Positronium as a Probe of Polymer Free Volume: History
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Contributor:
Giovanni Consolati
,
Dario Nichetti
,
Fiorenza Quasso

Positron annihilation lifetime spectroscopy (PALS) is a valuable technique to investigate defects in solids, such as vacancy clusters and grain boundaries in metals and alloys, as well as lattice imperfections in semiconductors. Positron spectroscopy is able to reveal the size, structure and concentration of vacancies with a sensitivity of 10−7. In the field of porous and amorphous systems, PALS can probe cavities in the range from a few tenths up to several tens of nm. In the case of polymers, PALS is one of the few techniques able to give information on the holes forming the free volume. This quantity, which cannot be directly measured with macroscopic techniques, is correlated to important mechanical, thermal, and transport properties of polymers. It can be deduced theoretically by applying suitable equations of state derived by cell models, and PALS supplies a quantitative measure of the free volume by probing the corresponding sub-nanometric holes. The system used is positronium (Ps), an unstable atom formed by a positron and an electron, whose lifetime can be related to the typical size of the holes. When analyzed in terms of continuous lifetimes, the positron annihilation spectrum allows one to gain insight into the distribution of the free volume holes, an almost unique feature of this technique. The present paper is an overview of PALS, with emphasis on the experimental aspects. After a general introduction on free volume, positronium, and the experimental apparatus needed to acquire the corresponding lifetime, some of the recent results will be shown, highlighting the connections between the free volume as probed by positronium and structural properties of the investigated materials.

  • free volume
  • positron annihilation lifetime spectroscopy
  • polymers
  • membranes
  • biopolymers
  • composites

1. Introduction

Although not univocally defined [1], the concept of free volume can be used to explain various features of polymers. For instance, mechanical properties [2] are generally (negatively) correlated to the free volume fraction of the polymer. Indeed, the applied load tends to concentrate in the free volume instead of distributing among the molecules of the polymer, and this can result in the failure of the material. Therefore, a polymer with a reduced free volume fraction generally shows better mechanical properties [3]. Transport properties are of the utmost importance in polymer membranes used in molecular separation processes such as water desalinization or environmental remediation. In this case, too, the free volume holes in the polymer matrix are responsible for gas separation [4][5]. Another feature involving free volume is physical aging [6]: an amorphous polymer exposed to temperatures below the glass transition for prolonged periods of time shows an increment of the mass density, with decrement of the molecular configurational energy (enthalpy relaxation). When the polymer is cooled from temperatures sufficiently high, the relationship between specific volume and temperature is linear and the structure is in energetic equilibrium. Continuing to cool the polymer, a deviation from linearity is noted in a restricted temperature range: on passing from the rubbery to the glassy state, the specific volume decreases at a rate lower than above the glass transition. The curve of the specific volume shrinkage runs parallel (but higher) to that of the same material ideally crystalline at equilibrium and resulting from the thermal contraction. This behavior is explained in terms of a reduction in the free volume. Struik [7] defined the physical aging using the concept of free volume, showing that large mechanical deformations can result in rejuvenation of the glass. Furthermore, a fundamental linear correlation between the glass transition temperature and free volume fraction has been derived at the molecular level [8]. Roughly speaking, the free volume is the difference between the total volume and ‘occupied’ volume. The different possible definitions of the latter [9] bring about the ambiguity of the idea of free volume. Indeed, when the occupied volume is formulated in terms of the Van der Waals volume [9], the free volume attains the maximum limiting value. On the other hand, in other models [10] the occupied volume also comprises the volume swept out by a molecular segment due to thermal vibration (‘fluctuation volume’ [9]). The difference between the total volume and this ‘vibrational’ volume is excess free volume, which allows movements of polymer segments. Free volume holes refer to this excess free volume, which has been incorporated in some cell models [11].
Holes can be examined by various probes, such as photochromic labels [12], fluorescence molecules [13], or techniques such as small angle XRD [14]. Positron annihilation lifetime spectroscopy (PALS) uses positronium (Ps), a bound electron-positron state, to gain information about the size and distribution of free volume holes (information about the physical properties of Ps can be found in a review by Berko and Pendleton [15]). This non-destructive technique is based on the fact that Ps is repelled from the molecules of the polymer, due to the exchange repulsion between the Ps electron and surrounding electrons and localizes into the open spaces of the host structure. In this sense it is a ‘seeker’ of free volume holes. Since these last have unequal sizes, the probability that Ps tunnels from the hole where it is confined to neighboring holes is negligible [16]. Therefore, Ps can effectively probe the size of the holes, as long as its lifetime is less than that of the cavity hosting it. Ps is a suitable probe even for small cavities, since its size is the same as hydrogen; however, it is much lighter (by a factor of about 2000). As a consequence, quantum effects have to be taken into consideration.
Positron annihilation techniques are not limited to polymer systems, but have been widely used in almost every field of materials science, e.g., to study and identify lattice imperfections in semiconductors [17][18] and metals [19]; understand structural changes occurring in age-hardenable alloys [20]; investigate damages induced by neutron irradiation [21] and electron irradiation, [22] as well as ion implantation [23], to get insight into defects formed in ceramics [24] and explore surfaces [25].

