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Temperature, pressure and source-irradiation effects in positronium formation in some solid long-chain alkanes
Chemical Physics 295 (2003) 243–253 www.elsevier.com/locate/chemphys
Temperature, pressure and source-irradiation effects in positronium formation in some solid long-chain alkanes T. Goworek *, R. Zaleski, J. Wawryszczuk Institute of Physics, Maria Curie-Sklodowska University, Pl. Marii Curie-Sklodowskiej 1, Lublin 20-031, Poland Received 23 April 2003; accepted 25 August 2003
Abstract Positron lifetime spectra in solid n-alkanes C19 H40 , C20 H42 and C21 H44 were measured. The effect of irradiation by positron source (increase of ortho-positronium intensity I3 ), known from polymer studies, was found to appear also in alkanes; at about 100 K it can lead to doubling the intensity. The time constant of I3 rise in n-eicosane increased with temperature, reaching the value of 60 h, while it decreased (down to 1 h) in its odd-numbered neighbours. The difference in ortho-Ps lifetimes between odd- and evennumbered alkanes can be explained by different thickness of interlamellar layer in the crystal structure. The variation of I3 after application of high pressure was also observed, but in this case the intensity decreased with time. 2003 Elsevier B.V. All rights reserved.
1. Introduction According to the spur model [1,2], representing the viewpoint of radiation chemistry, the atom of positronium is formed in condensed matter by binding the slowed down positron with one of quasi-free electrons which it produced in a series of preceding ionization acts. The probability of positronium (Ps) formation is related to the number of available electrons near the last ionization site. Positronium exists in two substates, depending on mutual spin orientation of involved particles. The triplet substate with parallel spins (ortho-positronium, o-Ps) is a relatively long-living species, thus easier to observe than its singlet counterpart. Already by the end of the eighties it was noticed that the intensity of o-Ps component in the lifetime spectrum of positrons in polymers was not stable, rising with time, particularly at low temperatures [3,4]. That intensity, I3 , can be assumed as the measure of Ps formation probability; in the absence of Ps transformation processes o-Ps represents 3/4 of the total positronium formed. In 1998, Wang et al. [5] proposed to ascribe I3 variation to the changes of the density of trapped electrons, finally the experiment by Hirade et al. [6,7], consisting in *
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0301-0104/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2003.08.019
bleaching with visible light, demonstrated that the surplus of I3 was induced entirely by the reaction of positrons with the reservoir of trapped electrons deposited in the sample by the positrons injected previously. The processes of trapping the electrons created in polymers during irradiation were studied intensively by the ESR and optical methods [8], nowadays supplemented by positron techniques (see, e.g. [9]). A thorough comparison of the ESR and positron annihilation data is given in the paper by Hirade et al. [10]. The studies of trapped electrons were oriented mainly toward polymers, although there are also some data about saturated hydrocarbon glasses [11]. In polymers the traps were found to be located mainly in the crystalline regions. Electron trapping was observed by Kayser (quoted in [8]) also in polycrystalline n-tritriacontane. Thus, if the effect appears also in crystalline media, the molecular crystals seem to be a simpler object to study. In particular long-chain alkanes, very similar in structure to polyethylene, can be of interest. Variation with time of positronium formation in organic crystalline media at low temperatures was observed on various occasions (e.g. [12,13]); a study of the discussed effect in n-alkanes was presented only in the paper by Levay et al. [14], which was unfortunately published before the appearance of the electron trapping concept. The aim of the present study was to collect more detailed data
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on positronium formation in some n-alkanes (paraffins) as a function of temperature and high pressure, taking into account the effects of electron trapping. The objects of the study were n-eicosane (C20 H42 ) and its odd-numbered carbon chain neighbours: n-nonadecane (C19 H40 ) and n-heneicosane (C21 H44 ). In even-numbered alkane C20 H42 , only one solid phase exists, while in odd C19 H40 and C21 H44 there is an additional rotator phase, in the range of temperatures from the melting point Tm down to about Tm 10 K [15]. The transition from the soft, waxy rotator phase to rigid one can be executed also by applying the pressure [16].
then the development of I3 with time was observed; the spectra were collected in 2-h intervals. Type B: like in the run Type A, the samples were cooled to 103 K and stored at that temperature up to the saturation of I3 (or, at least, approaching to it), then the temperature was increased stepwise and the PAL spectra were measured during 100 min at each step. Type C: the samples were cooled from the temperature Ti near the melting point to predetermined final temperature, one PAL spectrum was recorded during 1 h and the temperature returned to Ti for about 0.5 h; then the cycle was repeated for next final temperature.
2. Experimental
3. Results and discussion
The samples of C19 H40 and C20 H42 (from Sigma– Aldrich) were of 99% purity (mass spectrometric test showed that the impurity level was rather 0.5%), C21 H44 – 98%; all of them were used without further refinement. Positron annihilation lifetime spectra (PALS) were measured using a standard fast-slow delayed coincidence setup with BaF2 c detectors. The resolution time was about 210 ps, FWHM. In all measurements the same positron source, 0.90 MBq of 22 Na in a Kapton envelope, and the same geometry were used, thus the results for various samples and measurement series could be compared. The sandwich of positron source between two polycrystalline paraffin layers was placed in a small brass container fixed at the tip of a copper rod which could be heated or serve as a cold finger of a cryostat. The temperature was measured by a thermocouple inserted directly into the sample container. Selection of temperature sequence, stabilization and spectra recording were performed by a programmed controller. In the measuring chamber the air pressure was kept at the level of 0.5 Pa. The PALS spectra were assumed to consist of three exponential components convoluted with the instrumental resolution curve plus constant random coincidence background. The shortestlived component was ascribed to the decay of singlet para-Ps, the intermediate one – to the annihilation of free positrons, the longest-lived one to the decay of ortho-Ps. The spectra were analysed using the LT programme [17]. In order to reduce the number of free parameters the para- and ortho-intensity ratio was fixed, 1:3, according to the statistical weights of these states. Correction for positron absorption in the source envelope was also applied. The rate of data collecting was about 150 coincidences/s (70/s in the measurements in high pressure chamber). Three kinds of the measurement series were performed: Type A: the samples were cooled from the temperature Ti several kelvin below the melting point, to predetermined temperature at a rate of about 5 K/min and
3.1. n-Eicosane Change of I3 with time was observed in the runs Type A, for several storage temperatures. The results are shown in Fig. 1. One could suppose that, like in polymers, at sufficiently high temperature (near 300 K) the electron traps were empty (or disappeared), so immediately after fast cooling the concentration of trapped electrons should be negligible. The curves representing I3 increase can be approximated rather well by an exponential: I3 ¼ I3 max Itrap expðt=HÞ;
ð1Þ
where I3 max is the maximal (saturation) intensity, I3 max Itrap ¼ I30 is the ‘‘zero-dose’’ intensity and H is the time constant of I3 rise. The higher the temperature, the
I3, %
20
15
10
5
0 0
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90
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Fig. 1. n-Eicosane. I3 increase during sample storage at: diamonds – 103 K, squares – 153 K, dots – 193 K and crosses – 233 K. Reference value of I3 is that measured directly after cooling from 303 K. The curves are exponentials fitted to the experimental points. Uncertainty limits in this figure and all subsequent ones are smaller than the size of symbols (unless marked otherwise).
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Fig. 3. Temperature-dependence of o-Ps lifetime s3 in n-eicosane. The symbols relate to respective series of measurements from Fig. 2.
The inherent intensity I30 , corresponding to non-irradiated material, can be determined in the run Type C collecting the PALS spectrum in a possibly short time (t < H, in this case 60 min). These points are marked in Fig. 2 by crosses. The lifetime of o-Ps in the medium is determined by the size of the free space (void) in which it is trapped. Shortening the lifetime compared to its vacuum value is caused by pick-off process, i.e., annihilation of Ps positron with a strange electron picked from the surrounding medium. The most popular Tao–Eldrup model [19] gives a simple equation for the pick-off process rate as a function of the radius of spherical void. In solid alkanes, like in polymers, the lifetime of o-Ps, s3 , does not depend on the concentration of trapped electrons, i.e., on sample storage time at a fixed temperature. Independently of the start point and direction of temperature changes, s3 values lie along the same curve (Fig. 3). The series of s3 values collected in the run Type C is shifted slightly down, by 20 ps. It seems that after rapid change of temperature, the medium structure seen by positrons needs a measurable time to become stable (note that it is not the result of delay in reaching the new sample temperature; s3 is larger for higher temperature, while after the step we observe smaller s3 ). 3.2. Odd-numbered alkanes C19 H40 and C21 H44
45 40
I3, %
1.6
τ3, ns
larger is the time constant H: about 30 h at 103 K and 60 h at 230 K. The set of curves from Fig. 1 looks very similar to that for polypropylene obtained by Ito et al. [7]. Note that the increase of intensity Itrap in n-eicosane at 103 K is over 20% (relative increase Itrap =I30 by 110%), while in polypropylene [7] it amounts 7% (Itrap =I30 ¼ 37%). The electron traps are deep; the optical absorption experiments with polymers give that depth in a broad band 0.5–3 eV [18], however, the ESR studies have shown that the trapped electrons are vulnerable to thermal effects at the temperatures for which kT is much lower than the trap depth. The population of some of them begins to decrease with time already at liquid nitrogen temperature and the distribution of traps sensitive to low temperatures looks continuous in a broad range of temperatures [11]. In the ESR experiments the production of trapped electrons by external radiation source and the observation of the irradiation results were separate, while in our case the production and observation were simultaneous; during the sequence of measurements the dose absorbed in the sample rose. Nevertheless, accumulating the trapped electrons up to the saturation value of I3 and then increasing the temperature in the run Type B we should observe the changes of DI3 ¼ I3 I30 as a result of activation of the particular thermal decay processes. The measurements of this kind for n-eicosane were repeated for three storage temperatures. The results are shown in Fig. 2. Independently of the initial temperature the I3 ðT Þ curves are almost identical; those for higher start-point temperatures are slightly shifted down due to still incomplete saturation after a long storage. The intensity looks constant in certain ranges of temperature and reduces by diffuse steps at 110 K, 180 K and 260 K. Above the room temperature the intensity seems to be time-independent. The steps suggest the existence of three groups of trapped electrons subjected to different mechanisms of trap emptying, however, the last step near 260 K predominates.
