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Positronium formation in low-density polyethylene (LDPE)
Physics Letters A 313 (2003) 223–230 www.elsevier.com/locate/pla
Positronium formation in low-density polyethylene (LDPE) Chunqing He a,∗ , Takenori Suzuki a , V.P. Shantarovich b , Lin Ma c , Masaru Matsuo c , Kenjiro Kondo a , Yasuo Ito d a Radiation Science Center, High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan b Semenov Institute of Chemical Physics, Russian Academy of Sciences, 4 Kosygin str., Moscow 117334, Russia c Department of Textile and Apparel Science, Faculty of Human Life and Environment, Nara Women’s University, Nara 630-5806, Japan d RCNT, University of Tokyo, JAERI, Tokai-Mura, Ibaraki 319-1106, Japan
Received 30 March 2003; received in revised form 6 May 2003; accepted 7 May 2003 Communicated by R. Wu
Abstract Positron annihilation lifetime spectroscopy has been applied to low-density polyethylene (LDPE) as a function of the temperature and also the elapsed time in the dark as well as in the light. Positronium formation below, around and above the glass transition temperature of LDPE was investigated. It was found that far below the glass transition temperature, Tg , the o-Ps intensity increased with the elapsed time in the dark. It was interesting to observe an initial rapid increase in the o-Ps intensity, followed by a continuous decrease with the elapsed time, especially at around Tg . Above Tg , the o-Ps intensity decreased very slowly with the elapsed time at 250 K and almost did not decrease at room temperature. Furthermore, no increase, but only a decrease, in the o-Ps intensity with the elapsed time was seen in light. Possible explanations of these effects were discussed. A simple calculation of the activation energy of the trapped electrons was suggested. 2003 Elsevier Science B.V. All rights reserved. PACS: 78.70.Bj; 71.60.+z; 61.80.Ba; 61.80.Fe; 65.90.+i Keywords: Positronium formation; Trapped electron; Free radical; Irradiation
1. Introduction After injection into the condensed matter, positrons lose their kinetic energy within a few picoseconds. Positrons diffuse in the media, and finally annihilate * Corresponding author. Present address: Research Reactor In-
stitute, Kyoto University, Kumatori-cho, Sennan-gun, Osaka 5900494, Japan. E-mail address: [email protected] (C. He).
with electrons of the materials, emitting two γ -rays. In polymers, a positron might annihilate from a positronium (Ps) state, a bound state of a positron and an electron. A positronium has two spin states: a singlet state (para-Ps, p-Ps) and a triplet state (ortho-Ps, o-Ps). The ratio of the formation probability of p-Ps to o-Ps is 1:3. The intrinsic lifetime of p-Ps is 0.125 ns, while that of o-Ps is 142 ns in a vacuum. However, the lifetime of o-Ps in polymers might be shortened to a few nanoseconds because Ps can be trapped in molecular
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packing defects, e.g., free-volumes, and the positron of o-Ps can annihilate with an electron from the inner wall of the free volumes, which is called “pick-off annihilation”. Positron annihilation lifetime spectroscopy has been accepted to be a novel probe of micro-voids in polymers and porous materials [1,2]. The positron annihilation lifetime spectrum in polymers usually contains three exponentially decaying components. The longest-lived component and the corresponding intensity are attributed to pick-off annihilation of o-Ps localized in a nano-scale hole. The effective radius of relatively small hole (R < 1 nm) can be estimated from the pick-off annihilation rate of o-Ps according to a semi-empirical quantum-mechanical model developed by Tao [3], Eldrup et al. [4] and Nakanishi et al. [5]. Sizes of the large holes can be calculated from an extension model of Shantarovich [6], Goworek [7,8] and Gidley [9,10], or another calibrated equation [11]. Although the o-Ps lifetime has been successfully correlated to the size of the free volume present in polymers, its intensity has been found to be influenced by many factors, such as the temperature [12,13], positron irradiation [12–16], electric field [17] and polar group [17–19]. It is of great importance to well understand positronium formation in polymers, especially in low temperature polymers. There are several basic models of Ps formation. One of them is the epithermal model (or the Ore model), which assumes a simple electron abstraction reaction by a “hot” positron e+ ∗ as e+ ∗ + M → Ps + M+ .
(1)
In terms of the Ore model, positrons with energy E, that fulfill the following energy condition can form a positronium: EI − Eb E EI ,
(2)
where EI is the ionization energy of the media and Eb is the binding energy of Ps. This model was used to explain positronium formation in the noble gases. Positronium formation in condensed media was suggested to take place through combination of the thermalized positron with one of the track electrons produced by e+ ionization of the medium in the terminal part of the e+ [20,21]. Later on, the Onsager-type formulation of the recombination mechanism (the spur model [21]) was widely accepted
to describe the Ps formation in condensed matters. However, the positron terminal spur is a blob with a radius of some tens of angstroms and a small number of ion–electron pairs [22]. The blob size is about 30 to 40 Å [23]. An alternative approach (diffusion recombination model or blob model [23–25]) was developed for consideration of the electric field effect on Ps formation and the nature of the terminal positron spur was taken into account. The formation of a Ps was suggested to take place prior to its localization in a free volume [6,26]. In the present Letter, positronium formation in LDPE was extensively studied versus the temperature, elapsed time and light on/off. The results enable one to extensively understand the temperature dependence of positron formation in some polymers.