2. Positronium in Polymers

Ps is an unstable system, subjected to annihilation. Para-Ps (p-Ps, antiparallel spins) and ortho-Ps (o-Ps, parallel spins) are the two sublevels of ground state Ps, characterized by the different spin orientation of the two particles [26]. In a vacuum, p-Ps has a lifetime of 125 ps; o-Ps lifetime is much longer, 142 ns. In a material, p-Ps is scarcely influenced by the environment and changes in its lifetime are small; on the other hand, in a hole, o-Ps undergoes interactions with surrounding electrons and o-Ps can annihilate, in addition with its own electron, also with an outer electron in relative singlet spin state. This process is called ‘pickoff’, and it is responsible for the decrease in o-Ps lifetime with respect to its value in vacuo up to two orders of magnitude, depending on the overlap between the wave functions of the positron and surrounding electrons [27].
The free volume hole size can be estimated from o-Ps lifetime as supplied by the experiment; to this purpose holes have to be modeled within a suitable geometry. This necessary step allows one to convert the raw results of a measurement into quantitative information. In fact, many experimental techniques use conventional geometries: for instance, in porosimetry, pores are often approximated to cylinders. Concerning PALS, the most popular model for polymers is the spherical one [28][29], although other geometries have been applied, e.g., ellipsoidal cavities [30] have been used to frame the free volume holes in semicrystalline polymers subjected to tensile deformation. Starting from different geometries for the holes, differences in sizes (of the order of 20–30%) are obtained for the same o-Ps lifetime [31].
Tao [28] and later Eldrup [29] found a relationship between o-Ps lifetime and the size of the cavity hosting Ps. Their model assumes a spherical void with effective radius R. This Ps trap has a potential well with finite depth; however, for convenience of calculation, an infinite depth is assumed, with a corresponding increase in the radius to R + ΔR, ΔR being an empirical parameter [32] that describes the penetration of the Ps wave function into the bulk. The electron density is zero for r < R and constant for r > R. The probability p to find Ps inside the bulk polymer is:
p = 4 π ∫ R + Δ R ∞ ψ r 2 r 2 d r = Δ R R + Δ R + 1 2 π sin 2 π R R + Δ R
where [33]:
ψ r = 1 2 π R + Δ R sin π r R + Δ R r
inside the well and zero outside.
The annihilation rate of o-Ps in the bulk state is 𝜆0≅2  0≅2 ns−1, that is, the spin-averaged annihilation rate of p-Ps (8 ns−1) and o-Ps (0.007 ns−1) in a vacuum. The o-Ps pickoff decay rate 𝜆𝑝 (ns−1) is therefore λ0 p and the relationship with the hole radius R is:
λ p = λ 0 Δ R R + Δ R + 1 2 π s i n 2 π R R + Δ R
The o-Ps lifetime τ3 as determined by the experiment is the reciprocal of the pickoff decay rate, if researchers neglect the contribution of the intrinsic decay rate λi = 1/142 ns−1, which is typically two orders of magnitude less than 𝜆𝑝. Equation (3) is almost universally adopted to infer the average size of the holes. However, the irregular shape of real holes makes it appropriate to ask whether the spherical model is the most suitable to deduce: (a) the sizes of the holes; (b) the variation in free volume fraction versus a physical variable, such as the temperature.
Another parameter that is available from PALS measurements is o-Ps intensity I3, which represents the normalized amount of o-Ps in the positron annihilation lifetime spectrum.
It is usually related to the number density of holes, N, in the sense that a linear relation between N and o-Ps intensity I3 is assumed [34]. Accordingly, in most of the studies the free volume fraction is written as:
f = CI3vh                 
where vh is the volume of the spherical hole as obtained by o-Ps lifetime using the Tao-Eldrup Equation (3) and C is a structural constant. Often a relative free volume, I3vh, is used in the discussion of the results. Researchers will come back to this guess later, by discussing some of the results obtained by our group.
In the following Table 1, the lifetimes and intensities of some polymers are shown, together with the corresponding sizes of free volume holes, in spherical approximation. Sometimes, two long components corresponding to Ps in bigger and smaller holes are found.
Table 1. Ps lifetime and intensity of some polymers. The radius of the spherical hole is also shown. LDPE: low density polyethylene, HDPE: high density polyethylene, PMMA: poly (methyl methacrylate), PET: poly (ethylene terephthalate), SBR: styrene/butadiene rubber, PTFE: poly tetrafluoroethylene, PDMS: polydimethylsiloxane, PTMSP: poly [1-(trimethyl-silyl)propine].
Polymer τ3 (ns) R3 (nm) I3 (%) τ4 (ns) R4 (nm) I4 (%) Ref.
LDPE 2.55 ± 0.01 0.33 21.1 ± 0.4       [35]
HDPE 2.38 ± 0.04 0.32 19.7 ± 0.3       [36]
Nylon-6 1.55 ± 0.02 0.24 24.5 ± 0.04       [37]
PMMA 1.92 ± 0.01 0.28 23.6 ± 0.02       [38]
PET 1.66 ± 0.03 0.25 20.4 ± 0.3       [39]
Cis1,4-polybutadiene 2.614 ± 0.005 0.34 39.45 ± 0.07       [40]
SBR 2.50 ± 0.02 0.33 34.2 ± 0.9       [41]
PDMS 3.27 ± 0.03 0.39 30.3 ± 0.5       [42]
Nafion 3.27 ± 0.01 0.39 6.33 ± 0.06       [43]
PTFE 1.12 ± 0.09 0.19 9.6 ± 0.9 3.92 ± 0.02 0.43 13.8 ± 0.2 [44]
PTMSP 4.7 ± 0.3 0.47 9.7 ± 0.5 13.8 ± 0.1 0.79 31.1 ± 0.6 [45]

This entry is adapted from the peer-reviewed paper 10.3390/polym15143128

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