24 J. Nucl. Radiochem. Sci., Vol. 2, No. 1-2, 2001
2 - the second alternative appears when ν3>>λ −γ3. In thiscase, I3 independent on concentration of EFV, i.e. I3 ~ Q and
can not be used for calculations of concentration of elementary
free volumes N3.
The choice between the two alternatives can be done on
experiment by creating or removing elementary free volumes
N3 artificially and by measuring corresponding intensities I3.
We have done such experiments for the cross-linked
olygoetheracrylates 10 : I 3 decreased with increasing of thenumber of links. We refer here also to the experiments of
Suzuki and Goworek,11 where compression or cooling of thesample, which are also supposed to decrease the number ofEFV, gave strong inhibition of Ps formation (I 3 intensity).
Thus we get some arguments in favor of the first alternative (I3is a function of the trapping rate ν3 and of the concentration ofelementary free volumes N3), and the most correctly would be
to use eqs 3, 4 describing intermediate situation. Additional
argument has been obtained, on our opinion, by Dauwe et al.12
who used age-momentum-correlation technique AMOC for thestudies of poly(methylmetacrylate) PMMA. This techniqueenables one to estimate o-Ps lifetime till the moment of final
localization (trapping). This lifetime came out to be very long
(0.54 ± 0.04 ns). It would be difficult to suppose that Ps atom
is formed inside of the EFV and it takes such a long time forstabilization inside of this cavity. We suggest that theobserved lifetime is a time of positronium diffusion fromdeeper to the deepest trap till the moment of final localization,
and we describe this motion by some effective diffusion coefficient Dfo−Ps (see eq 4).
Now, what can we say about the value of this coefficient?
This value can be found in the experiments, when the numberof EFV is known because, for example, you are creating them(experiment with porous poly(phenylene oxide) PPO, 13 orchanging the size of some of them, using a known amount ofcross-linking agent. You calculate the trapping rate from annihilation characteristics using the equationo−Psfν 3 = [(λo−Ps− γ 3) (4I 3 /3)]/(Q−4I 3 /3),fstyrene) PTMSS, poly(4-methyltrimethylsilylstyrene)
PTMSMS, poly(vinyldimethylphenylsilane) PVPDMS, poly
(vinyltrimethylsilane) PVTMS, copolymers of 2,2-bifluoromethyl-4,5-difluoro-1,3-dioxole and tetrafluoroethylene
with different ratio of the components (AF1600 and AF2400),poly(trimethylsilylpropylne) PTMSP, poly(1-phenyl-2-ptriisopropylsilylphenylacetylene) PIPSDPA, poly(1-phenyl2-p-triphenylsilylphenylacetylene) PPSDPA, and Nafion-117.
After using CONTIN program to the spectra, obtained by
PAL measurements, these systems show a smooth transition
from substances with unimodal distribution of o-Ps lifetimes(unimodal distribution of elementary free volumes) to thosewith bimodal distribution (Figure 1). This result shows thatcomplex (non-statistical) distribution of elementary freevolumes is a general feature of many polymer compounds.
The second effect, which is typical for these substances, is astrong positronium quenching (shortening of o-Ps lifetime) byatmospheric oxygen in situation, when the number of oxygen
atoms N OX is smaller, than the number of elementary freevolumes. These two facts must be discussed together. 14
Positronium quenching enables determination of the diffusion
coefficient of localized Ps D lo−Ps (this coefficient can bedifferent from diffusion coefficient of nonlocalized, justformed Ps Dfo−Ps, which we considered above) from the equation
VAC
λ OX= 4π Dlo−Ps (RPs+ROX) NOX .
4 − λ4(8)

Thus, for Ps quenching by nitrobenzene in polycarbonate(7)

and, further on, determine Dfo−Ps.
Such experiments gave Dfo−Ps~10−5 cm2/s, which is much
lower than diffusion coefficient of free e + (Df+~10−1 cm2/s).
Therefore, we conclude that trapped particle is not positron,
which could form positronium after trapping, but non-localized
positronium.
All these facts support, to our mind, the described above
approach to calculations of the number of EFV and, correspondingly, of the specific free volume on the basis ofmeasurements of PAL spectra.
3. Positron Annihilation in Glassy Polymer Membrane
Materials
A group of polymers, selected for this study, attract attention as candidate materials for gas-separation membranes.
Many of them are characterized (using PATFIT program) by along o-Ps lifetime (>10ns), which is probably due to bulky
silyl groups, typical for their structure. It came out that, inspiteof generally-accepted homogeneity of the glass structure, someof these compounds have bimodal size-distribution of elementary free volume or FVE of two types. We were first to reportsuch characteristic for poly(trimethylsilylpropyne) PTMSP.
Now we are going to discuss similar results for three groups ofsuch polymers, prepared in the form of thin (~1 µm) sheets .
(1)- silicon bearing polystyrenes;(2)- polyacetylenes;(3)- perfluorinated polymers.
These polymers include poly(4-trifluoropropyl dimethylsilylstyrene) PFPDMSS, poly(4-trimethylsilyl-

Figure 1. Probability density function λα (λ) of positron annihilation
lifetimes τ = λ−1, obtained by CONTIN program from PAL spectra ofsome of the membrane materials (for explanation of abbreviations see
the text). Long-lived components are related to Ps and, using (5), canbe transformed to size-distribution of EFV.

