THE JOURNAL OF CHEMICAL PHYSICS 139, 204901 (2013)

Polarization of positronium in amorphous polar polymers: A case study G. Consolatia) and F. Quasso Department of Aerospace Science and Technology, Politecnico di Milano, via La Masa, 34, Milano 20156, Italy

(Received 24 September 2013; accepted 6 November 2013; published online 22 November 2013) The features of positronium in an amorphous copolymer (polyvinyl acetate-crotonic acid) in a range of temperatures including the glass transition were investigated by means of positron annihilation lifetime spectroscopy. In particular, para-positronium lifetime was found to be longer than in a vacuum and to decrease with the temperature. This was attributed to the electron density at the positron (contact density), which is lower than in vacuo due to the presence of polar groups in the copolymer. A three quantum yield experiment confirmed the lifetime results. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4832321] I. INTRODUCTION

Positron Annihilation Lifetime Spectroscopy as a tool to get information on the microscopic structure of non metallic solids, such as polymers1, 2 or porous systems,3 is based on the annihilation features of the positron-electron bound system, positronium (Ps). In particular, the ground state triplet sublevel (ortho-Ps, o-Ps, with parallel spins of the two particles) has a reduced lifetime with respect to o-Ps in vacuo (142 ns), due to interactions with the surrounding electrons which supply an additional channel, called “pickoff,”4 for annihilation. This allows one to obtain a typical size of the cavities hosting Ps when the shape of these holes is framed within a suitable geometry.5–8 Ps in condensed matter can differ with respect to Ps in a vacuum also for the electron density at the positron, or contact density,9 |ψ(0)|2 , where ψ is the Ps wavefunction. In vacuum:9 1 |ψ(0)|2 = , (1) π n2 a03 where n is the principal quantum number and a0 is the Bohr radius of Ps. Both the intrinsic o-Ps and para-Ps (p-Ps) decay rates λt and λs , respectively, as well as the hyperfine splitting W, the energy separation between the two ground state sublevels, are proportional to the contact density:9, 10  α 3 h2 α 2 h2 4  2 2 |ψ(0)| |ψ(0)|2 , π , λ = − 9 t π m2 c 9π 2 m2 c 56 π μ2 |ψ(0)|2 , W = (2) 3 where m is the electron mass, h is the Planck constant, c is the speed of light, α is the fine structure constant, and μ is the magnetic moment of the electron and positron. By introducing the pickoff decay rate λp , the total decay rates for ortho (λ3 ) and para Ps (λ1 ) in condensed matter can be written:11 λs =

λ3 = ηλt + λp ,

λ1 = ηλs + λp ,

(3)

a) Author to whom correspondence should be addressed. Electronic mail:

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where η = |ψ(0)|2matter / |ψ(0)|2vacuum is the relative contact density. This phenomenological parameter takes into account possible modifications of the Ps contact density with respect to that in vacuo through interactions with the surrounding matter and confinement effects. It is generally found12 that η < 1 which corresponds to a swollen Ps, although values >1 are not ruled out.13, 14 In the case of Ps in condensed matter, the average distance between positron and electron can differ with respect to the value in vacuum; in particular, it increases if the particles are spaced out, e.g., for polarization effect. Consequently, the Bohr radius in Eq. (1), equal to 2/3 of the average electron-positron distance, can increase, too. This justifies, at least qualitatively, values of η < 1. Values of η around 0.80 are quite common; smaller values can be expected in the presence of polar compounds, since the microscopic electric fields present in the material act in opposite way on the electron and the positron and the electron density at the positron should decrease. According to Eq. (3), the pickoff process and, overall, a relative contact density lower than unity may change the p-Ps decay rate in such a way that the lifetime of this Ps sublevel may significantly differ with respect to the value in a vacuum (125 ps). In particular, as a consequence of the previous contention the effect should be revealed in a polar medium. In this work we show positron annihilation lifetime data in an amorphous copolymer made by vinyl acetate and crotonic acid, in a range of temperatures including the glass transition, which support this view. This material was chosen since both the components exhibit an electric dipole moment. From the analysis of the positron spectra we obtained the trend of the relative contact density versus the temperature. The most popular method to obtain the relative contact density is to use the Zeeman effect on Ps: in the presence of a magnetic field the Ps lifetime of the triplet sublevel with magnetic quantum number m = 0 is quenched.15 For this reason the method is known as “magnetic quenching”; by means of it anomalous magnetic effects on Ps formed in different media were discovered.12, 16 However, the method has the drawback to be rather laborious and time consuming. It is also possible to deduce the contact density from the analysis of the lifetime spectrum, in the absence of magnetic field, if both the ortho and para Ps lifetimes are

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clearly detectable.17 This is the method used in the present work. On the other hand, also the detection of the three quantum yield allows one to estimate the relative contact density, when the Ps features in the investigated medium (lifetime of ortho and para sublevels as well as the corresponding intensities) are known.17 We made use of this method, too, in order to get another, independent value of the relative contact density. Since the three quantum yield technique is less common than the lifetime spectroscopy, we will expound its basic principle in Sec. II C.

