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Mechanism of positronium formation in liquid hydrocarbons and electrons solvation
Radiat. Phys. Chem. 1978. Vol. 11, pp. 145--149. Pergamon Press.
Printed in Great Britain.
M E C H A N I S M OF POSITRONIUM FORMATION IN LIQUID HYDROCARBONS AND E L E C T R O N S SOLVATION V. L. BUGAENKO, V. M. BYAKOV, V. I. GRAFUTIN, O. V. KOLDAEVA and E. V. MINAICHEV Institute of Theoretical and Experimental Physics. Moscow, USSR (Manuscript received 14 January 1977; received [or publication 10 October 1977) AMtract--It is demonstrated that addition of alcohols to liquid hydrocarbons leads to a decrease in the probability of positronium formation. The effective concentration region coincides with that of electron solvation due to alcohol polymer formation. These results support the model in which positronium is formed by recombination of positron and non-solvated electron. INTRODUCTION UNTIL RECENTLY the only model for positronium (Ps) formation was the Ore model. In this it is assumed that the positron, during its slowingdown, may form positronium by abstracting an electron from one of the surrounding molecules. More recently, various versions of a recombinational mechanism of positronium formation have been discussed. According to this mechanism, positronium is formed at the end of a positron track by recombination with one of the ejected electrons. These models are discussed in ref. (1). In earlier articles "-~ it was proposed that in polar liquids (water, alcohols) positronium is formed by recombination of the positron with "presolvated ''~s' electrons. The aim of this paper is to demonstrate that in non-polar liquids (hydrocarbons) positronium formation occurs in the same manner, by recombination of positrons with unsolvated electrons. We have investigated the influence of alcohol concentration on the positronium formation probability, P, in some hydrocarbons. It is known that alcohol molecules are not electron acceptors, but small concentrations in hydrocarbons lead to very effective electron solvation. If positronium formation in hydrocarbons proceeds by recombination of a positron with the unsolvated electron, the probability of positronium formation should vary inversely with the yield of solvated electrons. EXPERIMENTAL Compounds used in this investigation were of the highest available purity. Solutions were deaerated by freezing and pumping. Positron lifetime measurements were carried out by the usual delayed coincidence method. Experimental details have been described earlier) 6~
RESULTS AND DISCUSSION Measured probabilities P([ROH]) in hydrocarbon + alcohol mixtures (n-hexane + methanol, n-hexane + ethanol, n-hexane + n-propanol, iso-octane + n-propanol) relative to Po, the probability in pure hydrocarbon, are presented in Figs 1--4. If the same data are shown as functions of static dielectric const.ant ~ of the mixture (Fig. 5), it can be seen that P begins to decrease at alcohol concentrations which increase ~ above the value 1.8 for pure hydrocarbons. This figure also demonstrates that variations of solvated electron yield and positronium formation probability occur in the same concentration region. This suggests that t h e probability of positronium formation is determined by the effectiveness of the electron solvation process. On the other hand,
,
I
I
I
I
1
0.1
0.2
0.3
0.4
0.5
4r--
I.Oq
0.9
O~ 0.8 0. 0.7
0.6
0..'
Mole
fraction
.0
alcohol
FIG. !. Relative positronium formation probability in nhe×one-methanol mixture at 298 K. 145
t
I
1
I
1.0¢
q,-
i 1.0'1
I: T - 2 9 B K 2:. T - 2 4 8 K
0.9
0.9~
0.8~
o0.8 O. O. 0.7
0.7
--
O.E
0.6
--
0.5
I
T
0.1
0.2
~t
t
0.3
Mole
0.4
fraction
o!5 o
t
i
I
0.~
0.2
Mole
alcoi~ol
FIG. 2. Relative positronium formation probability in nhexane--ethanol mixtures: (I) T = 298 K; (2) T = 248 K.
t
I
0.5
t
I
I
I
4-
0.3
0.4
0.5
froc?ion
'1~ I.O
alcohol
FIG. 4. Relative positronium formation probability in 2,2,4-trimethylpentane-n-propanol mixture at 298 K.
~.oo
~"
I
I
_
I
t
I
.oo
1.0 (
0.75 - -
0.75 %
o.g
2
•~ 0.