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Fig. 2. n-Eicosane. Temperature-dependence of I3 : diamonds – for the samples stored 90 h at 103 K; squares – 117 h at 193 K; circles – 138 h at 233 K and crosses – each experimental point I3 measured directly after cooling from 303 K.
The accumulation of trapped electrons in n-nonadecane and n-heneicosane was investigated in the same way as described in the previous section. The results of the runs Type A are shown in Figs. 4 and 5. The lifetime s3 , like in n-eicosane, does not depend on irradiation dose. The results of runs Type B are shown in Figs. 6 and 7. I3 does not decrease by steps like in eicosane (Fig. 2), but changes non-monotonously. Especially, in n-heneicosane the value of I3 reaches 45% at the temperature as high as 200 K; at lower temperatures the saturation intensity is by 9% smaller; this effect is well visible also in Fig. 5. It means that temperature increase not only empties the traps, but can also be a factor augmenting positronium
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TEMPERATURE, K
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Fig. 4. n-Nonadecane. I3 increase during sample storage at: diamonds – 103 K, squares – 158 K, dots – 198 K and crosses – 223 K.
16
Fig. 7. n-Heneicosane. Temperature-dependence of I3 : diamonds – for the samples stored 45 h at 103 K and crosses – each experimental point I3 measured directly after cooling from 309 K.
formation on the trapped electrons. In both odd-numbered alkanes strong decrease of the trapping effect begins near 230 K, and then the intensity reaches a kind of plateau before approaching to the rigid-rotator phase transition point. In that plateau the irradiation effect is not reduced to nil, but the Itrap value does not exceed 4%; full disappearance of Itrap is observed in the rotator phase. 3.3. Time constant of I3 rise
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TIME, h Fig. 5. n-Heneicosane. I3 increase during sample storage at: diamonds – 103 K, squares – 158 K, dots – 198 K and crosses – 283 K.
ph. tr.
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I3, %
40 35 30 25 20 15 80
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Fig. 6. n-Nonadecane. Temperature-dependence of I3 : diamonds – for the samples stored 70 h at 103 K and crosses – each experimental point I3 measured directly after cooling from 299 K.
The intensity increase DI3 ðtÞ is approximately exponential, and the value of H from Eq. (1) can be used as a measure of the rate of I3 accumulation. From one series to another H is not exactly reproducible, nevertheless it shows the trends of change and the time scale of the discussed process. Like in many polymers, in n-eicosane H rises with temperature. In its odd-numbered neighbours, on contrary, H is generally shorter and reduces with increase of the storage temperature (Fig. 8). At 103 K the values of H are of the same order of magnitude, but at 220 K the rise of I3 in C19 H40 and C21 H44 occurs 50 times faster than in C20 H42 . At such a rapid I3 rise certain accumulation of trapped electrons occurs already during cooling the sample in the runs ‘‘C’’ (cooling can last more than half an hour); it gives a peculiar shape of the I3 ðT Þ curve in Fig. 6. In the range 240–280 K the time constant in oddnumbered alkanes is reduced to the values of about 1 h. In order to observe the initial part of I3 rise, the time of spectrum collection was reduced to 30 min (further shortening would increase too much the statistical errors). The results for n-nonadecane at 275 K are shown in Fig. 9. Fitting an exponential to I3 vs. storage time ts data is a rough approximation. The fitted I3 values for ts near zero are, as a rule, higher than experimental ones. The initial part of I3 rise is very steep and if, for sake of parametrization the I3 (ts ) dependence, we add the next exponential, that other time constant would be 20 min only. The tendency of I3 rise faster than expected for single exponential at the initial stage of electron trapping is discernible also on the uppermost curves in Fig. 5.
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70 2.4
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Fig. 10. ortho-Ps lifetime s3 as a function of temperature: open squares – n-nonadecane; diamonds – n-eicosane; dots – n-heneicosane and crosses – n-docosane (data taken from [20]). Temperature up series.
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0 100
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TEMPERATURE, K Fig. 8. Time constant of I3 rise as a function of temperature: squares – n-eicosane; diamonds – n-nonadecane and dots – n-heneicosane.
30
I3 , %
29 28 27 26 25 24 23 0
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4
6
8
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Fig. 9. I3 increase during n-nonadecane sample storage at 275 K (after annealing at 299 K). Continuous line – one exponential fit, H ¼ 1:5 h and dashed line – two exponentials fit, H1 ¼ 2:6 h, H2 ¼ 0:36 h.
than these for even ones. Near the room temperature we observe the lifetimes of about 1.4 ns (even-numbered) and 1.6 ns (odd-numbered alkanes). The long chain alkanes exhibit a lamellar close packing of molecules with parallel long axes. The interlamellar void thickness d0 at temperatures near rigidrotator transition is equal to 0.125 nm in even-numbered paraffins, and 0.195 nm in odd-numbered ones [21,22]. If one supposes that Ps locates in the interlayer voids, the rectangular version of the pick-off model [23,24], with two dimensions extended to infinity, predicts the o-Ps lifetime in even-carbon molecules not exceeding 1.0 ns, i.e., less than seen in the experiment. Thus, free volumes allowing o-Ps to survive longer can be created when the crystal structure contains certain disorder. In the low temperature phase the only kind of disorder are longitudinal displacements of rigid-body molecules (Fig. 11). Longitudinal shift of molecules does not change the average d0 but creates locally an additional free volume. One can assume that the free space accommodating o-Ps
(a)
(b)
3.4. Positronium lifetime in rigid and rotator phases
d0 In the rigid (low temperature) phase of the alkanes investigated here the lifetime and ‘‘zero dose’’ intensity I30 of ortho-Ps component vary smoothly in a broad range from 100 K to the phase transition point. The slope of lifetime vs. temperature curve, in its roughly linear part 140–240 K, is 1.4 ps/K for n-eicosane and nheneicosane; for n-nonadecane the slope is slightly larger, 2.4 ps/K. The experimental o-Ps lifetimes s3 for our alkanes in a limited range below the melting point are shown in Fig. 10. It is seen that the lifetimes for odd-numbered paraffins are systematically longer (by about 200 ps)
Fig. 11. Schematic structure of long-chain alkanes: (a) ideal crystal; (b) crystal with rigid-body displacements of molecules.
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2.0 1.6 1.2 0.8 0.4 0
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1
Fig. 12. ortho-Ps lifetime in cylindrical channel with R ¼ 0:25 nm as a function of the cylinder length, calculated from extended Tao–Eldrup model. The parameter D is assumed 0.166 nm.
