The quasi-ballistic model of electron mobility in liquid hydrocarbons

Radiat. Phys. Chem.Vol. 47, No. 3, pp. 349-351, 1996

Pergamon

0969-806X(95)00113-1

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THE QUASI-BALLISTIC MODEL OF ELECTRON MOBILITY IN LIQUID HYDROCARBONS A. M O Z U M D E R Department of Chemistry and Biochemistry, and the Radiation Laboratory, University of Notre Dame, Notre Dame, IN 46556, U.S.A. AImtraet--A phenomenological theory of low-mobility liquid hydrocarbons is developed which includes electron ballistic motion in the quasi-free state, in competition with diffusion and trapping. For most low-mobility liquids the theory predicts consistently the effective mobility and activation energy, in agreement with experiments, using quasi-free mobility and trap density respectively as ~ 100 cm: v-~ sand ~ 10~9cm-3. Field dependence of mobility is theoretically of quadratic type for relatively small fields, agreeing approximately with experimental data for n-hexane. Electron scavenging with "good" scavengers occurs via the quasi-free state at nearly diffusion-controlled rate; however the effect of large mean free path is seen clearly.

INTRODUCTION AND THE MODEL In view of the large mean free path, scattering of quasi-free electrons often competes with trapping in low-mobility liquids. When this ballistic motion is incorporated in the two-state model the effective mobility is given by (Mozumder, 1993): /~r I = (~u)~ l + ( / 1 ) / l , where ( P ) r = (e/m)T~/ (~,+z/) is called the ballistic mobility and (IA)F=IAqfZf/(Zt + Z f) is the usual trap-controlled mobility. Here /.tqf is the mobility in the quasi-free state and zf, respectively z, are the mean lifetimes in the quasi-free and trapped states. Using detailed balancing (AscareUi and Brown, 1960) for the ratio of trapping (kft = z/-l) and detrapping (ktf = Zt- l ) rates and a brownian motion model with a harmonic potential for detrapping, it may be shown that (Mozumder, 1995): ktf = (Eo/h) exp(-- Eo/ka T);

kft = nth2eo(2nmka T) -3/2. (1) In equation (1) eo is the binding energy of the electron in the trap, nt is the density of traps, T is the absolute liquid temperature and other symbols have usual significance. It is the purpose of the present paper to investigate the validity of the quasi-ballistic model, within the context of low and intermediate mobility liquids ( < 10 cm 2 v -l s - l ) , in respects o f : (I) the magnitude of mobility and the activation energy, (II) the fielddependence of mobility and (III) the rate of electronscavenger reaction. RESULTS AND DISCUSSION It has been shown that the ballistic mobility ( / ~ ) r and the usual trap-controlled mobility (/~)F have activation energies Eo + 3k a T and Eo - 2ks T

respectively (Mozumder, 1993). It is then expected that the activation energy for the effective mobility Pe~ will be close to the binding energy Eo. This has now been verified for a number of liquid hydrocarbons in which the electron mobility is relatively low (Mozumder, 1995). Figure 1 shows the relationship between the quasi-free mobility /lqf and the trap density n, in cyclohexane at 300 K where the experimental value of mobility, 0.45 cm 2 v -1 s -l according to Dodelet and Freeman (1972), has been matched with present theoretical calculation. The binding energy Eo = 0.111 eV generates an activation energy Ea=0.126eV, in almost total agreement with the experimental value 0.125 eV, which may be fortuitous (Dodelet and Freeman, 1972). This implies that if &f is around 100 cm 2 V - l S - l then nt in this liquid would be ~ 1.2 x 1019cm -3. Other allowed combinations of (#qf, nt) are indicated by the curve in Fig. 1. Undoubtedly similar matching can be obtained using the lower experimental determination of mobility in this liquid, 0.24cm2 v -~ s - l , by Nyikos et al. (1977). External electric field E increases electron mobility in low-mobility liquids both by increasing detrapping rate ktf (E) and by decreasing trapping rate ka (E). The former, due to supply of energy from the external field, may be expressed as (Mozumder, 1995): ktf(E)/ka(o) = sinh2/2 "~ 1 + t r E 2 for relatively small E, with 2 =eEa/kaT and a =e2a2/6k2.T 2, a being the trap size. Trapping rate kft decreases with E due to increased effective velocity over thermal. Using a simple kinetic model with trapping rate inversely proportional to velocity it has been shown (Mozumder, 1995) that: kft(E)/kft(o)= I - y E 2 for relatively small E, where y = (e ~6ks T) (ig)T (o)/ktf(o) and ( p ) r ( o ) , respectively ktf(o) are the ballistic mobility and detrapping rate at E = 0. So far we have ignored the effect of the external field on the quasi-free mobility #qf which, by analogy with the 349

A. Mozumder

350

400 --

k~t). Comparing this with the expression exp(-k~ncs t), the effective scavenging rate is given by

k~ff = kfsktf/(kft .-~ ktf) 300

"7 200

10

0.8

I 1.0

I 1.2

I 1.4

I 1.6

n t (1019cm -3)

Fig. 1. Relationship between quasi-free mobility/~qfand trap density nt according to the present model for cyclohexane. The calculated /~er is matched with experimental value (Dodelet and Freeman, 1972) of 0.45 em2v -I s-' at ~300 K with trap binding energy 0.111 eV resulting in an activation energy for mobility 0.126 eV.