II. EXPERIMENTAL A. Material

We used a commercial copolymer (VINAFLEX CR25) formed by vinyl acetate (VA, 99.2% in weight) and crotonic acid (CA), supplied by Vinavil S.p.A. (Milano, Italy). Figure 1 reports the structural formulas of the two components. It is obtained by means of polymerization in aqueous suspension and subsequent desiccation in air. A slab of the material was prepared by compression moulding at a temperature of 343 K for a few minutes, followed by cooling at room temperature by means of a water cooling system. Square samples with a surface area of 4 cm2 were cut out from the slab. Electric dipole moment of the copolymer is expected to be high, analogously to those ones of polyvinyl acetate (1.57 D)18 and of crotonic acid (2.13 D).19

B. Positron annihilation lifetime spectroscopy

The positron source (22 Na, activity 0.5 MBq) was enR foils (DuPontTM , veloped between two identical Kapton thickness 7 μm each), which were sealed each other by a cyanoacrylate glue. The source-support assembly was inserted into the sample in a typical “sandwich” configuration. The whole was placed in the center of a small copper container in direct contact with a temperature controller; stability of the temperature, monitored by a thermocouple, was ensured within 1 K. Positron annihilation lifetime spectra were collected through a conventional fast-fast coincidence set-up having a resolution of about 250 ps. Each spectrum contained about 3 × 106 counts; analyses were carried out through the computer code LT,20 with a suitable correction for the positrons annihilated in the Kapton.

FIG. 1. Structural formulas of vinyl acetate (a) and crotonic acid (b).

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C. Three quantum yield

The three quantum yield was obtained by means of the apparatus described in Ref. 21, able to simultaneously measure the three quantum annihilation events and the background. It consisted of three identical channels, formed by an integral line—a 5.1 cm × 5.1 cm NaI(TI) scintillator coupled to a photomultiplier tube—and a preamplifier, followed by an amplifier and a timing single channel analyzer. Four outgoing, identical signals from this last stage fed the same number of coincidence units, whose output triggered a counter and timer. The scintillators were arranged at 120◦ from each other; their symmetry axes were coplanar and in the same plane of the source. Each scintillator was surrounded by a truncated conical lead shield to make certain that the crystals could not “see” each other. The opening of the shield allowed the corresponding scintillator to be seen by the source under a solid angle of 0.20 sr. The gamma rays collected by the scintillators were processed through the timing single channel analyzer, in order to accept quanta in the range 280–420 keV, that is, a region centered around 340 keV (one third of the total annihilation energy, 1022 keV). The resolving time interval T of the system could cover a continuous range from 25 to 110 ns. The measurements were carried out at some fixed values of T, in order to simultaneously get the three quantum true coincidences and the background. Details on the procedure can be found in Ref. 21. The measurements were carried out alternatively in beryllium, where Ps does not form and in the investigated copolymer, to obtain the relative three quantum annihilation rate in the organic material, by using the metal as a reference.

III. RESULTS AND DISCUSSION

Positron annihilation lifetime spectra were at first analysed in three components without constraints, by requiring that the longest one was represented in terms of a lifetime distribution; resolution function was fitted by two Gaussians. However, we did not succeed to obtain significant information on the shortest component, which should correspond to p-Ps; in fact, large, random variations in the intensity occurred, in the range 3%–40%, on passing from a spectrum to the subsequent one. Furthermore, the relative uncertainty associated to the lifetime often attained 100%, in correspondence to the lowest intensities, which made impossible to give a physical interpretation of the component. We also attempted an analysis in terms of three components and two lifetime distributions, corresponding to the longest and the intermediate component. In this case, the standard deviation of the intermediate distribution was very low, near to zero. In addition, the presence of this new fitting parameter implied an increase of the uncertainties of all the other quantities (lifetimes, intensities, and standard deviations). Since aim of our work was the investigation of the Ps contact density, the presence of a distribution with a very small standard deviation in the intermediate component should have a negligible influence on the results. Therefore, we dropped out also this procedure. Then, we re-analyzed the spectra in three components by letting a single lifetime distribution corresponding to the longest

