6557
J. Phys. Chem. 1995,99, 6557-6562
The Quasi-Ballistic Model of Electron Mobility in Liquid Hydrocarbons: Application to Electron Scavenging A. Mozumder Radiation Laboratory' and the Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, Indiana 46556 Received: October 24, 1994; In Final Form: February 7, 1995@
In the two-state, quasi-ballistic model electron scavenging occurs in the quasi-free state with a specific rate k:, related to the overall effective rate kFff by ki = kiff(ktf kft)/ktf, where kft and ktf are respectively the rates of trapping and detrapping. kEff is compared with experimental values for five common scavengers in seven liquid hydrocarbons, and the reaction probability of the final chemical step is determined by an analysis based on k:. In general, both kf and the reaction probability decrease with mobility for all scavengers, partly due to long mean free path of the electron in the quasi-free state and partly due to the inefficiency of reaction. Reaction radius is estimated to be -1 nm for most scavengers. Few reactions are found to be diffusioncontrolled. Possible exceptions are low-mobility liquids (e.g., n-hexane) in which almost all scavenging reactions, except for biphenyl, appear to be diffusion-controlled.
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Introduction One of the important classes of reactions of the excess electron in hydrocarbon liquids is with a homogeneously distributed solute, generally called the electron scavenging reaction. In earlier studies only relative rates were determined while in certain cases ingenious methods were designed to arrive at absolute rates.'-3 Since the 1970s absolute rates of electron scavenging have been routinely measured using the techniques of pulse radiolysis, and a compendium of such rates now exists: The nature of scavenging reactions may be attachment, dissociative, or nondissociative processes. While for some scavengers such as aromatic hydrocarbons or COZthe reaction may be reversible, here we are dealing with only relatively fast, irreversible reactions leading to the disappearance of excess electrons. Measured specific rates of such reactions usually vary in the range from -lolo to -lOI4 M-' s - I . ~ - * By an analysis of the experimentally measured rate using a realistic theoretical model, it should be possible to investigate its dependence on the electron mobility, the energy at the bottom of the conduction band (VO),etc. In a specific case it would also be possible to discern whether the reaction is transport-controlled or not. These are the aims of the present paper. The logical necessity of a viable electron transport model in describing the reactions of the excess electron has long been r e c o g n i ~ e d . ~ - However, '~ a general agreement seems to be lacking among the experimentalists. The two-state model of electron transport, according to which the excess electron is found in the trapped or in the quasi-free state with definite probabilities, was introduced to explain the magnitude and temperature dependence of electron mobility in low-mobility liquid hydrocarbon^.'^-'^ According to Allen et al.,5.6 both electron transport and reaction may be attributed to the quasifree state although the electron resides most of the time in the trapped state for low-mobility liquids. Yakovlev et al.,'O.'' on the other hand, favor electron reaction in the trapped state while
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The work described herein was supported by the Office Basic Sciences of the United States Department of Energy. This is contribution No. NDRL3771 from the Notre Dame Radiation Laboratory. 'Abstract published in Advance ACS Absrracrs, April 1, 1995.
they do not entirely rule out reaction in the quasi-free state. Recently, we have introduced a modification of the two-state model, called the quasi-ballistic modelI5 in which the electron motion in the quasi-free state is conceived to be partly diffusive and partly ballistic, being determined by a competition between the processes of velocity randomization and trapping. The effective mobility in this model is given by
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where (u)r = (e/m). (ktf/kft)/(kft ktf) is the ballistic mobility, and (u)~ = ,u,fktf/(kft ktf) is the usual trap-controlled mobility. Here p,f is the mobility in the quasi-free state, and kft and ktf are respectively the rates of trapping and detrapping. Ballistic mobility originates only from random cycles of trapping and detrapping irrespective of the mobility in the quasi-free state. The usual trap-controlled mobility is proportional to the probability Pf of finding the electron in the quasi-free state given by Pf = ktf/(kft ktf). Given p,f, ktf, and kft, the effective mobility can be calculated using eq 1. The ratio kft/ktfis given by Ascarelli and BrownI6 from a detailed balancing consideration as
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kfJktf = nth3(2nmkBT)-3'2 exp(6dkJ')
(2)
where nt is the trap density, EO is the binding energy in the trap, and other symbols have usual significances. MozumderI5 uses a random walk model with a harmonic potential, giving, together with eq 2,
k,,= (Edh) exp(-cdk,T);
k, = n,h2~o(2nmkBT)3-3'2 (3)
For a vast majority of liquid hydrocarbons at or near room temperature, it has been found that p,f * 100 cm2 V-I s-l and nt E l O I 9 cm-3 give satisfactory agreement between calculated and measured electron m0bi1ity.I~ In this paper we proceed on the premise that the electron scavenging reaction occurs mainly in the quasi-free state although, except for very high-mobility liquids, the electron residues mostly in the trapped state. To us this seems to be the
Q022-3654/95/2099-6557$09.OQ/O 0 1995 American Chemical Society
6558 J. Phys. Chem., Vol. 99, No. 17, 1995
Mozumder
qualitatively correct picture for low- and immediate-mobility liquids @,ff < 10 cm2 V-' s-I) and not necessarily incorrect for high-mobility liquids. Since the transport in the quasi-free state occurs with long mean free path consequent upon high quasifree mobility, the diffusion-controlled rate requires a significant correction for the mean free path effect. Such corrections for electron-ion recombination rate in high-mobility liquids have already been made and consistently compared with experiment.'*.I9 Similar correction for the scavening reaction, which generally does not involve a long-range force, is indicated in the next section where the theory of scavenging rate in the quasiballistic model is developed. In the following sections we compare our results with experiments for five common scavengers (cc14, C2H5Br, biphenyl, SF6, and N20) in seven hydrocarbon liquids and draw conclusions regarding the efficiency of the final step of the reaction. N
The Scavenging Rate Denoting the probabilities of finding the electron in the quasifree, trapped, or scavenged state respectively by nf,nt, and %, the kinetics of scavenging may be described, with nf nl n, = 1, as follows.