0.50
3.50
oz
#
0,0
0.25
3.25
0.7
0
5
0.6 C
I
o.~ 0
I0 Static
O.t
I 0.2
Mole
l 0.3
fract'ion
0.4
0.5
alcohol
20
25
FIG. 5. Probability P ( ~ ) of positronium formation relative to that (P0) in pure n-hexan¢ and yield Ge-,ol (Q)m~ of solvated electrons relative to that (Go) in pure ethanol; as a function of static dielectric constant for n-hexane-
ethanol mixture at 298 K.
FIG. 3. Relative positronium formation probability in nhexane-n-propanol mixtures at 298 K.
there is no connection with the decrease in electron mobility. The latter takes place mainly in another alcohol concentration i'egion where P = Po (Fig. 6). This fact will be discussed below. Comparison of data from Figs 3 and 4 for npropanol in n-hexane and iso-octane shows that differences in the structure of hydrocarbons, and perhaps differences in mobility/z= of electrons or in the yield Ga of free ions, influence slightly the form of P-dependences in alcohol solutions. Let us consider in more detail the P-dependences on alcohol concentration. In all curves presented in Figs 1-4, three parts may be distinguished: (i) the region of comparatively small
t5
dielectric c o n s t a n t
,.c ~
~
~
,.oo.°
O.fl
3.8 ~.
o 0.6 ::L
-- 3.6
0.2
- - 0.2 ( ~
1
0
I
0.05
Mole
fraction
O.
alcot~ol
FIG. 6. Relative values of electron mobility ~/~.o (I-1)~:4~
PIPo (~) and G=-,o,IGo (O) (see Fig. 5) in n-hexaneethanol
m i x t u r e at 298 K v e r s u s
mole fraction alcohol.
147
Mechanism of positronium formation in liquid hydrocarbons and electrons solvation concentrations (-< several mole %), in which P slightly increases or remains constant: (ii) the region of moderate concentrations, which is characterized by a sharp decrease in P ; and (iii) the region of high concentration where P is insignificantly different from the value in pure alcohol. It may be noticed that similar P-dependences were found by P. Jansen et alJ 9~ for mixtures of pyrrolidine and n-heptane. The explanation proposed for the increase in P at small alcohol concentrations is as follows. According to the model, "-7) at the end of the positron track, the recombination proceeds: e÷ + e-..+ p s .
It competes, for example in the case of hexane, with the reaction e- + C~H ~'.(-")C6H*,. In the presence of an alcohol whose proton affinity (8.1-8.2eV "L~) is somewhat higher than that of the hydrocarbons, proton transfer from the primary cation is possible. This process leads to the formation of solvated hydrogen ions:* C 6 H h + C 2 H s O H -~ C,_HsOH2 ÷ + C~I-I,3.
As the rate constants of dry electron with protons solvated in water and alcohols are rather small, (7'8't3) this process inhibits the ion-electron recombination and thereby leads to an increase in P. A similar effect (and of the same order of magnitude) is observed in water when about 1 M ammonium or urea is added, c5) The explanation of the sharp decrease in P, and increase in solvated electron yield in the alcohol concentration region --- 10 mole%, is based on the fact that alcohols disolved in hydrocarbons are not present simply as monomers, but form small groups (associates or polymers) of several molecules. The model of the hydrated electron ct4) predicts that such polymers with n ~>4, may serve as appropriate traps for electron solvation. Partial fractions of different polymers in ,-heptane as functions of the ethanol mote fraction c=~ are shown in Fig. 7. Similar values of polymer fractions (n > 2) are found in some other papers (see for example Ref. (10)). Comparison with our positronium formation probability data shows *Kroh and Piekarska "2) observed that the addition of 0.1 Methanol in n-hexane and in iso-octane increases the yield of positive ions escaping intra-track recombination several times.