2. Experimental Positron annihilation in low-density polyethylene (LDPE G201, crystallinity 55.3%, Sumitomo Chemical Co. Ltd.) has been studied using “fast–fast” coincidence positron annihilation lifetime spectroscopy (PALS). The positron source, 22 Na, was prepared by depositing about 3.7 MBq of 22 NaCl on a 7 µm thick Kapton foil of 1 cm2 , being covered after drying by another foil of the same size. The diameter of the deposit was 1 to 1.5 mm. The sample-source in a sandwichgeometry was fixed on the cooling finger of a helium cryostat (CW303, Iwatani Plantech Co.). Samples could be irradiated by visible light through a glass window on the sample chamber. The temperatures could be controlled in the range of 30 to 370 K by a temperature control device (TCU-4, Iwatani Plantech Co.). For experiments conducted at fixed temperatures as a function of the elapsed time, the sample for each measurement was kept at room temperature for a few hours prior to being cooled down to the chosen temperature; for the temperature dependence experiment, the sample was cooled down to 30 K from RT in about 1 hour and then warmed up at a rate of about 5 K per hour. Each positron annihilation lifetime spectrum was collected within one hour. In some of previous studies, the lifetime spectra in semi-crystalline polymers were resolved into four components [27,28]. However, the integral statistic in our experiments was not enough to resolve the fourth and the third components in LDPE,
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especially at low temperature. Our purpose was to investigate the total o-Ps formation. Therefore, all of the lifetime spectra were resolved into three components using the PATFIT [29] program.
3. Results and discussion 3.1. Positronium formation in LDPE in the dark We measured the positronium formation in LDPE in the dark at different temperatures. Before each measurement the sample was heated up to room temperature for a few hours for “annealing” the trapped electrons in the samples. The results of the o-Ps lifetimes and their intensities for LDPE in the dark at different temperatures with the elapsed time are shown in shown Fig. 1. The long-lived Ps lifetime maintained a constant at each temperature. At 30 K, the o-Ps intensity was found to increase significantly to a high value of 42% in about 12 hours and saturate; at higher temperatures of 150 K, 200 K and 225 K, it took a few hours to reach a maximum values of ca. 39.5%, 37% and 33%, respectively. The most important reactions involved in the terminal spur are shown as follows: e+ + M → M + + e− + e+ ,
creation of ions,
(3)
e + M → R˙ + e + R , +
˙
+
creation of free radicals, +
∗
(4)
+
e +M→M +e , creation of excited states, +
e + e− spur + ∗
→ Ps,
Ps formation,
e + M → Ps + M+ , −
+
∗
Ps formation,
(5) (6) (7)
˙
e + M → M → R˙ + R , recombination and creation of free radicals, −
(8)
−
e + R˙ → R , electron capture by free radicals.
(9)
Once formed, a Ps diffuses and tends to localize in cavities in the polymers. However, in a low temperature polymer matrix, a large number of secondary electrons due to positron irradiation are trapped by some shallow trapping sites, for instance, free volumes or
Fig. 1. Intensities of o-Ps in dark varying as a function of the elapsed time at various temperatures below Tg . The solid symbols are for the o-Ps intensities and the open symbols are for the corresponding o-Ps lifetimes.
folds in the polymer chains. Besides, the positron mobility in PE is rather high [30,31], and estimations of the positron diffusion length in LDPE were found to be very large at low temperature, e.g., ca. 79 nm at 77 K. Therefore, positrons that diffuse out of the spur attain chances to encounter previously trapped electrons ( e− tr ) and form positroniums through e+ + e− tr → Ps,
Ps formation.
(10)
The o-Ps intensity (I3 ) at low temperature in the dark depends on the irradiation effects of the positron source, itself, especially on accumulation of the weakly bound electrons, i.e., trapped electrons [32,33]. Two processes including the electrons in the terminal spur and the trapped electrons outside of the terminal spur, which are expressed in Eqs. (6) and (10), are responsible for Ps formation in low-temperature LDPE in the dark. Therefore, the o-Ps intensity (I3 ) is a function of the Ps formation probability, the number of Ps trapping sites, the concentration of the trapped electrons. Furthermore, it could be seen that the saturation values for the o-Ps intensity decrease with increasing the temperature. This decrease possibly resulted from the thermal decay of trapped electrons and/or some kind of molecular motion involving the surrounding environment of the trapped electrons, which caused the electrons to be mobile. From this point of view, it is possible to estimate the thermal activation energy of the trapped electrons, which will be discussed in detail in Section 3.3.
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Fig. 2. Intensities of o-Ps in dark varying as a function of the elapsed time at various temperatures near Tg . The solid symbols are for the o-Ps intensities and the open symbols are for the corresponding o-Ps lifetimes.
Fig. 3. Intensities of o-Ps in dark varying as a function of the elapsed time at various temperatures above Tg . The solid symbols are for the o-Ps intensities and the open symbols are for the corresponding o-Ps lifetimes.