is a cylindrical one, 0.5 nm in diameter (molecular diameter), and the length being the sum of chain displacement and interlayer void thickness. Fig. 12 shows the o-Ps lifetime in a cylindrical void as a function of the cylinder length, calculated from the Tao–Eldrup model for cylindrical geometry and larger void sizes [25], 1=s3 ¼ k3 ¼ kb ðkb k1 ÞP ;
ð2Þ
where kb , k1 are the o-Ps decay rates in the bulk (kb ¼ 2/ ns) and in an infinitely long cylinder, while " X L 1 2pnD sin P¼ þ L þ 2D np L þ 2D n !# h2 n2 exp 2 16me ðL þ 2DÞ kT ! X h2 n2 ; ð3Þ exp 2 16me ðL þ 2DÞ kT n where L, h, me , k, T are: the cylinder length, PlanckÕs constant, electron mass, Boltzmann constant and temperature, respectively. In all estimates given here the parameter D related to the penetration of Ps wavefunction into the bulk is assumed 0.166 nm; that value is well tested for small voids and organic media. According to the model calculations the experimentally found lifetimes correspond to the cylinders 0.32 and 0.40 nm long; their difference is just the same as odd–even difference in d0 spacing; the average displacement is close to that of two-carbon segment (the length of one C–C segment of the molecule is 0.127 nm [21]). The model connecting the pick-off lifetime with the void size used here is rather a rough approximation, thus the numbers given above cannot be taken as a result of precise calculation. Nevertheless, the location of Ps at the irregularities on the surface of interlamellar layer seems to be proved. Greater slope of s3 ðT Þ curve in n-nonadecane could be tentatively explained by the reduction of d0 with lowering the temperature for this
particular alkane, however, there are no crystallographic data which could confirm this supposition. The lifetime s2 of freely annihilating positrons does not experience these even–odd variations, it is determined mainly by the density of the medium varying smoothly along the homologous series. For the three alkanes studied here average s2 in the range 280–290 K is: (318 1) ps, (319 1) ps, (318 1) ps, from nonadecane to heneicosane, i.e., practically constant. Also the p-Ps lifetime s1 in the rigid phase in all our experiments does not show any peculiarities, remaining in the limits 120–140 ps. Transition from the rigid to rotator phase in n-nonadecane occurs at (294.7 0.5) K, in n-heneicosane at (304.2 0.5) K; literature data are 295.2 and 305.6 K, respectively [26]. At the point of phase transition in oddnumbered alkane the lifetime is almost doubled: s3 ¼ ð2:28 0:01Þ ns in n-nonadecane and (2.50 0.01) ns in n-heneicosane. The rotator phase is characterized by an orientational disorder of the molecules around their long axes (the name rotator follows from activation of the motion around that axis, although it is rather a series of jumps by a certain angle than continuous rotation [26]). At the transition to the rotator phase also the intramolecular defects are created [27]. There are several types of such defected molecules (conformers). At low temperature the alkane molecules have a rod-like all-trans configuration; the flexibility of chains does not appear as long as the overall molecule length is below 10 nm (chain length 75–80 carbon atoms [28]). Above the transition point the ‘‘end-gauche’’, ‘‘double-gauche’’ and ‘‘kink’’ conformers appear (Fig. 13); the planar all trans chain can be transformed into non-planar structure. According to the estimates by Maroncelli et al. [27], in nnonadecane the concentration of end-gauche defects is (5–10)%, while of kink defects (10–20)%, in n-heneicosane these concentrations are slightly higher. The kinked structure creates a long empty channel in the lamella allowing o-Ps to live longer. As it is seen from Fig. 12, the lifetime observed at the transition point corresponds to a very long cylinder (length tending to infinity). There are two possibilities:
(b)
(c) (d)
Fig. 13. Structures of the long-chain alkane conformers: from top to bottom: all-trans, end-gauche, kink, double-gauche. Hydrogen atoms are not shown for simplicity.
m. p.
2.5
τ 3, ns
• o-Ps annihilates in the vacancies present in the crystal structure; • the transverse section of free space near the kinked molecule is larger than that occupied by all-trans molecule. Only the second variant is acceptable: in the first one there is no place for further extension of the lifetime with temperature. In the narrow range of temperatures above the solid-rotator transition point both o-Ps parameters, lifetime (Fig. 10) and intensity, rise rather rapidly, which means further distortion of the rotator phase structure. That is supported by crystallographic data; the second moment of electron density distribution (precisely: of electron density deficit in the interlamellar layer) in the rotator phase rises with temperature; the average width of the layer rises too, as the non-planar conformers are shorter than the straight all-trans ones [21]. The growth of the number of intramolecular defects per one carbon chain is also reported by Yesook Kim et al. [29] studying the IR spectra. The change of I3 intensity in the run ‘‘B’’ at the phase transition point is not so distinct as that of lifetime because the effect is obscured by the remnants of electron trapping process, still present below the phase transition point. In particular, in the reversed run ‘‘B’’ (temperature lowering from above the transition point in 1 K steps; 100 min/spectrum), the step of I3 is hard to notice. Only after shortening the spectrum collection time, a dip of I3 can be observed. The fragment of I3 ðT Þ dependence near the rigid-rotator phase transition is shown in Fig. 14 in an extended scale. Quick growth of I3 in time (it was seen earlier that the time constant H is about 1 h) results in creation of a narrow minimum only near 293 K.
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Fig. 15. Temperature-dependence of o-Ps lifetime s3 in n-eicosane in the region below the melting point (diamonds). A similar dependence near the rigid-rotator transition in n-octacosane [30] is also shown (dots).
An interesting difference in behaviour of s3 at approaching the solid–liquid and solid–rotator transition points can be observed. In n-eicosane s3 gradually rises in the range of about 10 K below the melting point (Fig. 15), while the transitions rigid solid–rotator are not preceded by s3 variation. It seems that in n-eicosane some structural changes appear before the melting. Xray studies can detect the conformers at a relatively large concentration, while positronium is much more sensitive to the presence of free space defects; their concentration at the level 0.1% can produce quite a large o-Ps component [31]. Thus, if the formation of non-planar intramolecular defects begins below the transition point, the lifetime change can be observed. Note that in the next even alkane, C22 H46 , the rotator phase is already formed, barely 1 K below melting [16]. 3.5. Effect of high pressure
80
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Fig. 14. I3 intensity near rigid-rotator transition in n-nonadecane observed in ‘‘type B’’ measurements. Full squares – temperature up (initial point 103 K, 100 min/point) and open squares – temperature down (from 297 K, 20 min/point). The figure shows also the pressure dependence of I3 (diamonds and dashed line). Pressure is shown at the top of the figure frame (inverted). The choice of pressure scale is explained in the legend of Fig. 18.
The transition rotator-rigid can be realized also by applying the high pressure. The pressure chamber containing the sample-source sandwich was kept at 300 K in the case of n-nonadecane, and 310 K for n-heneicosane. At these temperatures, before applying the pressure, the samples were in the rotator phase; then, the pressure was increased in steps up to 450 MPa (the time of spectrum collection – 2 h). Transition to the rigid phase at the temperatures indicated above occurs at p 20 MPa, as it could be expected from the measurements by the classic methods [16]. The values of s3 in n-nonadecane and n-heneicosane as a function of pressure are shown in Fig. 16, the intensities I3 in Fig. 17. The s3 ðpÞ dependence looks like mirror image of the curves from Fig. 10. In Fig. 18 a comparison of s3 lifetime dependence on temperature and pressure in a broad range of both factors in n-nonadecane is shown; the abscissa scales are chosen to get 1 K equivalent to 2.25 MPa. The curves look identical in the whole range of temperatures and pressures. Similar comparison of intensities I3 at the
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τ 3, ns
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Fig. 16. ortho-Ps lifetime s3 in n-nonadecane (open squares) and nheneicosane (dots) as a function of pressure.
I3, %
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Fig. 17. ortho-Ps intensity I3 in n-nonadecane (open squares) and nheneicosane (dots) as a function of pressure.
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temperature change. For n-nonadecane, the step of I3 at the transition under pressure amounts 7.5%, while in temperature-up measurements at normal pressure it does not exceed 3.5%. In Fig. 14 the experimental data from pressure experiment are added for easier comparison. The measurements described above were performed at the temperature near the melting point, where the electron trapping phenomena are not observed or play a marginal role. Application of pressure produced the effects similar to lowering the temperature (phase transition, decrease of lifetimes and intensity I3 ), thus it seemed worthwhile to check whether high pressure can restrain the escape of electrons from the traps. If so, under high pressure one should expect the reappearance of I3 variation with time. The pressure of 255 MPa was applied to n-heneicosane sample and a sequence of spectra measured in 2-h intervals. Contrary to expectations, the intensity I3 diminished with time (Fig. 19). Similar measurements were repeated at 130 and 380 MPa. The time constant of I3 decrease varied from 3.7 h for 130 MPa to 9.4 h for 380 MPa, i.e., rose with the increase of the pressure. The values of I3 shown in Fig. 17 (the measurement analogous to run ‘‘B’’) are close to asymptotic ones for very long time of storage under pressure. A run equivalent to type ‘‘C’’ was performed with n-heneicosane: after each experimental point the pressure was reduced to zero (return to the rotator phase was immediate) and the sample was kept without pressure 20 min before applying next selected p value. The results for run types ‘‘B’’ and ‘‘C’’ are shown in Fig. 20 (intensity I3 lifetimes s2 and s3 ). The lifetimes s3 are, like after lowering the temperature, independent of the storage time, however,
(a)
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choice of abscissa scale as above reveals a difference: after the transition to rigid phase induced by pressure I3 is lower than that appearing in the experiment with
τ2, ns
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Fig. 18. n-Nonadecane. Comparison of ortho-Ps lifetimes as a function of temperature and pressure. The scale of abscissa for pressure (top of the figure frame) is chosen to match the phase transition point and to get the same slope of s3 ðT Þ and s3 ðpÞ dependences. Full squares – temperature up; open squares – temperature down and diamonds – pressure.
0.32
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0.31 0
10
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Fig. 19. n-Heneicosane. Changes of intensity I3 and lifetime s2 (free positrons) with the time of sample storage at the pressure 255 MPa. Temperature 310 K.