Positron Annihilation Study
15

J. Nucl. Radiochem. Sci., Vol. 2, No. 1-2, 2001 R25
o−Ps
l

−6

Hirata et al. had found D ~10 cm2/s. In our case, for
PTMSP, it came out one order of magnitude higher, about thesame as D fo−Ps . This is probably due to highly developed
system of free volumes in the membrane materials.
As we sad, observation of the two long-lived positronium
components τ 3, τ 4 actually means existence of the two types of
positronium traps (EFV). For homogeneous distribution of
defects it can be described by eq 9:14
dPf /dt = − (λ f +ν3+ν4) Pf ,dP3 /dt = ν3 Pf − (γ3 +ν34) P3 ,dP4 /dt = ν4 Pf + ν34 P3 − γ4 P4 ,Pf (0) = 3Q/4,
P3 (0) = 0,
P4 (0) = 0,

log D = ad 2+ b.

(10)

Parameter A is also a function of I3 and I4. This approachgives N3 and N4 on the level 1019−1020 cm−3, while concentration of oxygen atoms is below 1018 cm−3. It means that, inorder to get quenching, the same Ps atom must move from hole
to hole and stay there, sometimes with oxygen (if oxygen istrapped in the given hole) and sometimes without it. Now, thequestion comes up: if Ps moves from hole to hole, why can we
resolve distinctly bimodal distribution of positronium lifetimes
in such polymer glasses as PTMSP, PIPSDPA and someothers? The answer can be only one: the distribution ofelementary free volumes in such systems is inhomogeneous,
and, once formed, positronium atom is not able to go out of thelimits of the micro-region of this inhomogeneous structure. To
be more accurate, kinetic eq 9 must be changed on the equations, describing separately positronium annihilation in the
micro-regions with large pores and in those with smaller pores.
Correspondingly, initial conditions (eq 9) should be changed,
because there is a competition between two types of theregions for the initial emergence of Ps. Roughly assuming thatthese probabilities are proportional to intensities I 3 and I 4 ,respectively, we get equations similar to eq 1:
dPf /dt = − (λo−Ps+νi) Pf ,fdPi /dt = νi Pf − γi Pi ,

(13)

(9)

which is similar to (1) and includes also a possibility of detrapping and transition from the trap “3” to the trap “4”. In thesimplest case ν34 = 0

ν 3 = (I3 /A)[(λ f − γ 3)(3Q/4 − I 4 )+(λ f − γ 4)I 4 ],
ν 4 = (I4 /A)[(λ f − γ 4)(3Q/4 − I 3 )+(λ f − γ 3)I 3 ].

Furthermore, results on the size and amount of elementary
free volumes were compared with permeation and selectivityof thin layers of the mentioned polymer glasses. It was foundthat namely N4, v4 (but not N3 , v3) are responsible for permeation (Figure 2).10 Permeation selectivity a is a coefficient ofproportion between diffusion coefficients D of different gases
in a given polymer and square of kinetic diameter d of penetrating molecules:

A nice correlation was found between the selectivity coefficient a and the size of elementary volume v 4 in the studiedmaterials. This correlation is useful for express determinationof membrane selectivity on the basis of PAL measurements,
instead of time-consuming and difficult measurements of penetration for different gases.
Finally we conclude that positron annihilation lifetime
measurements can be a useful instrument for investigations ofmembrane materials, providing the conditions of positronium
formation and trapping are estimated correctly.

Pf (0) = 3Qξi /4, (11)Pi (0) = 0,
Figure 2. Correlation between the size of elementary free volumes vf3
and vf4 (Å3), calculated as 4πR33 /3 and 4πR43 /3 from the lifetimes τ3 andτ4 in the PAL spectra of the substances, listed in the text (PATFIT
data, Eq 5), and their permeation P for oxygen, measured in Barrer.

where i =3,4 and ξ 3 = I3 /(I3+I4); ξ 4 = I4 /(I3+I4).
Correspondingly,
νi = (4Ii /3)(λ f − γ i )/(Qξi − 4Ii /3) = 4π Dfo−PsNi R i .

(12)

It is assumed that Q, λ f and Dfo−Ps are identical for the bothtypes of micro-phases.
The fractional free volume, calculated in this way, forexample, for PTMSP as (N3v3 + N4v4), attains 20% which isnormal for the systems with highly developed free volume.
Thus, we are able to make two conclusions:(a) In many polymer membrane materials (PTMSP,PIPSDPA, etc.) size-distribution of elementary free volumes is
non-statistical and looks complicated (sometimes bimodal).
This conclusion correlates well with the recent results of molecular modeling.16(b) Distribution of elementary free volumes in glassy
polymer matrix is inhomogeneous. Estimations of o-Ps diffusion length L, using the equation L = (6D o−Ps
) 1/2 , gives theleffective size of microregions with different EFV about 300 Å.
Remarkably that the problem of non-homogeneous characterof such polymer glasses was discussed previously by someauthors on the basis of different data.17, 18

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