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FIG. 2. Lifetime spectra corresponding to the lowest (297 K, black dots) and the highest (362 K, red dots) investigated temperatures.

component, and by introducing a constraint, that is, a ratio 1:3 for the intensities of the shortest and the longest components, respectively, in order to evidence a p-Ps component. Indeed, such a requirement corresponds to the statistical ratio between the singlet and triplet Ps sublevels; it could change only in the presence of ortho-para conversion processes,22 which are excluded in the present study. Statistical test of the goodness of fit was satisfactory: for each spectrum the χ 2 always resulted in the range 0.95–1.09. The longest lifetime component is attributed to the decay of o-Ps trapped in the free volume holes, whose distribution is mirrored by the corresponding distribution of o-Ps lifetimes. In Figure 2 the spectra corresponding to the lowest and highest temperatures are represented. Figures 3 and 4 show the two parameters of o-Ps distribution, that is, centroid τ 3 and standard deviation σ 3 23 as a function of the temperature. In the lower temperature region there is a moderate increase of the lifetime τ 3 , while a 3 is found above 331 K, which correchange of the slope dτ dT sponds to the glass transition. This last was found, as usual, by fitting the o-Ps lifetimes corresponding to low and high temperatures ranges in terms of two straight lines (shown in Figure 3). Their intersection supplies a conventional value of the glass transition temperature. Also σ 3 (Figure 4) increases with the temperature above the glass transition region, while it remains almost constant below it; this may be interpreted in terms of an increase of the dispersion of the average size

FIG. 3. Behaviour of the o-Ps lifetime τ 3 as a function of the temperature.

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FIG. 4. The standard deviation σ 3 of the o-Ps lifetime distribution versus the temperature.

of the holes with the temperature in the rubbery phase of the polymer. The fitting in terms of straight lines (Figure 4) gives Tg = 327 K, in good agreement with the value found with the lifetimes. Figure 5 shows the values of o-Ps intensity at the investigated temperatures. A slight increase with the temperature is observed, which can be ascribed to increased Ps formation probability in the free volume holes, which have larger average sizes at higher temperatures. The lifetimes of the other two components are shown in Figure 6. The middle component (squares) is attributed to positrons not forming Ps: its average value is 0.405 ± 0.009 3 = 2.4 × 10−4 ns and its increase with the temperature ( dτ dT ns/K) is very small. It could mirror the occupied volume expansion of the copolymer,24 since no change of the slope is found around the glass transition; indeed, the occupied volume is supposed to be insensitive to the glass transition.25, 26 The shortest component (circles in Figure 6) is due to p-Ps annihilations. Contrarily to the other two components it shows a slight decrease with temperature, more marked above the glass transition. Its value, around 0.2 ns, is rather different with respect to p-Ps lifetime in vacuo: according to Eq. (3), it could be due to a low value of the relative contact density. In the following, we aim to show that this interpretation is in agreement with our experimental data.

FIG. 5. o-Ps intensity I3 versus the temperature in the investigated PVA-CA copolymer.

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J. Chem. Phys. 139, 204901 (2013) ηλ +λ /372

It results: I3 ( t λ3p ). A second contribution is due to pPs, which decays only in two quanta in a vacuum, but in a medium it can decay in three quanta by means of the pickoff λ /372 process: I33 p λ1 . The last contribution comes from positrons not forming Ps and annihilate in three quanta: (1 − 43 I3 )/372. Therefore, the ratio P/P0 results:17     ηλt + λp /372 1 λp /372 4 P + 1 − I3 . = 372I3 + P0 λ3 3 λ1 3 (5)

FIG. 6. p-Ps (circles) and free positron (squares) lifetimes versus the temperature in the PVA-CA copolymer.

Knowledge of o-Ps and p-Ps lifetimes allows us to get the value of the relative contact density η at the various investigated temperatures. Indeed, from Eq. (3) it is easily obtained: − τ3 λ1 − λ3 τ η= = 1 . λs − λt λs − λt 1

1

(4)

Figure 7 reports the relative contact density data versus the temperature. In spite of a rather large scattering (according to Eq. (4) η is obtained from the difference between two experimentally determined decay rates) an increasing trend with the temperature is evident. The value supplied by the linear fit at room temperature is η = 0.55, which is in good agreement with that obtained from the three quantum method. This last is based on the fact that the rate of three quantum annihilations, P, in a medium where Ps is formed is raised with respect to a metal, P0 , where no Ps occurs. In a 1 t = 372 , by using metal, the relative three quantum yield is 3λ λs Eq. (2) (the factor 3 results from the average over the relative spin populations27 ). In a medium which allows for Ps formation the relative three quantum yield is the sum of three terms. The first one is due to o-Ps and is equal to the formation probability times the probability that o-Ps decays in three quanta through the intrinsic decay rate, ηλt /λ3 , or by pickoff, (λp /372)/λ3 ; the two contributions sum each other.