0.34 0.32 0.3
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h Rmin ¼ pffiffiffiffiffiffiffiffiffiffiffi ; 4 2mV0
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251
them; they are the same voids. If the reduction of void size occurs, it should refer to all voids localized in the vicinity of vibrating end-methyl groups, however, DI3 reduction is not accompanied by shortening of s3 . One can imagine the situation when the void represents a trap for positronium, but not for electrons. The minimal radius at which the energy level in the spherical well exists is
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300
400
Fig. 20. n-Heneicosane. Lifetimes s2 , s3 and intensity I3 as a function of pressure, measured during the run type ‘‘B’’ (diamonds) and type ‘‘C’’ (open circles).
one observes the change of free annihilation lifetime s2 , its rise with the time elapse (Fig. 19(b)). This last effect was not seen in the experiments at low temperatures. The mechanism reducing I3 is unknown to us, it can be rather a relaxation than trapped electron effect. 3.6. Origin of electron traps It is commonly assumed that electron traps are free spaces in the polymer structure. Disappearance of trapped electrons is ascribed to the activation of molecular motions; in particular, methyl group vibrations in the vicinity of voids make the electron density in them to increase, the size of free spaces is reduced; the zero point energy rises and trapping stops to be energetically favourable. The effect of DI3 reduction with temperature can be due to the escape of electrons or to the reduction of Ps formation probability. The decay of trapped electrons is well visible in the ESR measurements in polymers [11], thus the first of the two mentioned variants is more probable. From the viewpoint of the observed positronium properties, the radical reduction of void size by molecular motion seems not easy to accept. Constancy of s3 during irradiation means that the Ps accommodating voids are of the same size for positronium formed using trapped electron stores and without
ð4Þ
where m is the mass of trapped particle and V0 is the depth of the potential energy well. For positronium, the mass is twice of that of an electron, the energy V0 is different for electron and for neutral Ps atom. For certain R the condition (4) can be fulfilled for both particles, but the reduction of void size can lead to trapping Ps only (when R lies between Rmin for Ps and Rmin for electron). Such a condition can be fulfilled rarely, for a narrow range of R, and even it is, one should observe the shortening of o-Ps lifetime when the atomic motions are activated. However, detrapping of electrons is a common phenomenon and s3 change is not observed, thus, the interpretation presented above is not acceptable. The amplitude of vibrations is not so large to change the void dimensions, estimated in Section 3.3, to the extent reducing them to the values below Rmin It seems that at elevated temperatures all the traps still exist, but certain processes causing electron migration become activated. In polyethylene polymer, Keyser et al. [32] determined the activation energies for thermal decay of various groups of trapped electrons by ESR. For decays occurring in the range from 77 to 127 K, the activation energies Ea were from 0.14 to 0.22 eV; the ratio Ea =kT was constant and very high (over 20). In our measurements the biggest step on the I3 ðT Þ curve occurs in the range 200–240 K (it is approximately the same temperature as for the minimum of I3 in polymers [9]). In n-nonadecane, the intensity drops from maximal Imax 44.5% to Imin ¼ 27:5%. If it is the result of
ln[(Imax-I)/(I-Imin)]
τ2, ns
T. Goworek et al. / Chemical Physics 295 (2003) 243–253
1 0 -1 -2 -3 4.2
4.4
4.6
4.8
-3
Fig. 21. n-Nonadecane. Arrhenius plot for the fraction of o-Ps intensity related to the detrapping of electrons at the temperatures 200–240 K.
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an activation of electron migration processes one can try to make an Arrhenius plot [33] assuming proportionality of I3 to the density of the trapped electrons. Such a plot, Imax I H ¼ A exp ; ð5Þ I Imin kT is shown in Fig. 21. The activation enthalpy H is found 0.6 eV; H =kT is similar to that found for polymers.
4. Conclusions The rise of o-Ps component with irradiation time, well-known in polymers, appears also in long-chain alkanes, at least in those with n 20. The effect is large, leading sometimes to doubling the intensity observed in non-irradiated samples. A simple structure of the medium and well-defined defects make alkanes a good object to study the formation of positronium in the presence of trapped electrons. At our source activity the time constant H of I3 rise at 103 K is of the order of a day. With the increase of storage temperature, the even numbered n-eicosane behaves totally differently than odd numbered neighbours: the time constant in n-eicosane rises with temperature (like in polypropylene and other polymers), while in odd numbered it decreases down to few hours. It is to be checked whether such divergence continues through the whole homologous series of paraffins. The difference in properties of even and odd-numbered alkanes is known since 1877 [34] and reveals in various forms: popular matrices for molecular sieves – alkyltrimethyl bromides – exist with even alkyls only, one observes preferential abundance of odd chain biogenic alkanes [35] and preferential breaking into evennumbered fragments during combustion of paraffins as a rocket fuel [36], etc. The odd–even difference of o-Ps lifetimes is easily explainable by the crystallographic structure, but the divergence of time constants of I3 rise by almost two orders of magnitude remains difficult to explain. The radiation-chemical studies of Ps formation on pre-trapped electrons are in their initial stage and for the time being the set of time constants H determined here supplements the stores of data only. In odd numbered alkanes, between 230 and 280 K, Itrap is of the order of 4%, but the time constant of rise is 1 h. Very short time constants pose the problem whether the nature of I3 rise in this temperature range is the same as at low temperatures, or it is a different process. The yield of trapped electrons Gðe Þ does not depend strongly on temperature, also Ps formation probability changes slowly with temperature, as can be seen in Figs. 2, 6 and 7 (runs C), thus, the irradiation effects seem to be not sufficient to explain such fast changes. The shift of s3 ðT Þ curve for odd-numbered alkanes against the even ones allows us to localize the positro-
nium trapping centers in the vicinity of the interlamellar void (gap), swollen locally owing to longitudinal molecule displacements. The dependences of the o-Ps lifetime on temperature and pressure are similar. Variation of I3 is observed also after application of high pressure, but the direction of changes is opposite to that at low temperatures; it decreases by several percent with time constant 4–8 h.
Acknowledgements This work was supported by the Polish Committee for Research (KBN) Grant No. 5P03B 031 21.
References [1] O.E. Mogensen, J. Chem. Phys. 60 (1974) 998. [2] L.T. Bugayenko, V.M. Byakov, E.P. Kalazin, in: Symposium on Radiation Chemistry of Aqueous Systems Abstracts, Nauka, Moscow, 1973, p. 4. [3] P. Kindl, G. Reiter, Phys. Stat. Sol. (A) 104 (1987) 707. [4] T. Suzuki, T. Miura, Y. Oki, M. Numajiri, K. Kondo, Y. Ito, J. Phys. IV 3 (1993) 283. [5] C.L. Wang, T. Hirade, F.H.J. Maurer, M. Eldrup, N.J. Pedersen, J. Chem. Phys. 108 (1998) 4654. [6] T. Hirade, C.L. Wang, F.H.J. Maurer, M. Eldrup, N.J. Pedersen, in: Abstract book for 35th Annual Meeting on Radioisotopes in the Physical Science and Industries, June 30–July 1, Tokyo, 1998, p. 89. [7] Y. Ito, T. Hirade, E. Hamada, T. Suzuki, Y. Ito, Acta Phys. Polon. A 95 (1999) 533. [8] M. Dole (Ed.), The Radiation Chemistry of Macromolecules, Academic Press, New York, 1972. [9] T. Suzuki, C. He, V. Shantarovich, K. Kondo, Y. Ito, L. Ma, M. Matsuo, Mater. Res. Innovat. 7 (2003) 31. [10] T. Hirade, F.H.J. Maurer, M. Eldrup, Radiat. Phys. Chem. 58 (2000) 465. [11] J. Lin, K. Tsuji, F. Williams, J. Am. Chem. Soc. 90 (1968) 2766. [12] D. Lightbody, J.N. Sherwood, M. Eldrup, Chem. Phys. 93 (1985) 475. [13] T. Goworek, B. Jasi nska, J. Wawryszczuk, K. Ciesielski, J. Chem. Soc., Faraday Trans. 93 (1997) 1573. [14] B. Levay, M. Lalovic, H.J. Ache, J. Chem. Phys. 90 (1989) 3282. [15] J. Doucet, I. Denicol o, A.F. Craievich, C. Germain, J. Chem. Phys. 80 (1984) 1647. [16] H. Forsman, P. Anderson, J. Chem. Phys. 80 (1984) 2804. [17] J. Kansy, Nucl. Instrum. Methods A 374 (1996) 235. [18] R.M. Keyser, F. Williams, J. Phys. Chem. 73 (1969) 1623. [19] M. Eldrup, D. Lightbody, J.N. Sherwood, Chem. Phys. 63 (1981) 51. [20] T. Goworek, M. Chmiel, in: Proceedings of the 33rd Polish Seminar on Positron Annihilation, Opole University, 2002, p. 33. [21] A.F. Craievich, I. Denicol o, J. Doucet, Phys. Rev. B 30 (1984) 4782. [22] A. Craievich, J. Doucet, I. Denicol o, Phys. Rev. B 32 (1985) 4164. [23] B. Jasi nska, A.E. Koziol, T. Goworek, J. Radioanal. Nucl. Chem. 210 (1996) 617. [24] D.W. Gidley, W.E. Frieze, T.L. Dull, A.F. Yee, E.T. Ryan, H.-M. Ho, Phys. Rev. B 60 (1999) R5157. [25] R. Zaleski, J. Wawryszczuk, A. Bor owka, J. Goworek, T. Goworek, Microporous Mesoporous Mater. 62 (2003) 47.