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kf = -(As
+ kfl)nf + k g t
(4)
and At = --ktFt
+ kftltf
(5)
In eqs 4 and 5 dots represent time derivatives and As = g c s , where c, is the scavenger concentration and k: is the specific rate of scavenging in the quasi-free state. Using the initial condition nd0) nt(0)= 1 and the final condition n,(=)= 1, eqs 4 and 5 may be solved by Laplace transformation or otherwise, giving the time-dependent scavenging probability as
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not being scavenged per unit time is given by (1 k~C,/kft)-knk,j"f'+kfl'. Comparing this with the definitive expression exp( - k:c, l), one gets k,effc,=-
kftktf ln(1 ktf + ktf
+ k:c,/kft)
(9)
Usually kic,/kfl /[l- (1 - @)P(L>J= m
+ ... (10)
- 0 )
where e is the reaction probability at an encounter and P(L) is the first arrival probability after the pair come to rest at an average separation L following an unsuccessful encounter.2' Equation 10 is formally similar to Noyes-type theories20.2'except that explicit dependence of P(r0) and P(L) on fractal diffusion will be incorporated in the present case. From eq 8 of ref 18 we get an expression for P(r0) with R = and K DJ so that
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P(ro)= (aho)(1
-.
+ d/2r0)/(1 + d/2a)
(1 1)
where a is the encounter radius and d is a length parameter associated with the Takayasu22form for the distance dependence of fractal diffusivity at separation r, Le.,
D(r)-' = Do-'(l
+
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where gS = ASktf,a /3 = 1, k1f kft, and y = kdnd0). At a small scavenger concentration and designating ,6 < a, one gets a ktf kft and 8 , = A,kt$(kfl ktf). Taking the longtime limit of eq 6 such that exp(-Pt) >> exp(-at) % 0 (noting that p > kd,ff, we call it diffusion-controlled; if they are in the same order of magnitude, we may term the reaction as partially diffusion-controlled while, if kact > kd,ff. The apparent peaks might have been caused by calculational errors, and it can be stated that kc,or reaction efficiency generally falls with mobility. Figure 4 shows the variation of 7, the reaction efficiency with VO, obtained by eliminating peffusing eq 15. In this figure the data for tetramethylsilane (liquid VU) are not directly exhibited. However, it can be seen from Table l that the general trend of a decrease of 7 with mobility is also valid for this liquid with the possible exception of CC4 as scavenger in which case a small increase occurs. Also shown in Figure 4 is a linear relationship between the binding energy EO (eV) and Vo(eV); i.e., EO = 0.3174V0 0.1212, which may be helpful if dependence on the binding energy is desired. This relation results on eliminating peffbetween In peff= -44.1 lco 4.3245, obtained by fitting the data of ref 15, and a slightly modified form of the equation of Wada et al. (eq 15), given by HamillI2 as peff = 0.36 exp(-0.35Vdk~T). Expressing the limiting diffusivity by DO= (1/3)dU, where 6 is the mean thermal velocity and d = 1, the mean free path, one can write the diffusion-controlled rate in the quasi-free state as follows (cf. eq 13 et seq.)
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where the extreme right-hand side is obtained when d is >> 2a. This result is remarkable in that it is independent of the mean free path and therefore of the quasi-free mobility. It is reminiscent of gas kinetic collision frequency, differing only by a numerical factor.20 At T 300 K, fi = 1.15 x lo7 cm s-'; taking a 1 nm, we get (8n/e)Ua2 = 5.8 x l O I 4 M-' s-l, compared with 4.4 x lOI4 M-' s-l for the same value of a for CCb and biphenyl (Table l), using the full expression. For low-mobility liquids kft is >>ktf. From eq 8, one then obtains k y = (kt$kft)k%.With the same approximation the effective mobility is given byI5 peff A (e/m)(ktf/kft)(kft e/mp,f).-' Therefore, k:fflpeff= kt[(m/e)kft p,f-'I. Within a class of low-mobility liquids, kft does not vary muchI5 while pqfmay be considered to be sensibly constant for hydrocarbon liquids. If kf also varies little, then kif' will be proportional to perf. However, as shown by Funabashi and Magee, in many cases kf depends sensitively on VO and therefore indirectly on peff.8The ratio kT/peffcan also be written directly as kf(ktf/kfeeff), of which the factor within the parentheses has been fitted with the data of ref 15. For low-mobility liquids it varies very mildly with peff, given by (ktf/kffieff)= peff-0.094. This shows once again that the dependence of kifflpeffon peff should be traced to that of kt.