FIG. 7. Dependence of the relative contact density on the temperature in the PVA-CA copolymer.

The above formula is obtained by assuming that all the positrons emitted from the source annihilate in the material surrounding it. If k is the fraction of the positrons annihilated R support, the previous equation is modified as in the Kapton follows:      ηλt + λp /372 1 λp /372 4 P + 1 − I3 = 372I3 + P0 λ3 3 λ1 3 × (1 − k) + k.

(6)

Equation (6) contains the relative contact density η, which can be determined from the knowledge of k, of the pickoff decay rate and of the o-Ps intensity, as supplied by lifetime measurements. In fact, from the previous equation it is possible to deduce the value of η by treating it as a free parameter and by adjusting the ratio P/P0 to the value experimentally obtained. We measured the ratio P/P0 at two different temperatures, that is, 297 K and 332 K and we found the values 1.73 ± 0.12 and 2.15 ± 0.17, respectively. From them we deduced the relative contact density η, equal to 0.50 ± 0.07 at 297 K and to 0.62 ± 0.08 at 332 K. Although these values are not significantly different when the associated uncertainties are considered, nevertheless they are in agreement with the values of η evaluated from the fitting straight line displayed in Figure 7, which supplies 0.54 and 0.60 at 297 K and 332 K, respectively. The slight increase of the relative contact density with the temperature could be associated to the free volume holes expansion, which separates the electrons of the cavity walls from Ps. The increase of temperature implies an increase of the amplitude of vibration of the outermost layer of electrons belonging to the cavity walls, but the net effect on Ps is anyway an increased distance between positron and external, surrounding electrons, as highlighted by the increased o-Ps lifetime with temperature (Figure 3). This involves also a reduction of the intensity of the electric fields generated by the polar molecules and acting on Ps. We can obtain a rough estimate of such a reduction by using the Tao-Eldrup equation5, 6 to translate o-Ps lifetimes into average sizes of the cavity (approximated by a sphere). We find a radius equal to 2.79 Å and 3.02 Å at 297 K and 332 K, respectively. If we assume that the Ps induced dipole interacts with the permanent molecular dipoles through induced dipole forces (Debye forces), it is known that their intensity scales with the distance r as r−6 . Therefore, they would decrease by a factor (2.79/3.02)6 ; in other words, they would be about 40% weaker at 332 K than at 297 K. The smaller Ps polarization at higher temperatures is most likely the reason of the increased relative contact density.

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It would be interesting, as a check of our discussion, to carry out analogous measurements in a non polar polymer; indeed, we would expect a higher value of the relative contact density. Unfortunately, this is not easy, since it would be necessary to find identical—or, at least, very similar—lifetimes. In fact, the comparison would be carried out between polymers with similar size of the free volume holes: as pointed out in our discussion, small differences in the cavity size involve significant variations in the strength of interacting dipoles. An example of amorphous, non-polar polymer allowing Ps formation is atactic polypropylene (PPA), in which the relative contact density was determined28 and turns out to be 0.66 ± 0.04, as a result of a magnetic quenching experiment. Thus, it results that η is larger than in the polar PVACA copolymer. However, such a contention should be carefully considered: the measurements in PPA were carried out at room temperature, that is, above the glass transition, while the investigated copolymer was glassy at room temperature; furthermore, o-Ps lifetime in PPA at room temperature is 2.66 ns, which corresponds to larger cavities than in PVA-CA. Finally, we point out that, even in the presence of a nonpolar medium, a polarization of Ps can occur, owing to the interactions (London dispersion forces) among the induced, instantaneous dipoles of Ps and of the surrounding molecules. Since the strength of such interactions is weaker than the Debye forces, the resulting effect on the Ps contact density is nevertheless expected to be smaller, the other conditions being constant, than in the case of polar media. IV. CONCLUSIONS

In the present study we investigated the features of Ps formed in a polar copolymer, in particular we focused on the dependence of the relative contact density with the temperature. We obtained η from the comparison of the o-Ps and pPs decay rates. The p-Ps sublevel shows higher values of the lifetime with respect to that in vacuum, even at the highest investigated temperatures; the lifetime decreases with the temperature. This has been ascribed to an increase of the relative contact density, as results both from lifetime measurements and from the determination of the three quantum yield. Analyses of positron annihilation lifetime spectra are sometimes carried out by fixing the p-Ps lifetime to the value in vacuo (125 ps). We point out the risk associated to this procedure, since this value can be rather different from the

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real value found in the material under study. Such a constraint introduces a bias in the spectrum analysis with an uncontrolled influence on the other components.