T. Goworek et al. / Chemical Physics 295 (2003) 243–253 [26] F. Guillaume, J. Doucet, C. Sourisseau, A.J. Dianoux, J. Chem. Phys. 91 (1989) 2555. [27] M. Maroncelli, Song Ping Qi, H.L. Strauss, R.G. Snyder, J. Am. Chem. Soc. 104 (1982) 6237. [28] J. Pak, A. Boller, I. Moon, M. Pyda, B. Wunderlich, Thermochim. Acta 357 (2000) 259. [29] Yesook Kim, H.L. Strauss, R.G. Snyder, J. Phys. Chem. 93 (1989) 7520. [30] T. Goworek, in: Y.C. Jean (Ed.), Positron and Positronium Chemistry, World Scientific, Singapore, 1990, p. 533.
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Temperature, pressure and source-irradiation effects in positronium formation in some solid long-chain alkanes T. Goworek *, R. Zaleski, J. Wawryszczuk Institute of Physics, Maria Curie-Sklodowska University, Pl. Marii Curie-Sklodowskiej 1, Lublin 20-031, Poland Received 23 April 2003; accepted 25 August 2003
Abstract Positron lifetime spectra in solid n-alkanes C19 H40 , C20 H42 and C21 H44 were measured. The effect of irradiation by positron source (increase of ortho-positronium intensity I3 ), known from polymer studies, was found to appear also in alkanes; at about 100 K it can lead to doubling the intensity. The time constant of I3 rise in n-eicosane increased with temperature, reaching the value of 60 h, while it decreased (down to 1 h) in its odd-numbered neighbours. The difference in ortho-Ps lifetimes between odd- and evennumbered alkanes can be explained by different thickness of interlamellar layer in the crystal structure. The variation of I3 after application of high pressure was also observed, but in this case the intensity decreased with time. 2003 Elsevier B.V. All rights reserved.
1. Introduction According to the spur model [1,2], representing the viewpoint of radiation chemistry, the atom of positronium is formed in condensed matter by binding the slowed down positron with one of quasi-free electrons which it produced in a series of preceding ionization acts. The probability of positronium (Ps) formation is related to the number of available electrons near the last ionization site. Positronium exists in two substates, depending on mutual spin orientation of involved particles. The triplet substate with parallel spins (ortho-positronium, o-Ps) is a relatively long-living species, thus easier to observe than its singlet counterpart. Already by the end of the eighties it was noticed that the intensity of o-Ps component in the lifetime spectrum of positrons in polymers was not stable, rising with time, particularly at low temperatures [3,4]. That intensity, I3 , can be assumed as the measure of Ps formation probability; in the absence of Ps transformation processes o-Ps represents 3/4 of the total positronium formed. In 1998, Wang et al. [5] proposed to ascribe I3 variation to the changes of the density of trapped electrons, finally the experiment by Hirade et al. [6,7], consisting in *
Corresponding author. Tel.: +48-815376225; fax: +48-815376191. E-mail address: [email protected] (T. Goworek).
0301-0104/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2003.08.019
bleaching with visible light, demonstrated that the surplus of I3 was induced entirely by the reaction of positrons with the reservoir of trapped electrons deposited in the sample by the positrons injected previously. The processes of trapping the electrons created in polymers during irradiation were studied intensively by the ESR and optical methods [8], nowadays supplemented by positron techniques (see, e.g. [9]). A thorough comparison of the ESR and positron annihilation data is given in the paper by Hirade et al. [10]. The studies of trapped electrons were oriented mainly toward polymers, although there are also some data about saturated hydrocarbon glasses [11]. In polymers the traps were found to be located mainly in the crystalline regions. Electron trapping was observed by Kayser (quoted in [8]) also in polycrystalline n-tritriacontane. Thus, if the effect appears also in crystalline media, the molecular crystals seem to be a simpler object to study. In particular long-chain alkanes, very similar in structure to polyethylene, can be of interest. Variation with time of positronium formation in organic crystalline media at low temperatures was observed on various occasions (e.g. [12,13]); a study of the discussed effect in n-alkanes was presented only in the paper by Levay et al. [14], which was unfortunately published before the appearance of the electron trapping concept. The aim of the present study was to collect more detailed data
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on positronium formation in some n-alkanes (paraffins) as a function of temperature and high pressure, taking into account the effects of electron trapping. The objects of the study were n-eicosane (C20 H42 ) and its odd-numbered carbon chain neighbours: n-nonadecane (C19 H40 ) and n-heneicosane (C21 H44 ). In even-numbered alkane C20 H42 , only one solid phase exists, while in odd C19 H40 and C21 H44 there is an additional rotator phase, in the range of temperatures from the melting point Tm down to about Tm 10 K [15]. The transition from the soft, waxy rotator phase to rigid one can be executed also by applying the pressure [16].
then the development of I3 with time was observed; the spectra were collected in 2-h intervals. Type B: like in the run Type A, the samples were cooled to 103 K and stored at that temperature up to the saturation of I3 (or, at least, approaching to it), then the temperature was increased stepwise and the PAL spectra were measured during 100 min at each step. Type C: the samples were cooled from the temperature Ti near the melting point to predetermined final temperature, one PAL spectrum was recorded during 1 h and the temperature returned to Ti for about 0.5 h; then the cycle was repeated for next final temperature.
2. Experimental
3. Results and discussion
The samples of C19 H40 and C20 H42 (from Sigma– Aldrich) were of 99% purity (mass spectrometric test showed that the impurity level was rather 0.5%), C21 H44 – 98%; all of them were used without further refinement. Positron annihilation lifetime spectra (PALS) were measured using a standard fast-slow delayed coincidence setup with BaF2 c detectors. The resolution time was about 210 ps, FWHM. In all measurements the same positron source, 0.90 MBq of 22 Na in a Kapton envelope, and the same geometry were used, thus the results for various samples and measurement series could be compared. The sandwich of positron source between two polycrystalline paraffin layers was placed in a small brass container fixed at the tip of a copper rod which could be heated or serve as a cold finger of a cryostat. The temperature was measured by a thermocouple inserted directly into the sample container. Selection of temperature sequence, stabilization and spectra recording were performed by a programmed controller. In the measuring chamber the air pressure was kept at the level of 0.5 Pa. The PALS spectra were assumed to consist of three exponential components convoluted with the instrumental resolution curve plus constant random coincidence background. The shortestlived component was ascribed to the decay of singlet para-Ps, the intermediate one – to the annihilation of free positrons, the longest-lived one to the decay of ortho-Ps. The spectra were analysed using the LT programme [17]. In order to reduce the number of free parameters the para- and ortho-intensity ratio was fixed, 1:3, according to the statistical weights of these states. Correction for positron absorption in the source envelope was also applied. The rate of data collecting was about 150 coincidences/s (70/s in the measurements in high pressure chamber). Three kinds of the measurement series were performed: Type A: the samples were cooled from the temperature Ti several kelvin below the melting point, to predetermined temperature at a rate of about 5 K/min and
3.1. n-Eicosane Change of I3 with time was observed in the runs Type A, for several storage temperatures. The results are shown in Fig. 1. One could suppose that, like in polymers, at sufficiently high temperature (near 300 K) the electron traps were empty (or disappeared), so immediately after fast cooling the concentration of trapped electrons should be negligible. The curves representing I3 increase can be approximated rather well by an exponential: I3 ¼ I3 max Itrap expðt=HÞ;
ð1Þ
where I3 max is the maximal (saturation) intensity, I3 max Itrap ¼ I30 is the ‘‘zero-dose’’ intensity and H is the time constant of I3 rise. The higher the temperature, the
I3, %
20
15
10
5
0 0
30
60
90
120
150
Fig. 1. n-Eicosane. I3 increase during sample storage at: diamonds – 103 K, squares – 153 K, dots – 193 K and crosses – 233 K. Reference value of I3 is that measured directly after cooling from 303 K. The curves are exponentials fitted to the experimental points. Uncertainty limits in this figure and all subsequent ones are smaller than the size of symbols (unless marked otherwise).
T. Goworek et al. / Chemical Physics 295 (2003) 243–253
1.4 1.2 1.0 0.8 80
120
160
200
240
280
320
Fig. 3. Temperature-dependence of o-Ps lifetime s3 in n-eicosane. The symbols relate to respective series of measurements from Fig. 2.