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6562 J. Phys. Chem., Vol. 99, No. 17, 1995 Finally, we would like to comment briefly on electron scavenging by biphenyl in cyclohexane. On the basis of steadystate scavenging and the ratios of mobilities of electrons and ions, Rzad et al.’ evaluated that specific rate as 3.0 x 10” M-’ s-l. Theoretical analysis by Mozumder3produced a somewhat higher value, 1.0 x 10l2 M-’ s-l. Soon afterward Beck and Thomas,30using laser photolysis technique, determined that rate to be (2.6 f 0.4) x lo1*M-’ s - I , which is the value used in Table 1. Nevertheless, we believe the reaction to be partially diffusion-controlled, the encounter reaction probability being only 0.475. Conclusions
(1) In the two-state, quasi-ballistic model electron scavenging reaction occurs mainly in the quasi-free state. (2) For efficient scavengers (CC14, C2H5Br, biphenyl, SF6, NzO) the reaction radii lie in the interval 1.OO- 1.25 nm. (3) Most scavenging reactions in low-mobility liquids considered here are diffusion-controlled. (4) For the same scavenger, reaction efficiency generally decreases with effective mobility. References and Notes (1) Rzad, S. J.; Infelta, P. P.; Warman, J. M.; Schuler, R. H. J . Chem. Phys. 1970, 52, 3971. (2) Thomas, J. K.; Johnson, K.; Klippert, T.; Lowers, R. J . Chem. Phys. 1968, 48, 1608. (3) Mozumder A. J . Chem. Phys. 1971, 55, 3026. (4) Handbook of Radiation Chemistry;Tabata, Y., Ito, Y., Tagawa, S., Eds.; CRC Press: Boca Raton, F‘L, 1991; Chapter VII.
Mozumder ( 5 ) Allen, A. 0.;Holroyd, R. A. J . Phys. Chem. 1974, 78, 796. (6) Allen, A. 0.;Gangwar, T. E.; Holroyd, R. A. J. Phys. Chem. 1975, 79, 25. (7) Bakale, G.; Sowada, U.; Schmidt, W. F. J . Phys. Chem. 1975, 79, 3041; 1976, 80, 2556. (8) Wada, T.; Shinsaka, K.; Namba, H.; Hatano, Y. Can. J . Chem. 1977, 55, 2144. (9) Funabashi, K.; Magee, J. L. J . Chem. Phys. 1975, 62, 4428. (IO) Yakovlev, B. S.; Boriev, I. A.; Balakin, A. A. Int. J . Radiat. Phys. Chem. 1974, 6 , 23. (11) Balakin, A. A.; Boriev, I. A.; Frankevich, E. L.; Lukin, L. V.; Yakovlev, B. S. Can. J . Chem. 1977, 55, 2156. (12) Hamill, W. H. J . Phys. Chem. 1981, 85, 3588. (13) Davis, H. T.; Brown, R. G. Adv. Chem. Phys. 1975, 31, 329. (14) Allen, A. 0. Drift Mobilities and Conduction Band Energies of Excess Electrons in Dielectric Liquids; NSRDS-NBS Circular No. 58; US GPO: Washington, DC, 1976 and references therein. (15) Mozumder A. Chem. Phys. Lett. 1993, 207, 245. (16) Ascarelli, G.; Brown, S. C. Phys. Rev. 1960, 120, 1615. (17) Mozumder, A. Chem. Phys. Lett. 1995, 233, 167. (18) Mozumder, A. J . Phys. 1990, 92, 1015. (19) Mozumder, A. J . Chem. Phys. 1994, 101, 10388. (20) Noyes, R. M.; Prog. React. Kinet 1961, I 129. (21) Mozumder, A. J. Chem. Phys. 1978, 6, 1389. (22) Takayashu, H. J . Phys. Sac. Jpn. 1982, 51, 3057. (23) Tsurumi, S.; Takayasu. H. Phys. Lett. 1986, 113A, 449. (24) Rappaport, D. C. Phys. Rev. Lett. 1984, 53, 1965. (25) Sano, H. J . Chem. Phys. 1981, 74, 1394. (26) Rice, S. A.; Baird, J. K. J . Chem. Phys. 1978, 69, 1989. (27) Lbpez-Quinetella, M. A.; Bujin-Nliiiez, M. C. Chem. Phys. 1991, 157, 307. (28) Dodelet, J. P.; Freeman, G. R. Can. J . Chem. 1972, 50, 2667. (29) Itoh, K.; Munoz, R. C.; Holroyd, R. A. J. Chem. Phys. 1989, 90, 1128. (30) Beck, G.; Thomas, J. K. J . Chem. Phys. 1972, 57, 3649. JP942876W