It is of interest to note that a rapid increase occurred in the o-Ps intensity for low-temperature LDPE, followed by saturation. Then, a much slower decrease occurred, which appeared to become faster at a higher temperature near the glass transition temperature of LDPE (245 K). This was very distinct at 235 K, as demonstrated in Fig. 2. Discussing the reason of this effect, we must mention that some effects of irradiation, such as cross-linking [34] and intra-electric fields [35], may cause a decrease in Ps formation. However, a comparison between the values of the o-Ps intensities at room temperature for the beginning and for the end of the experiment shows that these effects are probably negligible here. The presumable reason is due to an accumulation of free radicals, since the concentration of trapped electrons in PE at 77 K was found to rise to a maximum value at a radiation dose of 3 × 1019 eV g−1 , which is about 4.8 kGy, and thereafter decrease with a further increase in dose [36]. The dose rate for the experiments in PE was calculated to be about 0.4 kGy/h for our positron source. Detailed calculation can be found elsewhere [37]. Then, after 12 hours, the total dose was estimated to be 4.8 kGy. Though the absorbed dose in PE should be smaller, the result seems to suggest that positron source irradiation for about 12 hours gives essential concentration of the trapped electrons. By irradiating the polymer matrix, many types of free radicals are created and some of them decay slowly at low temperature. In PE, alkyl radicals are created in the temperature range below
243 K, and above this temperature these radicals convert into the allyl radical by encountering double bonds [38,39]. As a result, except for the reaction between the electrons in the terminal spur and the free radicals − ˙ e− spur + R → R ,
(11)
the accumulated stable free radicals participate in a competition of positronium formation through [40] − ˙ e− tr + R → R .
(12)
The above reaction is expected to be exothermic by about 1 eV and thermodynamically feasible [41]. Obviously, free radicals have a high mobility at a higher temperature and, correspondingly, a higher probability to encounter the secondary electrons (including electrons in the terminal spur, e− spur , and trapped electrons, − etr ). Thus, it seems reasonable to see that below Tg the speed of the decline of the o-Ps intensity as function of the elapsed time depends on the temperature, and this decline is stronger at higher temperature. The above discussion could well explain the minimum values for the o-Ps intensity (in Fig. 5) during a PALS experiment as a function of the temperature. As shown in Fig. 3, with further sample heating above Tg at 250 K, the o-Ps intensity was observed to gradually decrease from the beginning of the measurement, and at 295 K it almost maintains a constant value in a not too long time. The increase in the o-Ps intensity with the elapsed time was not observed any more, since above the glass transition temperature no
C. He et al. / Physics Letters A 313 (2003) 223–230
trapped electrons can form due to the high mobility of polymer molecular chains. The decrease in the oPs intensity with the exposure time above Tg can also be interpreted as being due to the accumulation of species (ions and free radicals) that inhibit Ps formation through trapping the precursors (e+ and/or e− ). Furthermore, the electron scavenging of free radicals becomes weaker at higher temperatures due to the recombination of free radicals, since the mobility of radicals is much higher above Tg . It is possible that some structural change, such as crosslinking caused by free radicals [34], could result in a decrease of o-Ps formation.
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be seen that o-Ps intensity decrease from about 20% to 15% in about 25 hours; at a higher temperature of 230 K, it took about 50 hours for I3 to attain that value. With further increasing the temperature to RT, the o-Ps intensity almost did not decrease with the elapsed time. Actually, it decreased very slowly at RT over a long period of PALS measurement (unpublished result). The results confirmed that the free radicals play an important role for o-Ps formation in LDPE. The same explanation seems to be valid for the results published in [16]. 3.3. Temperature dependence of the o-Ps intensity and activation energy of trapped electrons
3.2. Positronium formation in LDPE in visible light The o-Ps intensity in LDPE only decreases with the elapsed time under the visible-light conditions, as shown in Fig. 4. This is because the radiation effects are dominated by radical formation without the accumulation of trapped electrons. In visible light, we noted that an initial rapid decrease in the o-Ps intensity with the elapsed time is followed by a much slower decline. The decrease in the o-Ps intensity is due to scavenging of the electrons in the spur by free radicals (see reaction (11)). The mobility of free radicals is of the great important for the described above process, since a higher mobility of free radicals at higher temperatures can result in a decrease in their concentration through some chemical reactions, such as recombination and crosslinking. At 200 K, it can
Fig. 4. Intensities of o-Ps in visible light varying as a function of the elapsed time. The solid symbols are for the o-Ps intensities and the open symbols are for the corresponding o-Ps lifetimes.
The temperature dependences of the o-Ps lifetime and intensity in LDPE are shown in Fig. 5. The scattered solid points were obtained from the maximum values of the described above experiments conducted in the dark at different temperatures. The results agree well. The temperature dependence of the o-Ps lifetime is widely accepted to relate with the structure transition in polymers. However, the character of temperature dependence of the o-Ps intensity below Tg was also attributed to a change in the physical properties of polymers at low temperatures [27,28]. As we know now, the o-Ps intensity depends on the density of trapped electrons accumulated in the dark [32]. Ac-
Fig. 5. Temperature dependences of the o-Ps lifetime and its intensity in LDPE in dark and in light. The solid triangle and circles are for the o-Ps lifetimes and the maximum values of the o-Ps intensities obtained from the measurements in dark at chosen temperatures; the open triangle and circles are for the o-Ps lifetimes and their intensities obtained from the measurements in dark with increasing the temperature. The open squares are the o-Ps intensities measured in visible light with increasing the temperature.