a.0 o.8 ,~ c~ ", 0.4
o.z 0
0.005 0.01
O.OZ~ O.OS
0.1
O.ZS
0.5
1.0
X=
FIG. 7. Partial mole fraction X,/Xa of polymer ethanol associates in heptane versus mote fraction )CA of ethanol. (zS)
that the decrease of P and increase of solvated electron yield occurs in the alcohol concentration region where an increase in alcohol polymers (n > 2) takes place. In accordance with the idea of polymers as electron traps, the data (Fig. 2) demonstrating the influence of temperature on positronium formation probability show that a decrease of 50°C markedly increases the ability of alcohol to inhibit the positronium formation. This result is obviously a consequence of the increase of alcohol polymers at lower temperatures. The subsequent small variation in P in the alcohol concentration region 20-100 mole % is possibly caused by the weak influence of hydrocarbon molecules on the alcohol structure ('5) which gives a slight variation in the electron solvation time."6) According to the above interpretation of the experimental data, the sharp increase in P from alcohol mole fraction 40.2 to 40.05 indicates that at <0.1, the positronium formation time (the recombination time of e* and e-) is less than the electron solvation time, and at a mole fraction ->0.2, the two times are comparable. Chase and Hunt ~'~) found that the electron solvation time in pure alcohols (methanol, ethanol, ,-propanol) was - 1 0 - " s. In dilute solutions of these alcohols in hydrocarbons, this time will probably increase, due to the time required for the encounter of an electron with the alcohol polymers. The positronium formation time in methanol, ethanol and n-propanol has been estimated to be 10-'z-10- " s. cs' The positron-electron recombination time in a hydrocarbon may be estimated using data on the influence of an external electric field on the positronium formation probability in
148
V . L . BUGAENKOet al.
hydrocarbons. "s~ The marked effect of fields in which the electron gains energy - k T travelling - 1 0 0 ~ , , indicates that in the recombinational mechanism of positronium formation, thermal electrons take part, the average distance R between recombining positron and electron being comparable with the Onsager radius Ro = (eZlekT) (= 300 ,~ for n-hexane and iso-octane). If, in accordance with refs (19, 20, 26), the diffusion coefficient D of quasi-free electrons in hydrocarbons =2 c m ' s , using the drift approximation, the electron-positron recombination time is: R.
R: = R
Ro 6D
0.8
~.~ 0.6 d 0,
0.2
= 10-1: s,
6D
0.2
0.a
0.6
0.8
L0
X,/(X~*X~)
which is consistent with the foregoing interpretation of the experimental data. In this connection it may be noticed that the smaller effect of n-propanol in iso-octane than that in n-hexane, may be explained by the suggestion that the positronium formation time is shorter because of the higher mobility of quasi-free electrons.(26~* The value of the positronium formation time obtained indicates that the absence of connection between values of P, G(e,) on the one hand and those of electron mobility (demonstrated in Fig. 6) on the other hand, is due to the fact that the time of positronium formation is much shorter than the equilibrium time between quasi-free and solvated electrons. If this is the case, the dependence of positronium yield on alcohol concentration reflects processes in the picosecond time scale, whereas the decrease of electron mobility is connected with processes occurring in nanoseconds. Let C~ and C2 be the volume averaged concentrations of quasi-free and solvated electrons. Then if at t = 0 (terminal time of recombination processes in spurs) C~ = Ct °, C2 = 0, we have: (1)
C,(t)
c, 0
=
e_(,~÷,,),+ -
As ~x - - ~
[1
- -
e-~,,.,,,] .
.
Here At and As are rates of solvation and desolvation of electrons. The quasi-free electron frac*We have not discussed positron solvation in alcohols?TM The role of this process in positronium formation in mixture of liquids has been noticed, tz:~However, there are no quantitative data which allow the idea to be applied here. ~'After this investigation was finished we learned about analogous experiments of O. Mogensen, reported to the Fourth International Conference on Positron Annihilation (Helsing~r, Denmark, 23-26 August 1976).
FIG. 8. The quasi-free electron fraction Cl(t)/Ci ° in function of XJ(X~+ Xz) at various times. tions C,(t)/C, ° as functions of A,/CA,+A_-) are demonstrated for various times in Fig. 8. It may be seen from this figure and from (1) that at earlier times, higher values of X,/(;tt+Xs) are needed to drop the electron mobility, and therefore higher alcohol concentrations, if one supposes that A~ increases with increasing alcohol concentration. Thus the data of Fig. 6 may be considered as another argument in favour of the recombination mechanism of Ps formation discussed here, with the participation of unsolvated electrons. In addition, the above discussion shows that the data can give useful information about the properties of these liquid mixtures and particularly about the mechanism of solvation of electrons in them.f Acknowledgement--The authors are deeply grateful to Professor J. H. Baxendale for his critical remarks and kind help in preparing the manuscript.