high-mobility case, is expected to decrease with E. If we assume ~Uqf to be independent of E, take a ffi 1 nm and use other parameters for n-hexane at 300 K (Mozumder, 1993), we can calculate the effective mobility at an external field E by combining the trapping and detrapping rates. This would give an onset of field effect for this liquid, i.e. ~ 10% increase in #e~, at E = 65 kV/cm (Mozumder, 1995) which is somewhat lower than experimentally observed (Schmidt and Allen, 1970). To include the effect of E on #qf we make the plausible assumption that /~qf(E)/~qf(O) = (1 q - E I E ~ ) -t/2 where the critical field Ec is such that at trapping the excess kinetic energy acquired by the electron at this field is, on an average, 0.1 times the thermal energy. The onset of field effect on the effective mobility now occurs at 100 kV/cm in n-hexane, in agreement with experiment (Schmidt and Allen, 1970). Figure 2 shows the variation of excess relative mobility in n-hexane fitted quadratically to the external field according to the procedure outlined above. F o r homogeneous scavenging we make the reasonable assumption that while the electron can exist in the quasi-free or trapped states, scavenging occurs only from the quasi-free state with a specific rate k~. The probability b that an electron would be trapped before scavenging at scavenger concentration c~ is given by b f e x p ( - c ~ k ~ T f ) . During time t, the number of cycles of trapping and detrapping is V ~t/('Cfq-'~t). Therefore the probability that the electron will remain unseavenged at time t is b ~ = exp[-c~k~tktf/(ka + ktf)] (of. zf = kff I and Tt =

(2)

which admits of an intuitive interpretation. An elaborate time-dependent reaction scheme, upon solution using Laplace transform, reproduces the same expression for steady-state scavenging rate (Mozumder, 1995). For typical "good" scavengers, such as biphenyl, SFt, N20, CC14, C2HsBr etc., the measured specific rate of electron scavenging in low mobility liquids is ~ 1 0 ~2M -~ s -~ and there is a tendency of increase with the mobility (Tabata et al., 1991). For example kexpt for the e + ~2 reaction is 0.8 x l0 n M -1 s -l in n-hexane and is 1.3 x 10~3M-~s -~ in iso-octane (Tabata et al., 1991). In cyclohexane this reaction has a nominal measured rate 3.0 x 10 u M -~ s -l (Rzad et al., 1970) which was later rationalized to be 1.0 x 10 ~: M -~ s -~ in view of competition with geminate recombination (Mozumder, 1971). If we use k ~ = 1.0 x 101: M -l s -t and the values of ktf and kft appropriate for electron mobility in cyclohexane (Mozumder, 1993), i.e. ~ 0 . 4 c m 2 v -~ s -1, we obtain from equation (2) k ~ f l . 2 x 1014M-Is -I for the scavenging rate in the quasi-free state of this system. Because of very long mean free path of scattering in the quasi-free state, the diffusion-controlled rate requires a "fractal" correction attributable to diminished diffusivity at smaller separations (Mozumder, 1990). Another factor ~ indicating the efficiency of the chemical step of the reaction is in principle required if the reaction is not fully diffusioncontrolled. Combining these two effects we get (Mozumder, 1990, 1995) kfs = kD~/(1 + d/2ro)

(3)

where r o is the reaction radius, d for neutral scavengers is the electron mean free path and kv = 4nDro is the Debye rate. A further correction due to reencounters can be shown to be negligible for neutral scavengers since we anticipate d>>ro (Mozumder, 1995). Here D is the diffusion coefficient in the quasi-free state at large separation, the value of 0.5

x

~

--

•

/./ .I""

0.3 -

o.2

-

0.1 -

, )a'°

f

~e ~

l

I

I

I

l

2

3

4

(e/~c) 3 Fig. 2. Field effect on electron mobility in n-hexane. Circles denote calculated result by a procedure outlined in the text. Straight line is a quadratic fit of #c~(E)/#e~(o) - 1 vs (E/Ec)2 with Ec = 100 kV/cm; the slope is 0.120.

Electron mobility in liquid hydrocarbons which for #qf= 100cm2v-I s -t is 2.5crn2s -t, using the Nernst-Einstein equation at 300K. Taking r o = 0 . 5 n m , we compute kD=9.6 x 10~M-~s -~. The mean free path of scattering 3ka Tlqf/eO with mean velocity 0 = (3kBT/m) ~n is computed to be 6.53 rim, which is equated to d. Equation (3) then gives k[ = ct.1.27 x 1014M -~ s -~. Comparing this with the previously rationalized experimental value (k~fr= 1.0 x 10~2M-ts -I giving k [ = 1.2 x 10~4M -~ s -I ) and acknowledging the uncertainties in calculation and in the experiments, we conclude that ct ~_ 1.0 or that the reaction is diffusion-controlled in the quasi-free state. If instead we use the nominal experimental value k~~= 3 x 10lj M -l s -l for e + q~2 reaction in cyclohexane (Rzad et al., 1970) and repeat the above procedure we will get ct = 0.28 or that the reaction is partly diffusion-controlled. Similar calculation for other "good" scavengers generally indicate the electron scavenging reaction in the quasi-free state of low mobility liquids to be at near diffusioncontrolled limit (Mozumder, 1995). The shortcoming from the Debye rate is essentially due to the fractal effect originating from large mean free path and not so much from the inefficiency of the final step of the reaction (Mozumder, 1990).

Acknowledgements--The work described herein was supported by the Office of Basic Energy Sciences of the U.S. Department of Energy. This is Contribution No. NDRL-3749 from the Notre Dame Radiation Laboratory.

351 REFERENCES

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