The inherent intensity I30 , corresponding to non-irradiated material, can be determined in the run Type C collecting the PALS spectrum in a possibly short time (t < H, in this case 60 min). These points are marked in Fig. 2 by crosses. The lifetime of o-Ps in the medium is determined by the size of the free space (void) in which it is trapped. Shortening the lifetime compared to its vacuum value is caused by pick-off process, i.e., annihilation of Ps positron with a strange electron picked from the surrounding medium. The most popular Tao–Eldrup model [19] gives a simple equation for the pick-off process rate as a function of the radius of spherical void. In solid alkanes, like in polymers, the lifetime of o-Ps, s3 , does not depend on the concentration of trapped electrons, i.e., on sample storage time at a fixed temperature. Independently of the start point and direction of temperature changes, s3 values lie along the same curve (Fig. 3). The series of s3 values collected in the run Type C is shifted slightly down, by 20 ps. It seems that after rapid change of temperature, the medium structure seen by positrons needs a measurable time to become stable (note that it is not the result of delay in reaching the new sample temperature; s3 is larger for higher temperature, while after the step we observe smaller s3 ). 3.2. Odd-numbered alkanes C19 H40 and C21 H44
45 40
I3, %
1.6
τ3, ns
larger is the time constant H: about 30 h at 103 K and 60 h at 230 K. The set of curves from Fig. 1 looks very similar to that for polypropylene obtained by Ito et al. [7]. Note that the increase of intensity Itrap in n-eicosane at 103 K is over 20% (relative increase Itrap =I30 by 110%), while in polypropylene [7] it amounts 7% (Itrap =I30 ¼ 37%). The electron traps are deep; the optical absorption experiments with polymers give that depth in a broad band 0.5–3 eV [18], however, the ESR studies have shown that the trapped electrons are vulnerable to thermal effects at the temperatures for which kT is much lower than the trap depth. The population of some of them begins to decrease with time already at liquid nitrogen temperature and the distribution of traps sensitive to low temperatures looks continuous in a broad range of temperatures [11]. In the ESR experiments the production of trapped electrons by external radiation source and the observation of the irradiation results were separate, while in our case the production and observation were simultaneous; during the sequence of measurements the dose absorbed in the sample rose. Nevertheless, accumulating the trapped electrons up to the saturation value of I3 and then increasing the temperature in the run Type B we should observe the changes of DI3 ¼ I3 I30 as a result of activation of the particular thermal decay processes. The measurements of this kind for n-eicosane were repeated for three storage temperatures. The results are shown in Fig. 2. Independently of the initial temperature the I3 ðT Þ curves are almost identical; those for higher start-point temperatures are slightly shifted down due to still incomplete saturation after a long storage. The intensity looks constant in certain ranges of temperature and reduces by diffuse steps at 110 K, 180 K and 260 K. Above the room temperature the intensity seems to be time-independent. The steps suggest the existence of three groups of trapped electrons subjected to different mechanisms of trap emptying, however, the last step near 260 K predominates.
245
35 30 25 20 15 80
120
160
200
240
280
320
Fig. 2. n-Eicosane. Temperature-dependence of I3 : diamonds – for the samples stored 90 h at 103 K; squares – 117 h at 193 K; circles – 138 h at 233 K and crosses – each experimental point I3 measured directly after cooling from 303 K.
The accumulation of trapped electrons in n-nonadecane and n-heneicosane was investigated in the same way as described in the previous section. The results of the runs Type A are shown in Figs. 4 and 5. The lifetime s3 , like in n-eicosane, does not depend on irradiation dose. The results of runs Type B are shown in Figs. 6 and 7. I3 does not decrease by steps like in eicosane (Fig. 2), but changes non-monotonously. Especially, in n-heneicosane the value of I3 reaches 45% at the temperature as high as 200 K; at lower temperatures the saturation intensity is by 9% smaller; this effect is well visible also in Fig. 5. It means that temperature increase not only empties the traps, but can also be a factor augmenting positronium
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T. Goworek et al. / Chemical Physics 295 (2003) 243–253 50
ph. tr.
45
I3, %
40
I3, %
12
35 30 25
8
20 15 80
4
120
160
200
240
280
320
TEMPERATURE, K
0 0
10
20
30
40
50
60
70
Fig. 4. n-Nonadecane. I3 increase during sample storage at: diamonds – 103 K, squares – 158 K, dots – 198 K and crosses – 223 K.
16
Fig. 7. n-Heneicosane. Temperature-dependence of I3 : diamonds – for the samples stored 45 h at 103 K and crosses – each experimental point I3 measured directly after cooling from 309 K.
formation on the trapped electrons. In both odd-numbered alkanes strong decrease of the trapping effect begins near 230 K, and then the intensity reaches a kind of plateau before approaching to the rigid-rotator phase transition point. In that plateau the irradiation effect is not reduced to nil, but the Itrap value does not exceed 4%; full disappearance of Itrap is observed in the rotator phase. 3.3. Time constant of I3 rise
I3, %
12
8
4
0 0
10
20
30
40
50
TIME, h Fig. 5. n-Heneicosane. I3 increase during sample storage at: diamonds – 103 K, squares – 158 K, dots – 198 K and crosses – 283 K.
ph. tr.
45
I3, %
40 35 30 25 20 15 80
120
160
200
240
280
320
Fig. 6. n-Nonadecane. Temperature-dependence of I3 : diamonds – for the samples stored 70 h at 103 K and crosses – each experimental point I3 measured directly after cooling from 299 K.
The intensity increase DI3 ðtÞ is approximately exponential, and the value of H from Eq. (1) can be used as a measure of the rate of I3 accumulation. From one series to another H is not exactly reproducible, nevertheless it shows the trends of change and the time scale of the discussed process. Like in many polymers, in n-eicosane H rises with temperature. In its odd-numbered neighbours, on contrary, H is generally shorter and reduces with increase of the storage temperature (Fig. 8). At 103 K the values of H are of the same order of magnitude, but at 220 K the rise of I3 in C19 H40 and C21 H44 occurs 50 times faster than in C20 H42 . At such a rapid I3 rise certain accumulation of trapped electrons occurs already during cooling the sample in the runs ‘‘C’’ (cooling can last more than half an hour); it gives a peculiar shape of the I3 ðT Þ curve in Fig. 6. In the range 240–280 K the time constant in oddnumbered alkanes is reduced to the values of about 1 h. In order to observe the initial part of I3 rise, the time of spectrum collection was reduced to 30 min (further shortening would increase too much the statistical errors). The results for n-nonadecane at 275 K are shown in Fig. 9. Fitting an exponential to I3 vs. storage time ts data is a rough approximation. The fitted I3 values for ts near zero are, as a rule, higher than experimental ones. The initial part of I3 rise is very steep and if, for sake of parametrization the I3 (ts ) dependence, we add the next exponential, that other time constant would be 20 min only. The tendency of I3 rise faster than expected for single exponential at the initial stage of electron trapping is discernible also on the uppermost curves in Fig. 5.
T. Goworek et al. / Chemical Physics 295 (2003) 243–253
247
70 2.4
τ3 , ns
60
50
2.0
1.6
Θ, h
40 1.2 270
280
290
300
310
30
Fig. 10. ortho-Ps lifetime s3 as a function of temperature: open squares – n-nonadecane; diamonds – n-eicosane; dots – n-heneicosane and crosses – n-docosane (data taken from [20]). Temperature up series.
20
10
0 100
150
200
250
300
TEMPERATURE, K Fig. 8. Time constant of I3 rise as a function of temperature: squares – n-eicosane; diamonds – n-nonadecane and dots – n-heneicosane.
30
I3 , %
29 28 27 26 25 24 23 0
2
4
6
8
10
12
14
Fig. 9. I3 increase during n-nonadecane sample storage at 275 K (after annealing at 299 K). Continuous line – one exponential fit, H ¼ 1:5 h and dashed line – two exponentials fit, H1 ¼ 2:6 h, H2 ¼ 0:36 h.
than these for even ones. Near the room temperature we observe the lifetimes of about 1.4 ns (even-numbered) and 1.6 ns (odd-numbered alkanes). The long chain alkanes exhibit a lamellar close packing of molecules with parallel long axes. The interlamellar void thickness d0 at temperatures near rigidrotator transition is equal to 0.125 nm in even-numbered paraffins, and 0.195 nm in odd-numbered ones [21,22]. If one supposes that Ps locates in the interlayer voids, the rectangular version of the pick-off model [23,24], with two dimensions extended to infinity, predicts the o-Ps lifetime in even-carbon molecules not exceeding 1.0 ns, i.e., less than seen in the experiment. Thus, free volumes allowing o-Ps to survive longer can be created when the crystal structure contains certain disorder. In the low temperature phase the only kind of disorder are longitudinal displacements of rigid-body molecules (Fig. 11). Longitudinal shift of molecules does not change the average d0 but creates locally an additional free volume. One can assume that the free space accommodating o-Ps
(a)
(b)
3.4. Positronium lifetime in rigid and rotator phases
d0 In the rigid (low temperature) phase of the alkanes investigated here the lifetime and ‘‘zero dose’’ intensity I30 of ortho-Ps component vary smoothly in a broad range from 100 K to the phase transition point. The slope of lifetime vs. temperature curve, in its roughly linear part 140–240 K, is 1.4 ps/K for n-eicosane and nheneicosane; for n-nonadecane the slope is slightly larger, 2.4 ps/K. The experimental o-Ps lifetimes s3 for our alkanes in a limited range below the melting point are shown in Fig. 10. It is seen that the lifetimes for odd-numbered paraffins are systematically longer (by about 200 ps)
Fig. 11. Schematic structure of long-chain alkanes: (a) ideal crystal; (b) crystal with rigid-body displacements of molecules.