ACKNOWLEDGMENTS

The authors are grateful to Mrs. Maria Rosaria Pagano and to Elena Macerata, PhD., (Politecnico di Milano) for their technical assistance as well as to Mr. Tito Zanetta (Vinavil S.p.A.) for supplying the copolymer. 1 Y.

C. Jean, Positron Spectroscopy of Solids, edited by A. Dupasquier and A. P. Mills, Jr. (IOS Press, Amsterdam, 1995), pp. 563–580. 2 G. Dlubek, Polymer Physics: From Suspensions to Nanocomposites and Beyond, edited by L. Utracki and A. Jamieson (John Wiley and Sons, Inc., New York, 2010), pp. 421–472. 3 W. Schmidt, Handbook of Porous Solids, edited by F. Schüth, K. S. W. Sing, and J. Weitkamp (Wiley-VCH, Weinheim, 2002), pp. 506–532. 4 A. Dupasquier, Positron Solid-State Physics, edited by W. Brandt and A. Dupasquier (North-Holland, Amsterdam, 1983), p. 510. 5 S. J. Tao, J. Chem. Phys. 56, 5499 (1972). 6 M. Eldrup, D. Lightbody, and N. J. Sherwood, Chem. Phys. 63, 51 (1981). 7 B. Jasinska, A. E. Koziol, and T. Goworek, J. Radioanal. Nucl. Chem. 210, 617 (1996). 8 G. Consolati, J. Chem. Phys. 117, 7279 (2002). 9 V. I. Goldanskii, At. Energy Rev. 6, 3 (1968). 10 S. Berko and H. N. Pendleton, Annu. Rev. Nucl. Part. Sci. 30, 543 (1980). 11 A. Dupasquier, P. De Natale, and A. Rolando, Phys. Rev. B 43, 10036 (1991). 12 G. Consolati, J. Radioanal. Nucl. Chem. 210, 273 (1996). 13 T. McMullen and M. T. Stott, Can. J. Phys. 61, 504 (1983). 14 W. Brandt, S. Berko, and W. W. Walker, Phys. Rev. 120, 1289 (1960). 15 O. Halpern, Phys. Rev. 94, 904 (1954). 16 G. Consolati and F. Quasso, J. Phys.: Condens. Matter 2, 3941 (1990). 17 G. Consolati and F. Quasso, Appl. Phys. A 52, 295 (1991). 18 S. Mashimo, R. Nozaki, S. Yagihara, and S. Takeishi, J. Chem. Phys. 77, 6259 (1982). 19 Handbook of Chemistry and Physics, edited by David R. Lide, 85th ed. (CRC Press, Boca Raton, FL, 2004), Chap. 9, p. 45. 20 J. Kansy, Nucl. Instrum. Methods Phys. Res. A 374, 235 (1996). 21 G. Consolati, Rev. Sci. Instrum. 69, 3155 (1998). 22 O. E. Mogensen, Positron Annihilation in Chemistry, Springer Series in Chemical Physics Vol. 58 (Springer-Verlag, Berlin, 1995), p. 176. 23 Y. C. Jean and Q. Deng, J. Polym. Sci. B 30, 1359 (1992). 24 A. A. Bondi, Physical Properties of Molecular Crystals, Liquids and Glasses (John Wiley and Sons, New York, 1961), p. 255. 25 P. Bandzuch, J. Kristiak, O. Sausa, and J. Zrubcova, Phys. Rev. B 61, 8784 (2000). 26 G. Consolati, M. Levi, L. Messa, and G. Tieghi, Europhys. Lett. 53, 497 (2001). 27 A. Ore and J. L. Powell, Phys. Rev. 75, 1696 (1949). 28 G. Consolati and F. Quasso, J. Phys. C 21, 4143 (1988).

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Polarization of positronium in amorphous polar polymers: a case study.

The features of positronium in an amorphous copolymer (polyvinyl acetate-crotonic acid) in a range of temperatures including the glass transition were...
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