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cordingly, the variation of the o-Ps intensity vs temperature has been associated with the molecular motion and structural transitions when the PALS experiment was conducted in the dark after pre-irradiation [16,42]. As shown in Fig. 5, the apparent transition temperatures for LDPE were estimated to be 143 K and 245 K for the γ - and β-transitions, respectively. Small deviation (a few K) of the above mentioned temperatures from those given in [16] are probably due to some systematic errors of our measurements. The concentration of the trapped electrons was determined by the production due to the source irradiation and their detrapping. The number of trapped electrons, Ne , can be described by the following kinetic equation: N˙ e = J − Ne /τd ,
Ne (0) = 0,
e− tr
(13) νd−1 eE/ kB T
production and τd = where J is the is the their detrapping time (νd is typical oscillation frequency, E is the activation energy of detrapping and kB is the Boltzmann constant). The solution of the above equation is Ne (t) = J τd 1 − exp(−t/τd ) . (14) It is reasonable to assume that the additional positronium formation (I3 ) due to the accumulation of trapped electrons is proportional to the concentration of trapped electrons I3 = I3D (T , t) − I3L (T , t) = CNe = CJ νd−1 eE/ kB T 1 − exp(−t/τd ) ,
(15)
or ln I3 = C +
E + ln 1 − exp(−t/τd ) , kB T
(16)
where I3D and I3L are the o-Ps intensity in dark and in the visible light at temperature T at a given measurement time, respectively; C and C are constants. The measurement time t is much longer than τd , Eq. (16) can be rewritten as E . ln I3 = C + (17) kB T The above equation enables us to estimate the activation energy of trapped positrons in LDPE from the temperature dependence of the additional positronium formation. In the initial stage of increasing the temperature in the range from 30 K to about 138 K, the o-Ps
Fig. 6. Relation between I3 (T ) and 1/T .
intensity increases during experiment and finally saturates. The trapped electrons are strongly frozen in the structure of LDPE at such low temperatures. Between 143 K (γ -transition) and 245 K (β-transition, Tg ), the o-Ps intensity decreases gradually, since the trapped electrons are released due to some kind of molecular motion involving the surrounding environment of the trapped electrons. Meanwhile, the free radicals and the stored electrons attain a higher probability and are able to recombine. The process competes with the Ps formation. It seems that the rapid decrease in the o-Ps intensity between about 200 K and 245 K is mainly due to the electron scavenging by the free radicals, since the main polymer chains start to move above 200 K [16]. Above the glass transition temperature, the o-Ps intensities for LDPE in the dark and in the visible light are almost same and the continuous increase in them with elevating the temperature might be explained as being due to the thermal expansion of the polymer matrix. The temperature dependence of the additional o-Ps intensity (I3 ) is plotted in another scale in Fig. 6. The experimental data for the measurement with increasing the temperature and that at chosen temperature were fitted according to Eq. (17). Both of the measurements gave the same values for the activation energies of the trapped electrons, which were estimated to be 0.006 eV and 1.17 eV in the γ - and β-transition regions, respectively. The estimated activation energies for the γ -transition are not consistent with some type of rotation about the C–C bond. However, the estimated activation energy (1.17 eV) of trapped electrons
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for the temperature region of β-transition could be comparable with the activation energy for β-transition of the LDPE, which was determined to be 114 kJ/mol [43], i.e., 1.18 eV. Hence, the activation energies for the β-transition of some polymers might be estimated from the positron annihilation experiments and it is possible that the electron detrapping results from some kind of molecular motion involving the surrounding environment of the trapped electrons, which cause the electron to be mobile. Recently, the Ps formation data in presence of an external electric field has been theoretically calculated by using the diffusion recombination model [25], which is very appropriate for consideration of the processes of blobs. Very good agreement between the experimental data and the theoretical calculation was obtained. Thus, it is expected to interpret our Ps formation data within this model when we take into account the trapped electrons. Further experimental and theoretical works on the Ps formation in polymers will be continued in the future.
4. Conclusion Positronium formation has been studied in LDPE at various temperatures as a function of the elapsed time. The experiments were conducted in the dark as well as in the light. In the dark, it was found that the o-Ps formation in low-temperature LDPE was affected by trapped electrons as well as free radicals. However, above Tg it might only be affected by the free radical formation. In the light, o-Ps formation was observed to decrease over a wide temperature range due to an inhibition of Ps formation by the free radicals. The mobility of free radicals naturally plays an important role on the inhibition process. The stability of trapped electrons and the effect of positronium formation were found to correlate with phase transitions in the polymer structure. According to the results, one could estimate the activation energy of detrapping the weakly bound electrons and the activation energies for the β-transition of some polymers. The obtained results indicated that PALS data are useful in studying polymers. However, theoretical interpretation of the Ps formation data in low temperature LDPE can be done when taking account of the trapped electrons
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within the diffusion recombination model, which will be continued.
Acknowledgements One of the authors (C.Q. He) would like to thank Prof. T. Shibata, the head of Radiation Science Center, for his support to conduct research work at KEK. The work was supported by a Grant-in-Aid of the Japanese Ministry of Education, Culture, Sports, Science and Technology.