REFERENCES 1. O. E. MOGENSEN,J. chem. Phys. 1974: 60. 998; H. J. ACHE, E. R. WILDand L. J. BARTAL.Proceedings of the 4th Symposium on Radaition Chemistry, Keszthely (Hungary), 1-5 June 1976. 2. L. T. BUGAENKO,V. M. BVAKOVand E. P. KAtJAZtN. Symposium on Radiation Chemistry o[ Aqueous Systems (Abstracts), Nauka, Moscow, 1973, p. 5. 3. V. M. BYAKov, Int. J. Radiat. Phys. Chem. 1976, 8, 238. 4. V. M. BYAKOV,V. I. GOLDANSKIIand V. P. SHASTAROVlCH,Doklady A N USSR, 1974, 219. 633. 5. V. M. BYAKov, V. I. GRAFo'rIN and O. V. KOLDAEVA,Radiat. Phys. Chem. 1977. 10, 239. 6. V. M. BYAKOV.V. I. Gi~AFUTIN,E. V. /~NAEHEV, F. G. NEmr,oRov, Yu. V. OBUKHOVand O. P. SrEPANOVA,Khim. Vys. Energ. 1977, 11,323.
Mechanism of positronium formation in liquid hydrocarbons and electrons solvation 7. V. M. BYAKOV,V, L GRAFUTIN, O. V. KOLD.-~EVA,E. V. MINAICHEV, F. G. NICHIPOROV, YU. V. OBUKHOV and O. P. STEPANOVA,Chem. Phys. 1977, 2-1, 91. 8. G. R. FREEMAN, Proceedings of the 4th Synposium on Radiation Chemistry, Keszthely (Hungary), I-5 June 1976. 9. P. JANSEN, M. ELDRUP, O. E. MOGENSEN and P. PAGSBERG, J. chem. Phys. 1974, 6, 265. 10. A. N. FLETCHER, J. phys. Chem. 1972, 76, 2562. II. L. V. GURVlCH, G. V. KARACHEVTSEV, V. N. KONDRATJER, YU. A. LEBEDEV, V. A. MEDVEDEV, V. K. POTAPOV and Yu. S. KHODEEV, Energii razryva himicheskih svyazei. Potenciaty ionizacii i srodstvo k electronu, Nauka, Moscov, 1974. 12. J. KROH and J. PIEKARSKA,Bull. Acad. polon. Sci., Ser. ScL Chem. 1974, 22, 167. 13. K. Y. LAM and J. W. HUNT, Int. J. Radiat. Phys. Chem. 1975, 7, 317. 14. V. M. BYAKOV, B. S. KIJACHKO and A. A. OVCHINNIKOV, Conligurational model of hydrated electron, Preprint ITEP-27, 1973; ITEP-25, 1974; ITEP-4, 1976; Moscow. 15. C.B. KRISHNAN and H. L. FRIEDMAN,J. phys Chem. 1971, 75, 3598.
149
16. J. A. KENNEY-WALLACEand S. D. JONACH. Chem. Phys. Lett. 1976, 39. 596. 17. W. J. CHASE and J. W. HUNT, J. phys. Chem. 1976. 79, 2835. 18. O. A. ANISXMOV, A. M. RAITSIMRING and Yu. N. MOLIN, Zh. Eksp. Theor. Fiz. (Letters) 1975, 22, 197. 19. J.-P. DODELET, X. SHIMAKA and G. R. FREEMAN. Can. J. Chem. 1976, $4, 744. 20. L. NYLKOS. E. ZADER and R. SCHILLER, Proceedings of the 4th International Symposium on Radiation Chemistry. Keszthely (Hungary), 1-5 June 1976. 21. O. E. MOGENSEN and V. P. SHANTAROVICH, Chem. Phys. 1974, 6, I00. 22. R. E. WILD and H. J. ACHE, J. chem. Phys. 1976, 65, 247. 23. B. J. BROWN, N. T. BARKER and D. F. SANGSTER, Aust. J. Chem. 1973, 26, 2089. 24. G. BECK and J. K. THOMAS, J. chem. Phys. 1972, 57, 3649. 25. H. C. VAN NESS, TON VAN WINKLE, H. H. RICHTOL and H. B. I"IOLLINGERf. phys. Chem. 1967, 71, 1483. 26. A. O. ALLEN and R. A. HOLROYD, J. phys. Chem. 1974, 78, 796.