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2.0 1.6 1.2 0.8 0.4 0
0.2
0.4
0.6
0.8
1
Fig. 12. ortho-Ps lifetime in cylindrical channel with R ¼ 0:25 nm as a function of the cylinder length, calculated from extended Tao–Eldrup model. The parameter D is assumed 0.166 nm.
is a cylindrical one, 0.5 nm in diameter (molecular diameter), and the length being the sum of chain displacement and interlayer void thickness. Fig. 12 shows the o-Ps lifetime in a cylindrical void as a function of the cylinder length, calculated from the Tao–Eldrup model for cylindrical geometry and larger void sizes [25], 1=s3 ¼ k3 ¼ kb ðkb k1 ÞP ;
ð2Þ
where kb , k1 are the o-Ps decay rates in the bulk (kb ¼ 2/ ns) and in an infinitely long cylinder, while " X L 1 2pnD sin P¼ þ L þ 2D np L þ 2D n !# h2 n2 exp 2 16me ðL þ 2DÞ kT ! X h2 n2 ; ð3Þ exp 2 16me ðL þ 2DÞ kT n where L, h, me , k, T are: the cylinder length, PlanckÕs constant, electron mass, Boltzmann constant and temperature, respectively. In all estimates given here the parameter D related to the penetration of Ps wavefunction into the bulk is assumed 0.166 nm; that value is well tested for small voids and organic media. According to the model calculations the experimentally found lifetimes correspond to the cylinders 0.32 and 0.40 nm long; their difference is just the same as odd–even difference in d0 spacing; the average displacement is close to that of two-carbon segment (the length of one C–C segment of the molecule is 0.127 nm [21]). The model connecting the pick-off lifetime with the void size used here is rather a rough approximation, thus the numbers given above cannot be taken as a result of precise calculation. Nevertheless, the location of Ps at the irregularities on the surface of interlamellar layer seems to be proved. Greater slope of s3 ðT Þ curve in n-nonadecane could be tentatively explained by the reduction of d0 with lowering the temperature for this
particular alkane, however, there are no crystallographic data which could confirm this supposition. The lifetime s2 of freely annihilating positrons does not experience these even–odd variations, it is determined mainly by the density of the medium varying smoothly along the homologous series. For the three alkanes studied here average s2 in the range 280–290 K is: (318 1) ps, (319 1) ps, (318 1) ps, from nonadecane to heneicosane, i.e., practically constant. Also the p-Ps lifetime s1 in the rigid phase in all our experiments does not show any peculiarities, remaining in the limits 120–140 ps. Transition from the rigid to rotator phase in n-nonadecane occurs at (294.7 0.5) K, in n-heneicosane at (304.2 0.5) K; literature data are 295.2 and 305.6 K, respectively [26]. At the point of phase transition in oddnumbered alkane the lifetime is almost doubled: s3 ¼ ð2:28 0:01Þ ns in n-nonadecane and (2.50 0.01) ns in n-heneicosane. The rotator phase is characterized by an orientational disorder of the molecules around their long axes (the name rotator follows from activation of the motion around that axis, although it is rather a series of jumps by a certain angle than continuous rotation [26]). At the transition to the rotator phase also the intramolecular defects are created [27]. There are several types of such defected molecules (conformers). At low temperature the alkane molecules have a rod-like all-trans configuration; the flexibility of chains does not appear as long as the overall molecule length is below 10 nm (chain length 75–80 carbon atoms [28]). Above the transition point the ‘‘end-gauche’’, ‘‘double-gauche’’ and ‘‘kink’’ conformers appear (Fig. 13); the planar all trans chain can be transformed into non-planar structure. According to the estimates by Maroncelli et al. [27], in nnonadecane the concentration of end-gauche defects is (5–10)%, while of kink defects (10–20)%, in n-heneicosane these concentrations are slightly higher. The kinked structure creates a long empty channel in the lamella allowing o-Ps to live longer. As it is seen from Fig. 12, the lifetime observed at the transition point corresponds to a very long cylinder (length tending to infinity). There are two possibilities:
(b)
(c) (d)
Fig. 13. Structures of the long-chain alkane conformers: from top to bottom: all-trans, end-gauche, kink, double-gauche. Hydrogen atoms are not shown for simplicity.
m. p.
2.5
τ 3, ns
• o-Ps annihilates in the vacancies present in the crystal structure; • the transverse section of free space near the kinked molecule is larger than that occupied by all-trans molecule. Only the second variant is acceptable: in the first one there is no place for further extension of the lifetime with temperature. In the narrow range of temperatures above the solid-rotator transition point both o-Ps parameters, lifetime (Fig. 10) and intensity, rise rather rapidly, which means further distortion of the rotator phase structure. That is supported by crystallographic data; the second moment of electron density distribution (precisely: of electron density deficit in the interlamellar layer) in the rotator phase rises with temperature; the average width of the layer rises too, as the non-planar conformers are shorter than the straight all-trans ones [21]. The growth of the number of intramolecular defects per one carbon chain is also reported by Yesook Kim et al. [29] studying the IR spectra. The change of I3 intensity in the run ‘‘B’’ at the phase transition point is not so distinct as that of lifetime because the effect is obscured by the remnants of electron trapping process, still present below the phase transition point. In particular, in the reversed run ‘‘B’’ (temperature lowering from above the transition point in 1 K steps; 100 min/spectrum), the step of I3 is hard to notice. Only after shortening the spectrum collection time, a dip of I3 can be observed. The fragment of I3 ðT Þ dependence near the rigid-rotator phase transition is shown in Fig. 14 in an extended scale. Quick growth of I3 in time (it was seen earlier that the time constant H is about 1 h) results in creation of a narrow minimum only near 293 K.
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2.0 1.5 1.0 270
280
290
300
310
320
330
340
Fig. 15. Temperature-dependence of o-Ps lifetime s3 in n-eicosane in the region below the melting point (diamonds). A similar dependence near the rigid-rotator transition in n-octacosane [30] is also shown (dots).
An interesting difference in behaviour of s3 at approaching the solid–liquid and solid–rotator transition points can be observed. In n-eicosane s3 gradually rises in the range of about 10 K below the melting point (Fig. 15), while the transitions rigid solid–rotator are not preceded by s3 variation. It seems that in n-eicosane some structural changes appear before the melting. Xray studies can detect the conformers at a relatively large concentration, while positronium is much more sensitive to the presence of free space defects; their concentration at the level 0.1% can produce quite a large o-Ps component [31]. Thus, if the formation of non-planar intramolecular defects begins below the transition point, the lifetime change can be observed. Note that in the next even alkane, C22 H46 , the rotator phase is already formed, barely 1 K below melting [16]. 3.5. Effect of high pressure
80
60
40
20
0
30
I 3, %
28 26 24 22 20 270
280
290
300
310
Fig. 14. I3 intensity near rigid-rotator transition in n-nonadecane observed in ‘‘type B’’ measurements. Full squares – temperature up (initial point 103 K, 100 min/point) and open squares – temperature down (from 297 K, 20 min/point). The figure shows also the pressure dependence of I3 (diamonds and dashed line). Pressure is shown at the top of the figure frame (inverted). The choice of pressure scale is explained in the legend of Fig. 18.
The transition rotator-rigid can be realized also by applying the high pressure. The pressure chamber containing the sample-source sandwich was kept at 300 K in the case of n-nonadecane, and 310 K for n-heneicosane. At these temperatures, before applying the pressure, the samples were in the rotator phase; then, the pressure was increased in steps up to 450 MPa (the time of spectrum collection – 2 h). Transition to the rigid phase at the temperatures indicated above occurs at p 20 MPa, as it could be expected from the measurements by the classic methods [16]. The values of s3 in n-nonadecane and n-heneicosane as a function of pressure are shown in Fig. 16, the intensities I3 in Fig. 17. The s3 ðpÞ dependence looks like mirror image of the curves from Fig. 10. In Fig. 18 a comparison of s3 lifetime dependence on temperature and pressure in a broad range of both factors in n-nonadecane is shown; the abscissa scales are chosen to get 1 K equivalent to 2.25 MPa. The curves look identical in the whole range of temperatures and pressures. Similar comparison of intensities I3 at the
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τ 3, ns
2.4 2.0 1.6 1.2 0.8 0
100
200
300
400
500
Fig. 16. ortho-Ps lifetime s3 in n-nonadecane (open squares) and nheneicosane (dots) as a function of pressure.