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Positronium formation in low-density polyethylene (LDPE) Chunqing He a,∗ , Takenori Suzuki a , V.P. Shantarovich b , Lin Ma c , Masaru Matsuo c , Kenjiro Kondo a , Yasuo Ito d a Radiation Science Center, High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan b Semenov Institute of Chemical Physics, Russian Academy of Sciences, 4 Kosygin str., Moscow 117334, Russia c Department of Textile and Apparel Science, Faculty of Human Life and Environment, Nara Women’s University, Nara 630-5806, Japan d RCNT, University of Tokyo, JAERI, Tokai-Mura, Ibaraki 319-1106, Japan
Received 30 March 2003; received in revised form 6 May 2003; accepted 7 May 2003 Communicated by R. Wu
Abstract Positron annihilation lifetime spectroscopy has been applied to low-density polyethylene (LDPE) as a function of the temperature and also the elapsed time in the dark as well as in the light. Positronium formation below, around and above the glass transition temperature of LDPE was investigated. It was found that far below the glass transition temperature, Tg , the o-Ps intensity increased with the elapsed time in the dark. It was interesting to observe an initial rapid increase in the o-Ps intensity, followed by a continuous decrease with the elapsed time, especially at around Tg . Above Tg , the o-Ps intensity decreased very slowly with the elapsed time at 250 K and almost did not decrease at room temperature. Furthermore, no increase, but only a decrease, in the o-Ps intensity with the elapsed time was seen in light. Possible explanations of these effects were discussed. A simple calculation of the activation energy of the trapped electrons was suggested. 2003 Elsevier Science B.V. All rights reserved. PACS: 78.70.Bj; 71.60.+z; 61.80.Ba; 61.80.Fe; 65.90.+i Keywords: Positronium formation; Trapped electron; Free radical; Irradiation
1. Introduction After injection into the condensed matter, positrons lose their kinetic energy within a few picoseconds. Positrons diffuse in the media, and finally annihilate * Corresponding author. Present address: Research Reactor In-
stitute, Kyoto University, Kumatori-cho, Sennan-gun, Osaka 5900494, Japan. E-mail address: [email protected] (C. He).
with electrons of the materials, emitting two γ -rays. In polymers, a positron might annihilate from a positronium (Ps) state, a bound state of a positron and an electron. A positronium has two spin states: a singlet state (para-Ps, p-Ps) and a triplet state (ortho-Ps, o-Ps). The ratio of the formation probability of p-Ps to o-Ps is 1:3. The intrinsic lifetime of p-Ps is 0.125 ns, while that of o-Ps is 142 ns in a vacuum. However, the lifetime of o-Ps in polymers might be shortened to a few nanoseconds because Ps can be trapped in molecular
0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00752-7
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packing defects, e.g., free-volumes, and the positron of o-Ps can annihilate with an electron from the inner wall of the free volumes, which is called “pick-off annihilation”. Positron annihilation lifetime spectroscopy has been accepted to be a novel probe of micro-voids in polymers and porous materials [1,2]. The positron annihilation lifetime spectrum in polymers usually contains three exponentially decaying components. The longest-lived component and the corresponding intensity are attributed to pick-off annihilation of o-Ps localized in a nano-scale hole. The effective radius of relatively small hole (R < 1 nm) can be estimated from the pick-off annihilation rate of o-Ps according to a semi-empirical quantum-mechanical model developed by Tao [3], Eldrup et al. [4] and Nakanishi et al. [5]. Sizes of the large holes can be calculated from an extension model of Shantarovich [6], Goworek [7,8] and Gidley [9,10], or another calibrated equation [11]. Although the o-Ps lifetime has been successfully correlated to the size of the free volume present in polymers, its intensity has been found to be influenced by many factors, such as the temperature [12,13], positron irradiation [12–16], electric field [17] and polar group [17–19]. It is of great importance to well understand positronium formation in polymers, especially in low temperature polymers. There are several basic models of Ps formation. One of them is the epithermal model (or the Ore model), which assumes a simple electron abstraction reaction by a “hot” positron e+ ∗ as e+ ∗ + M → Ps + M+ .
(1)
In terms of the Ore model, positrons with energy E, that fulfill the following energy condition can form a positronium: EI − Eb E EI ,
(2)
where EI is the ionization energy of the media and Eb is the binding energy of Ps. This model was used to explain positronium formation in the noble gases. Positronium formation in condensed media was suggested to take place through combination of the thermalized positron with one of the track electrons produced by e+ ionization of the medium in the terminal part of the e+ [20,21]. Later on, the Onsager-type formulation of the recombination mechanism (the spur model [21]) was widely accepted
to describe the Ps formation in condensed matters. However, the positron terminal spur is a blob with a radius of some tens of angstroms and a small number of ion–electron pairs [22]. The blob size is about 30 to 40 Å [23]. An alternative approach (diffusion recombination model or blob model [23–25]) was developed for consideration of the electric field effect on Ps formation and the nature of the terminal positron spur was taken into account. The formation of a Ps was suggested to take place prior to its localization in a free volume [6,26]. In the present Letter, positronium formation in LDPE was extensively studied versus the temperature, elapsed time and light on/off. The results enable one to extensively understand the temperature dependence of positron formation in some polymers.