Printed in Great Britain.
M E C H A N I S M OF POSITRONIUM FORMATION IN LIQUID HYDROCARBONS AND E L E C T R O N S SOLVATION V. L. BUGAENKO, V. M. BYAKOV, V. I. GRAFUTIN, O. V. KOLDAEVA and E. V. MINAICHEV Institute of Theoretical and Experimental Physics. Moscow, USSR (Manuscript received 14 January 1977; received [or publication 10 October 1977) AMtract--It is demonstrated that addition of alcohols to liquid hydrocarbons leads to a decrease in the probability of positronium formation. The effective concentration region coincides with that of electron solvation due to alcohol polymer formation. These results support the model in which positronium is formed by recombination of positron and non-solvated electron. INTRODUCTION UNTIL RECENTLY the only model for positronium (Ps) formation was the Ore model. In this it is assumed that the positron, during its slowingdown, may form positronium by abstracting an electron from one of the surrounding molecules. More recently, various versions of a recombinational mechanism of positronium formation have been discussed. According to this mechanism, positronium is formed at the end of a positron track by recombination with one of the ejected electrons. These models are discussed in ref. (1). In earlier articles "-~ it was proposed that in polar liquids (water, alcohols) positronium is formed by recombination of the positron with "presolvated ''~s' electrons. The aim of this paper is to demonstrate that in non-polar liquids (hydrocarbons) positronium formation occurs in the same manner, by recombination of positrons with unsolvated electrons. We have investigated the influence of alcohol concentration on the positronium formation probability, P, in some hydrocarbons. It is known that alcohol molecules are not electron acceptors, but small concentrations in hydrocarbons lead to very effective electron solvation. If positronium formation in hydrocarbons proceeds by recombination of a positron with the unsolvated electron, the probability of positronium formation should vary inversely with the yield of solvated electrons. EXPERIMENTAL Compounds used in this investigation were of the highest available purity. Solutions were deaerated by freezing and pumping. Positron lifetime measurements were carried out by the usual delayed coincidence method. Experimental details have been described earlier) 6~
RESULTS AND DISCUSSION Measured probabilities P([ROH]) in hydrocarbon + alcohol mixtures (n-hexane + methanol, n-hexane + ethanol, n-hexane + n-propanol, iso-octane + n-propanol) relative to Po, the probability in pure hydrocarbon, are presented in Figs 1--4. If the same data are shown as functions of static dielectric const.ant ~ of the mixture (Fig. 5), it can be seen that P begins to decrease at alcohol concentrations which increase ~ above the value 1.8 for pure hydrocarbons. This figure also demonstrates that variations of solvated electron yield and positronium formation probability occur in the same concentration region. This suggests that t h e probability of positronium formation is determined by the effectiveness of the electron solvation process. On the other hand,
,
I
I
I
I
1
0.1
0.2
0.3
0.4
0.5
4r--
I.Oq
0.9
O~ 0.8 0. 0.7
0.6
0..'
Mole
fraction
.0
alcohol
FIG. !. Relative positronium formation probability in nhe×one-methanol mixture at 298 K. 145
t
I
1
I
1.0¢
q,-
i 1.0'1
I: T - 2 9 B K 2:. T - 2 4 8 K
0.9
0.9~
0.8~
o0.8 O. O. 0.7
0.7
--
O.E
0.6
--
0.5
I
T
0.1
0.2
~t
t
0.3
Mole
0.4
fraction
o!5 o
t
i
I
0.~
0.2
Mole
alcoi~ol
FIG. 2. Relative positronium formation probability in nhexane--ethanol mixtures: (I) T = 298 K; (2) T = 248 K.
t
I
0.5
t
I
I
I
4-
0.3
0.4
0.5
froc?ion
'1~ I.O
alcohol
FIG. 4. Relative positronium formation probability in 2,2,4-trimethylpentane-n-propanol mixture at 298 K.
~.oo
~"
I
I
_
I
t
I
.oo
1.0 (
0.75 - -
0.75 %
o.g
2
•~ 0.