I3, %
28
24
20
16 0
100
200
300
400
500
Fig. 17. ortho-Ps intensity I3 in n-nonadecane (open squares) and nheneicosane (dots) as a function of pressure.
2.8
500
400
300
200
100
0
temperature change. For n-nonadecane, the step of I3 at the transition under pressure amounts 7.5%, while in temperature-up measurements at normal pressure it does not exceed 3.5%. In Fig. 14 the experimental data from pressure experiment are added for easier comparison. The measurements described above were performed at the temperature near the melting point, where the electron trapping phenomena are not observed or play a marginal role. Application of pressure produced the effects similar to lowering the temperature (phase transition, decrease of lifetimes and intensity I3 ), thus it seemed worthwhile to check whether high pressure can restrain the escape of electrons from the traps. If so, under high pressure one should expect the reappearance of I3 variation with time. The pressure of 255 MPa was applied to n-heneicosane sample and a sequence of spectra measured in 2-h intervals. Contrary to expectations, the intensity I3 diminished with time (Fig. 19). Similar measurements were repeated at 130 and 380 MPa. The time constant of I3 decrease varied from 3.7 h for 130 MPa to 9.4 h for 380 MPa, i.e., rose with the increase of the pressure. The values of I3 shown in Fig. 17 (the measurement analogous to run ‘‘B’’) are close to asymptotic ones for very long time of storage under pressure. A run equivalent to type ‘‘C’’ was performed with n-heneicosane: after each experimental point the pressure was reduced to zero (return to the rotator phase was immediate) and the sample was kept without pressure 20 min before applying next selected p value. The results for run types ‘‘B’’ and ‘‘C’’ are shown in Fig. 20 (intensity I3 lifetimes s2 and s3 ). The lifetimes s3 are, like after lowering the temperature, independent of the storage time, however,
(a)
2.0
22
1.6
I3, %
τ 3, ns
2.4
1.2 0.8 80
120
160
200
240
280
21 20 19
320
choice of abscissa scale as above reveals a difference: after the transition to rigid phase induced by pressure I3 is lower than that appearing in the experiment with
τ2, ns
18
Fig. 18. n-Nonadecane. Comparison of ortho-Ps lifetimes as a function of temperature and pressure. The scale of abscissa for pressure (top of the figure frame) is chosen to match the phase transition point and to get the same slope of s3 ðT Þ and s3 ðpÞ dependences. Full squares – temperature up; open squares – temperature down and diamonds – pressure.
0.32
(b)
0.31 0
10
20
30
40
Fig. 19. n-Heneicosane. Changes of intensity I3 and lifetime s2 (free positrons) with the time of sample storage at the pressure 255 MPa. Temperature 310 K.
0.34 0.32 0.3
τ3, ns
2.4
h Rmin ¼ pffiffiffiffiffiffiffiffiffiffiffi ; 4 2mV0
2.0
1.6
1.2
28
I 3, %
251
them; they are the same voids. If the reduction of void size occurs, it should refer to all voids localized in the vicinity of vibrating end-methyl groups, however, DI3 reduction is not accompanied by shortening of s3 . One can imagine the situation when the void represents a trap for positronium, but not for electrons. The minimal radius at which the energy level in the spherical well exists is
24
20
16 0
100
200
300
400
Fig. 20. n-Heneicosane. Lifetimes s2 , s3 and intensity I3 as a function of pressure, measured during the run type ‘‘B’’ (diamonds) and type ‘‘C’’ (open circles).
one observes the change of free annihilation lifetime s2 , its rise with the time elapse (Fig. 19(b)). This last effect was not seen in the experiments at low temperatures. The mechanism reducing I3 is unknown to us, it can be rather a relaxation than trapped electron effect. 3.6. Origin of electron traps It is commonly assumed that electron traps are free spaces in the polymer structure. Disappearance of trapped electrons is ascribed to the activation of molecular motions; in particular, methyl group vibrations in the vicinity of voids make the electron density in them to increase, the size of free spaces is reduced; the zero point energy rises and trapping stops to be energetically favourable. The effect of DI3 reduction with temperature can be due to the escape of electrons or to the reduction of Ps formation probability. The decay of trapped electrons is well visible in the ESR measurements in polymers [11], thus the first of the two mentioned variants is more probable. From the viewpoint of the observed positronium properties, the radical reduction of void size by molecular motion seems not easy to accept. Constancy of s3 during irradiation means that the Ps accommodating voids are of the same size for positronium formed using trapped electron stores and without
ð4Þ
where m is the mass of trapped particle and V0 is the depth of the potential energy well. For positronium, the mass is twice of that of an electron, the energy V0 is different for electron and for neutral Ps atom. For certain R the condition (4) can be fulfilled for both particles, but the reduction of void size can lead to trapping Ps only (when R lies between Rmin for Ps and Rmin for electron). Such a condition can be fulfilled rarely, for a narrow range of R, and even it is, one should observe the shortening of o-Ps lifetime when the atomic motions are activated. However, detrapping of electrons is a common phenomenon and s3 change is not observed, thus, the interpretation presented above is not acceptable. The amplitude of vibrations is not so large to change the void dimensions, estimated in Section 3.3, to the extent reducing them to the values below Rmin It seems that at elevated temperatures all the traps still exist, but certain processes causing electron migration become activated. In polyethylene polymer, Keyser et al. [32] determined the activation energies for thermal decay of various groups of trapped electrons by ESR. For decays occurring in the range from 77 to 127 K, the activation energies Ea were from 0.14 to 0.22 eV; the ratio Ea =kT was constant and very high (over 20). In our measurements the biggest step on the I3 ðT Þ curve occurs in the range 200–240 K (it is approximately the same temperature as for the minimum of I3 in polymers [9]). In n-nonadecane, the intensity drops from maximal Imax 44.5% to Imin ¼ 27:5%. If it is the result of
ln[(Imax-I)/(I-Imin)]
τ2, ns
T. Goworek et al. / Chemical Physics 295 (2003) 243–253
1 0 -1 -2 -3 4.2
4.4
4.6
4.8
-3
Fig. 21. n-Nonadecane. Arrhenius plot for the fraction of o-Ps intensity related to the detrapping of electrons at the temperatures 200–240 K.
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T. Goworek et al. / Chemical Physics 295 (2003) 243–253
an activation of electron migration processes one can try to make an Arrhenius plot [33] assuming proportionality of I3 to the density of the trapped electrons. Such a plot, Imax I H ¼ A exp ; ð5Þ I Imin kT is shown in Fig. 21. The activation enthalpy H is found 0.6 eV; H =kT is similar to that found for polymers.
4. Conclusions The rise of o-Ps component with irradiation time, well-known in polymers, appears also in long-chain alkanes, at least in those with n 20. The effect is large, leading sometimes to doubling the intensity observed in non-irradiated samples. A simple structure of the medium and well-defined defects make alkanes a good object to study the formation of positronium in the presence of trapped electrons. At our source activity the time constant H of I3 rise at 103 K is of the order of a day. With the increase of storage temperature, the even numbered n-eicosane behaves totally differently than odd numbered neighbours: the time constant in n-eicosane rises with temperature (like in polypropylene and other polymers), while in odd numbered it decreases down to few hours. It is to be checked whether such divergence continues through the whole homologous series of paraffins. The difference in properties of even and odd-numbered alkanes is known since 1877 [34] and reveals in various forms: popular matrices for molecular sieves – alkyltrimethyl bromides – exist with even alkyls only, one observes preferential abundance of odd chain biogenic alkanes [35] and preferential breaking into evennumbered fragments during combustion of paraffins as a rocket fuel [36], etc. The odd–even difference of o-Ps lifetimes is easily explainable by the crystallographic structure, but the divergence of time constants of I3 rise by almost two orders of magnitude remains difficult to explain. The radiation-chemical studies of Ps formation on pre-trapped electrons are in their initial stage and for the time being the set of time constants H determined here supplements the stores of data only. In odd numbered alkanes, between 230 and 280 K, Itrap is of the order of 4%, but the time constant of rise is 1 h. Very short time constants pose the problem whether the nature of I3 rise in this temperature range is the same as at low temperatures, or it is a different process. The yield of trapped electrons Gðe Þ does not depend strongly on temperature, also Ps formation probability changes slowly with temperature, as can be seen in Figs. 2, 6 and 7 (runs C), thus, the irradiation effects seem to be not sufficient to explain such fast changes. The shift of s3 ðT Þ curve for odd-numbered alkanes against the even ones allows us to localize the positro-
nium trapping centers in the vicinity of the interlamellar void (gap), swollen locally owing to longitudinal molecule displacements. The dependences of the o-Ps lifetime on temperature and pressure are similar. Variation of I3 is observed also after application of high pressure, but the direction of changes is opposite to that at low temperatures; it decreases by several percent with time constant 4–8 h.
Acknowledgements This work was supported by the Polish Committee for Research (KBN) Grant No. 5P03B 031 21.
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