2. Experimental Positron annihilation in low-density polyethylene (LDPE G201, crystallinity 55.3%, Sumitomo Chemical Co. Ltd.) has been studied using “fast–fast” coincidence positron annihilation lifetime spectroscopy (PALS). The positron source, 22 Na, was prepared by depositing about 3.7 MBq of 22 NaCl on a 7 µm thick Kapton foil of 1 cm2 , being covered after drying by another foil of the same size. The diameter of the deposit was 1 to 1.5 mm. The sample-source in a sandwichgeometry was fixed on the cooling finger of a helium cryostat (CW303, Iwatani Plantech Co.). Samples could be irradiated by visible light through a glass window on the sample chamber. The temperatures could be controlled in the range of 30 to 370 K by a temperature control device (TCU-4, Iwatani Plantech Co.). For experiments conducted at fixed temperatures as a function of the elapsed time, the sample for each measurement was kept at room temperature for a few hours prior to being cooled down to the chosen temperature; for the temperature dependence experiment, the sample was cooled down to 30 K from RT in about 1 hour and then warmed up at a rate of about 5 K per hour. Each positron annihilation lifetime spectrum was collected within one hour. In some of previous studies, the lifetime spectra in semi-crystalline polymers were resolved into four components [27,28]. However, the integral statistic in our experiments was not enough to resolve the fourth and the third components in LDPE,
C. He et al. / Physics Letters A 313 (2003) 223–230
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especially at low temperature. Our purpose was to investigate the total o-Ps formation. Therefore, all of the lifetime spectra were resolved into three components using the PATFIT [29] program.
3. Results and discussion 3.1. Positronium formation in LDPE in the dark We measured the positronium formation in LDPE in the dark at different temperatures. Before each measurement the sample was heated up to room temperature for a few hours for “annealing” the trapped electrons in the samples. The results of the o-Ps lifetimes and their intensities for LDPE in the dark at different temperatures with the elapsed time are shown in shown Fig. 1. The long-lived Ps lifetime maintained a constant at each temperature. At 30 K, the o-Ps intensity was found to increase significantly to a high value of 42% in about 12 hours and saturate; at higher temperatures of 150 K, 200 K and 225 K, it took a few hours to reach a maximum values of ca. 39.5%, 37% and 33%, respectively. The most important reactions involved in the terminal spur are shown as follows: e+ + M → M + + e− + e+ ,
creation of ions,
(3)
e + M → R˙ + e + R , +
˙
+
creation of free radicals, +
∗
(4)
+
e +M→M +e , creation of excited states, +
e + e− spur + ∗
→ Ps,
Ps formation,
e + M → Ps + M+ , −
+
∗
Ps formation,
(5) (6) (7)
˙
e + M → M → R˙ + R , recombination and creation of free radicals, −
(8)
−
e + R˙ → R , electron capture by free radicals.
(9)
Once formed, a Ps diffuses and tends to localize in cavities in the polymers. However, in a low temperature polymer matrix, a large number of secondary electrons due to positron irradiation are trapped by some shallow trapping sites, for instance, free volumes or
Fig. 1. Intensities of o-Ps in dark varying as a function of the elapsed time at various temperatures below Tg . The solid symbols are for the o-Ps intensities and the open symbols are for the corresponding o-Ps lifetimes.
folds in the polymer chains. Besides, the positron mobility in PE is rather high [30,31], and estimations of the positron diffusion length in LDPE were found to be very large at low temperature, e.g., ca. 79 nm at 77 K. Therefore, positrons that diffuse out of the spur attain chances to encounter previously trapped electrons ( e− tr ) and form positroniums through e+ + e− tr → Ps,
Ps formation.
(10)
The o-Ps intensity (I3 ) at low temperature in the dark depends on the irradiation effects of the positron source, itself, especially on accumulation of the weakly bound electrons, i.e., trapped electrons [32,33]. Two processes including the electrons in the terminal spur and the trapped electrons outside of the terminal spur, which are expressed in Eqs. (6) and (10), are responsible for Ps formation in low-temperature LDPE in the dark. Therefore, the o-Ps intensity (I3 ) is a function of the Ps formation probability, the number of Ps trapping sites, the concentration of the trapped electrons. Furthermore, it could be seen that the saturation values for the o-Ps intensity decrease with increasing the temperature. This decrease possibly resulted from the thermal decay of trapped electrons and/or some kind of molecular motion involving the surrounding environment of the trapped electrons, which caused the electrons to be mobile. From this point of view, it is possible to estimate the thermal activation energy of the trapped electrons, which will be discussed in detail in Section 3.3.
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Fig. 2. Intensities of o-Ps in dark varying as a function of the elapsed time at various temperatures near Tg . The solid symbols are for the o-Ps intensities and the open symbols are for the corresponding o-Ps lifetimes.
Fig. 3. Intensities of o-Ps in dark varying as a function of the elapsed time at various temperatures above Tg . The solid symbols are for the o-Ps intensities and the open symbols are for the corresponding o-Ps lifetimes.