0.50
3.50
oz
#
0,0
0.25
3.25
0.7
0
5
0.6 C
I
o.~ 0
I0 Static
O.t
I 0.2
Mole
l 0.3
fract'ion
0.4
0.5
alcohol
20
25
FIG. 5. Probability P ( ~ ) of positronium formation relative to that (P0) in pure n-hexan¢ and yield Ge-,ol (Q)m~ of solvated electrons relative to that (Go) in pure ethanol; as a function of static dielectric constant for n-hexane-
ethanol mixture at 298 K.
FIG. 3. Relative positronium formation probability in nhexane-n-propanol mixtures at 298 K.
there is no connection with the decrease in electron mobility. The latter takes place mainly in another alcohol concentration i'egion where P = Po (Fig. 6). This fact will be discussed below. Comparison of data from Figs 3 and 4 for npropanol in n-hexane and iso-octane shows that differences in the structure of hydrocarbons, and perhaps differences in mobility/z= of electrons or in the yield Ga of free ions, influence slightly the form of P-dependences in alcohol solutions. Let us consider in more detail the P-dependences on alcohol concentration. In all curves presented in Figs 1-4, three parts may be distinguished: (i) the region of comparatively small
t5
dielectric c o n s t a n t
,.c ~
~
~
,.oo.°
O.fl
3.8 ~.
o 0.6 ::L
-- 3.6
0.2
- - 0.2 ( ~
1
0
I
0.05
Mole
fraction
O.
alcot~ol
FIG. 6. Relative values of electron mobility ~/~.o (I-1)~:4~
PIPo (~) and G=-,o,IGo (O) (see Fig. 5) in n-hexaneethanol
m i x t u r e at 298 K v e r s u s
mole fraction alcohol.
147
Mechanism of positronium formation in liquid hydrocarbons and electrons solvation concentrations (-< several mole %), in which P slightly increases or remains constant: (ii) the region of moderate concentrations, which is characterized by a sharp decrease in P ; and (iii) the region of high concentration where P is insignificantly different from the value in pure alcohol. It may be noticed that similar P-dependences were found by P. Jansen et alJ 9~ for mixtures of pyrrolidine and n-heptane. The explanation proposed for the increase in P at small alcohol concentrations is as follows. According to the model, "-7) at the end of the positron track, the recombination proceeds: e÷ + e-..+ p s .
It competes, for example in the case of hexane, with the reaction e- + C~H ~'.(-")C6H*,. In the presence of an alcohol whose proton affinity (8.1-8.2eV "L~) is somewhat higher than that of the hydrocarbons, proton transfer from the primary cation is possible. This process leads to the formation of solvated hydrogen ions:* C 6 H h + C 2 H s O H -~ C,_HsOH2 ÷ + C~I-I,3.
As the rate constants of dry electron with protons solvated in water and alcohols are rather small, (7'8't3) this process inhibits the ion-electron recombination and thereby leads to an increase in P. A similar effect (and of the same order of magnitude) is observed in water when about 1 M ammonium or urea is added, c5) The explanation of the sharp decrease in P, and increase in solvated electron yield in the alcohol concentration region --- 10 mole%, is based on the fact that alcohols disolved in hydrocarbons are not present simply as monomers, but form small groups (associates or polymers) of several molecules. The model of the hydrated electron ct4) predicts that such polymers with n ~>4, may serve as appropriate traps for electron solvation. Partial fractions of different polymers in ,-heptane as functions of the ethanol mote fraction c=~ are shown in Fig. 7. Similar values of polymer fractions (n > 2) are found in some other papers (see for example Ref. (10)). Comparison with our positronium formation probability data shows *Kroh and Piekarska "2) observed that the addition of 0.1 Methanol in n-hexane and in iso-octane increases the yield of positive ions escaping intra-track recombination several times.