It is of interest to note that a rapid increase occurred in the o-Ps intensity for low-temperature LDPE, followed by saturation. Then, a much slower decrease occurred, which appeared to become faster at a higher temperature near the glass transition temperature of LDPE (245 K). This was very distinct at 235 K, as demonstrated in Fig. 2. Discussing the reason of this effect, we must mention that some effects of irradiation, such as cross-linking [34] and intra-electric fields [35], may cause a decrease in Ps formation. However, a comparison between the values of the o-Ps intensities at room temperature for the beginning and for the end of the experiment shows that these effects are probably negligible here. The presumable reason is due to an accumulation of free radicals, since the concentration of trapped electrons in PE at 77 K was found to rise to a maximum value at a radiation dose of 3 × 1019 eV g−1 , which is about 4.8 kGy, and thereafter decrease with a further increase in dose [36]. The dose rate for the experiments in PE was calculated to be about 0.4 kGy/h for our positron source. Detailed calculation can be found elsewhere [37]. Then, after 12 hours, the total dose was estimated to be 4.8 kGy. Though the absorbed dose in PE should be smaller, the result seems to suggest that positron source irradiation for about 12 hours gives essential concentration of the trapped electrons. By irradiating the polymer matrix, many types of free radicals are created and some of them decay slowly at low temperature. In PE, alkyl radicals are created in the temperature range below
243 K, and above this temperature these radicals convert into the allyl radical by encountering double bonds [38,39]. As a result, except for the reaction between the electrons in the terminal spur and the free radicals − ˙ e− spur + R → R ,
(11)
the accumulated stable free radicals participate in a competition of positronium formation through [40] − ˙ e− tr + R → R .
(12)
The above reaction is expected to be exothermic by about 1 eV and thermodynamically feasible [41]. Obviously, free radicals have a high mobility at a higher temperature and, correspondingly, a higher probability to encounter the secondary electrons (including electrons in the terminal spur, e− spur , and trapped electrons, − etr ). Thus, it seems reasonable to see that below Tg the speed of the decline of the o-Ps intensity as function of the elapsed time depends on the temperature, and this decline is stronger at higher temperature. The above discussion could well explain the minimum values for the o-Ps intensity (in Fig. 5) during a PALS experiment as a function of the temperature. As shown in Fig. 3, with further sample heating above Tg at 250 K, the o-Ps intensity was observed to gradually decrease from the beginning of the measurement, and at 295 K it almost maintains a constant value in a not too long time. The increase in the o-Ps intensity with the elapsed time was not observed any more, since above the glass transition temperature no
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trapped electrons can form due to the high mobility of polymer molecular chains. The decrease in the oPs intensity with the exposure time above Tg can also be interpreted as being due to the accumulation of species (ions and free radicals) that inhibit Ps formation through trapping the precursors (e+ and/or e− ). Furthermore, the electron scavenging of free radicals becomes weaker at higher temperatures due to the recombination of free radicals, since the mobility of radicals is much higher above Tg . It is possible that some structural change, such as crosslinking caused by free radicals [34], could result in a decrease of o-Ps formation.
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be seen that o-Ps intensity decrease from about 20% to 15% in about 25 hours; at a higher temperature of 230 K, it took about 50 hours for I3 to attain that value. With further increasing the temperature to RT, the o-Ps intensity almost did not decrease with the elapsed time. Actually, it decreased very slowly at RT over a long period of PALS measurement (unpublished result). The results confirmed that the free radicals play an important role for o-Ps formation in LDPE. The same explanation seems to be valid for the results published in [16]. 3.3. Temperature dependence of the o-Ps intensity and activation energy of trapped electrons
3.2. Positronium formation in LDPE in visible light The o-Ps intensity in LDPE only decreases with the elapsed time under the visible-light conditions, as shown in Fig. 4. This is because the radiation effects are dominated by radical formation without the accumulation of trapped electrons. In visible light, we noted that an initial rapid decrease in the o-Ps intensity with the elapsed time is followed by a much slower decline. The decrease in the o-Ps intensity is due to scavenging of the electrons in the spur by free radicals (see reaction (11)). The mobility of free radicals is of the great important for the described above process, since a higher mobility of free radicals at higher temperatures can result in a decrease in their concentration through some chemical reactions, such as recombination and crosslinking. At 200 K, it can
Fig. 4. Intensities of o-Ps in visible light varying as a function of the elapsed time. The solid symbols are for the o-Ps intensities and the open symbols are for the corresponding o-Ps lifetimes.
The temperature dependences of the o-Ps lifetime and intensity in LDPE are shown in Fig. 5. The scattered solid points were obtained from the maximum values of the described above experiments conducted in the dark at different temperatures. The results agree well. The temperature dependence of the o-Ps lifetime is widely accepted to relate with the structure transition in polymers. However, the character of temperature dependence of the o-Ps intensity below Tg was also attributed to a change in the physical properties of polymers at low temperatures [27,28]. As we know now, the o-Ps intensity depends on the density of trapped electrons accumulated in the dark [32]. Ac-
Fig. 5. Temperature dependences of the o-Ps lifetime and its intensity in LDPE in dark and in light. The solid triangle and circles are for the o-Ps lifetimes and the maximum values of the o-Ps intensities obtained from the measurements in dark at chosen temperatures; the open triangle and circles are for the o-Ps lifetimes and their intensities obtained from the measurements in dark with increasing the temperature. The open squares are the o-Ps intensities measured in visible light with increasing the temperature.