a.0 o.8 ,~ c~ ", 0.4
o.z 0
0.005 0.01
O.OZ~ O.OS
0.1
O.ZS
0.5
1.0
X=
FIG. 7. Partial mole fraction X,/Xa of polymer ethanol associates in heptane versus mote fraction )CA of ethanol. (zS)
that the decrease of P and increase of solvated electron yield occurs in the alcohol concentration region where an increase in alcohol polymers (n > 2) takes place. In accordance with the idea of polymers as electron traps, the data (Fig. 2) demonstrating the influence of temperature on positronium formation probability show that a decrease of 50°C markedly increases the ability of alcohol to inhibit the positronium formation. This result is obviously a consequence of the increase of alcohol polymers at lower temperatures. The subsequent small variation in P in the alcohol concentration region 20-100 mole % is possibly caused by the weak influence of hydrocarbon molecules on the alcohol structure ('5) which gives a slight variation in the electron solvation time."6) According to the above interpretation of the experimental data, the sharp increase in P from alcohol mole fraction 40.2 to 40.05 indicates that at <0.1, the positronium formation time (the recombination time of e* and e-) is less than the electron solvation time, and at a mole fraction ->0.2, the two times are comparable. Chase and Hunt ~'~) found that the electron solvation time in pure alcohols (methanol, ethanol, ,-propanol) was - 1 0 - " s. In dilute solutions of these alcohols in hydrocarbons, this time will probably increase, due to the time required for the encounter of an electron with the alcohol polymers. The positronium formation time in methanol, ethanol and n-propanol has been estimated to be 10-'z-10- " s. cs' The positron-electron recombination time in a hydrocarbon may be estimated using data on the influence of an external electric field on the positronium formation probability in
148
V . L . BUGAENKOet al.
hydrocarbons. "s~ The marked effect of fields in which the electron gains energy - k T travelling - 1 0 0 ~ , , indicates that in the recombinational mechanism of positronium formation, thermal electrons take part, the average distance R between recombining positron and electron being comparable with the Onsager radius Ro = (eZlekT) (= 300 ,~ for n-hexane and iso-octane). If, in accordance with refs (19, 20, 26), the diffusion coefficient D of quasi-free electrons in hydrocarbons =2 c m ' s , using the drift approximation, the electron-positron recombination time is: R.
R: = R
Ro 6D
0.8
~.~ 0.6 d 0,
0.2
= 10-1: s,
6D
0.2
0.a
0.6
0.8
L0
X,/(X~*X~)
which is consistent with the foregoing interpretation of the experimental data. In this connection it may be noticed that the smaller effect of n-propanol in iso-octane than that in n-hexane, may be explained by the suggestion that the positronium formation time is shorter because of the higher mobility of quasi-free electrons.(26~* The value of the positronium formation time obtained indicates that the absence of connection between values of P, G(e,) on the one hand and those of electron mobility (demonstrated in Fig. 6) on the other hand, is due to the fact that the time of positronium formation is much shorter than the equilibrium time between quasi-free and solvated electrons. If this is the case, the dependence of positronium yield on alcohol concentration reflects processes in the picosecond time scale, whereas the decrease of electron mobility is connected with processes occurring in nanoseconds. Let C~ and C2 be the volume averaged concentrations of quasi-free and solvated electrons. Then if at t = 0 (terminal time of recombination processes in spurs) C~ = Ct °, C2 = 0, we have: (1)
C,(t)
c, 0
=
e_(,~÷,,),+ -
As ~x - - ~
[1
- -
e-~,,.,,,] .
.
Here At and As are rates of solvation and desolvation of electrons. The quasi-free electron frac*We have not discussed positron solvation in alcohols?TM The role of this process in positronium formation in mixture of liquids has been noticed, tz:~However, there are no quantitative data which allow the idea to be applied here. ~'After this investigation was finished we learned about analogous experiments of O. Mogensen, reported to the Fourth International Conference on Positron Annihilation (Helsing~r, Denmark, 23-26 August 1976).
FIG. 8. The quasi-free electron fraction Cl(t)/Ci ° in function of XJ(X~+ Xz) at various times. tions C,(t)/C, ° as functions of A,/CA,+A_-) are demonstrated for various times in Fig. 8. It may be seen from this figure and from (1) that at earlier times, higher values of X,/(;tt+Xs) are needed to drop the electron mobility, and therefore higher alcohol concentrations, if one supposes that A~ increases with increasing alcohol concentration. Thus the data of Fig. 6 may be considered as another argument in favour of the recombination mechanism of Ps formation discussed here, with the participation of unsolvated electrons. In addition, the above discussion shows that the data can give useful information about the properties of these liquid mixtures and particularly about the mechanism of solvation of electrons in them.f Acknowledgement--The authors are deeply grateful to Professor J. H. Baxendale for his critical remarks and kind help in preparing the manuscript.
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