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cordingly, the variation of the o-Ps intensity vs temperature has been associated with the molecular motion and structural transitions when the PALS experiment was conducted in the dark after pre-irradiation [16,42]. As shown in Fig. 5, the apparent transition temperatures for LDPE were estimated to be 143 K and 245 K for the γ - and β-transitions, respectively. Small deviation (a few K) of the above mentioned temperatures from those given in [16] are probably due to some systematic errors of our measurements. The concentration of the trapped electrons was determined by the production due to the source irradiation and their detrapping. The number of trapped electrons, Ne , can be described by the following kinetic equation: N˙ e = J − Ne /τd ,
Ne (0) = 0,
e− tr
(13) νd−1 eE/ kB T
production and τd = where J is the is the their detrapping time (νd is typical oscillation frequency, E is the activation energy of detrapping and kB is the Boltzmann constant). The solution of the above equation is Ne (t) = J τd 1 − exp(−t/τd ) . (14) It is reasonable to assume that the additional positronium formation (I3 ) due to the accumulation of trapped electrons is proportional to the concentration of trapped electrons I3 = I3D (T , t) − I3L (T , t) = CNe = CJ νd−1 eE/ kB T 1 − exp(−t/τd ) ,
(15)
or ln I3 = C +
E + ln 1 − exp(−t/τd ) , kB T
(16)
where I3D and I3L are the o-Ps intensity in dark and in the visible light at temperature T at a given measurement time, respectively; C and C are constants. The measurement time t is much longer than τd , Eq. (16) can be rewritten as E . ln I3 = C + (17) kB T The above equation enables us to estimate the activation energy of trapped positrons in LDPE from the temperature dependence of the additional positronium formation. In the initial stage of increasing the temperature in the range from 30 K to about 138 K, the o-Ps
Fig. 6. Relation between I3 (T ) and 1/T .
intensity increases during experiment and finally saturates. The trapped electrons are strongly frozen in the structure of LDPE at such low temperatures. Between 143 K (γ -transition) and 245 K (β-transition, Tg ), the o-Ps intensity decreases gradually, since the trapped electrons are released due to some kind of molecular motion involving the surrounding environment of the trapped electrons. Meanwhile, the free radicals and the stored electrons attain a higher probability and are able to recombine. The process competes with the Ps formation. It seems that the rapid decrease in the o-Ps intensity between about 200 K and 245 K is mainly due to the electron scavenging by the free radicals, since the main polymer chains start to move above 200 K [16]. Above the glass transition temperature, the o-Ps intensities for LDPE in the dark and in the visible light are almost same and the continuous increase in them with elevating the temperature might be explained as being due to the thermal expansion of the polymer matrix. The temperature dependence of the additional o-Ps intensity (I3 ) is plotted in another scale in Fig. 6. The experimental data for the measurement with increasing the temperature and that at chosen temperature were fitted according to Eq. (17). Both of the measurements gave the same values for the activation energies of the trapped electrons, which were estimated to be 0.006 eV and 1.17 eV in the γ - and β-transition regions, respectively. The estimated activation energies for the γ -transition are not consistent with some type of rotation about the C–C bond. However, the estimated activation energy (1.17 eV) of trapped electrons
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for the temperature region of β-transition could be comparable with the activation energy for β-transition of the LDPE, which was determined to be 114 kJ/mol [43], i.e., 1.18 eV. Hence, the activation energies for the β-transition of some polymers might be estimated from the positron annihilation experiments and it is possible that the electron detrapping results from some kind of molecular motion involving the surrounding environment of the trapped electrons, which cause the electron to be mobile. Recently, the Ps formation data in presence of an external electric field has been theoretically calculated by using the diffusion recombination model [25], which is very appropriate for consideration of the processes of blobs. Very good agreement between the experimental data and the theoretical calculation was obtained. Thus, it is expected to interpret our Ps formation data within this model when we take into account the trapped electrons. Further experimental and theoretical works on the Ps formation in polymers will be continued in the future.
4. Conclusion Positronium formation has been studied in LDPE at various temperatures as a function of the elapsed time. The experiments were conducted in the dark as well as in the light. In the dark, it was found that the o-Ps formation in low-temperature LDPE was affected by trapped electrons as well as free radicals. However, above Tg it might only be affected by the free radical formation. In the light, o-Ps formation was observed to decrease over a wide temperature range due to an inhibition of Ps formation by the free radicals. The mobility of free radicals naturally plays an important role on the inhibition process. The stability of trapped electrons and the effect of positronium formation were found to correlate with phase transitions in the polymer structure. According to the results, one could estimate the activation energy of detrapping the weakly bound electrons and the activation energies for the β-transition of some polymers. The obtained results indicated that PALS data are useful in studying polymers. However, theoretical interpretation of the Ps formation data in low temperature LDPE can be done when taking account of the trapped electrons
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within the diffusion recombination model, which will be continued.
Acknowledgements One of the authors (C.Q. He) would like to thank Prof. T. Shibata, the head of Radiation Science Center, for his support to conduct research work at KEK. The work was supported by a Grant-in-Aid of the Japanese Ministry of Education, Culture, Sports, Science and Technology.
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