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230 Charged Particle and Photon Interactions with Matter

very large (∼10–45Å), and their temperature variation could not be justied by any model. For

example, the jump length in ethane increased rapidly from 10Å at 100K to 45Å at 200 K, while that

in propane was virtually independent of temperature. The situation is expected to be similar in other

low-mobility liquid hydrocarbons, as has been commented on by Davis etal. (1973) in the case of

n-hexane. On the other hand, Funabashi and Rao (1976) consider electron hopping over a uctuat-

ing energy barrier, instead of a constant, ε

. They were able to justify the eld-dependent mobility

in ethane and propane with a plausible distribution of an energy barrier and a reasonable value of

L= 7.5 Å. One must remember, however, that this model is one-dimensional. No three-dimensional

generalization has yet been made. This point is important because the electron can avoid a high bar-

rier in three dimensions by going around it. Funabashi (1974) points out that for the validity of the

hopping model, it is required that w

< 10

−1

. If we take L = 10Å, we get from Equation 9.20 μ ≤ 4cm

−1

at room temperature, which is the upper limit of validity of the hopping model.

9.4.3.2 two-state

odels

A fundamental assumption of two-state models is that an electron can exist in two different states

of vastly different mobilities. One important case, sometimes called the trapping model, was rst

formulated by Frommhold (1968) for electron motion in a gas, interrupted by random attachment–

detachment processes. It was then applied to liquid hydrocarbons by Minday etal. (1971, 1972). In

this model, an electron can exist either in a quasi-free state of high mobility or in a trapped state of

much lower mobility. Thermal equilibrium between these states is brought about by frequent and

random trapping and detrapping processes. This results in a probability, P

, of the electron of being

in the quasi-free state and a probability, P

≡ 1 − P

, of it being in the trapped state. In terms of the

mean lifetimes in the quasi-free and trapped states, denoted respectively by

f ft

−1

and

t tf

−1

, P

given by P

= τ

/ (τ

+ τ

). Here, k

and k

, respectively, are the rates of transition from the quasi-free

state to the trapped state and vice versa. In the simple trapping model, the overall mobility is given by

µ µ µ µ= + − ≈P P P

f qf f t f qf

( ) ,1 (9.21)

where μ

and μ

are the mobilities in the quasi-free and trapped states, respectively. The extreme

right-hand side of Equation 9.21 is obtained because, in most cases, the drift velocity is determined

by the motion in the quasi-free state even though the electron stays in this state only momentarily.

Nevertheless, this equation implies that in all situations, the lifetime in the quasi-free state is suf-

ciently

long for a stationary drift velocity to ensue.

The

two-state trapping model is intended for low- and intermediate-mobility liquids, in which

case P

≈ k

, since τ

>> τ

. This ratio of rates may be obtained from detailed balancing (Ascarelli

and Brown, 1960) as k

= (1/n

)(2πmk

3/2

exp(−ε

T), where n

is the trap density, m is the

electron mass, and ε

is the binding energy in the trap, that is, the amount by which the trapped-state

energy lies below the quasi-free energy, V

. From the magnitude and temperature dependence of the

observed mobility in many liquid hydrocarbons, the quasi-free mobility has often been taken as a

constant (μ

= 100cm

−1

) (Minday etal., 1971, 1972; Davis and Brown, 1975). Although there

is no fundamental reason for the constancy of the quasi-free mobility, some justication derives

from the same value in TMS, which is considered to be trap free. With a known or assumed value of

and the ratio of detrapping to trapping rates as given above, the mobility in the two-state trapping

model

may be written as follows:

µ µ π













−













n h

mk T

k T

3 2

( ) exp .

(9.22)

Equation

9.22 can be tted for electron mobility in a large class of hydrocarbons in which the elec-

tron

mobility is <10 cm

−1

. So-determined values of activation energy lie between 0.10 and

0.25eV with a few exceptions. Trap densities are on the order of ∼10

−3

(Allen, 1976; Munoz, 1991;

Electron Localization and Trapping in Hydrocarbon Liquids 231

Nishikawa, 1991; Mozumder, 1999, Chapter 10). Since the dominant temperature effect in Equation

9.22 is in the exponential term, the activation energy in this model equals the trap binding energy,

if ε

>> k

T. A thermodynamic interpretation of electron transport in the two-state model has been

given

by Baird and Reheld (1987).

Schmidt

(1977) rst pointed out that the validity of Equation 9.21 depends on a sufciently

long residence of the electron in the quasi-free state so that a steady drift velocity can be obtained.

Mozumder (1993, 1995) introduced the quasi-ballistic model to correct for the competition

between trapping and velocity randomization in the quasi-free state. Some conclusions of this

model may now be summarized as follows: (1) Typically, the activation energy slightly exceeds

the binding energy with ε

act

≈ 0.89. (2) For relatively high-mobility liquids, the trap-controlled

mobility dominates. For intermediate mobilities, both ballistic and trap-controlled mobilities

make comparable contributions. For very-low-mobility liquids (<0.1 cm

−1

), diffusion results

only by random trapping and detrapping, irrespective of motion in the quasi-free state. This is

an important concept, and it is required to explain the dependence of the free-ion yield on the

mobility (vide infra). (3) With few exceptions, agreement with experiments for normal mobility

and activation energy can be obtained (Mozumder, 1999, Chapter 10, and references therein),

with the binding energy between ∼ 0.1 and 0.28 eV and with the trap density between ∼0.1 and

1.0× 10

−3

. (4) A consistent correlation between free-ion yield and mobility has been obtained

by Mozumder (2002) using the quasi-ballistic model. The elastic- and inelastic-scattering mean

free paths of epithermal electrons (≤0.2 eV) determine the thermalization distance distribution

and, therefore, the free-ion yield. The quasi-free mobility (μ

) is also obtained from the same

cross-sections using the Lorentz model. The effective mobility is then calculated from the quasi-

free mobility, the trap concentration, and the binding energy. Thus, a relationship is established

between the free-ion yield and the mobility in the quasi-ballistic model (vide supra). When the

mobility is very low (<0.1 cm

−1

), diffusion is governed by random trapping and detrapping,

being independent of the quasi-free mobility. As found empirically by Jay-Gerin etal. (1993), the

free-ion yield is then virtually independent of mobility. This is clearly explained in the quasi-

ballistic model, but not in the usual trapping model. By comparing with experiment, it has been

found that the elastic mean free path, L, in low-mobility liquids is ∼1–5Å and the probability of

an inelastic collision per elastic scattering is ∼0.1–0.3 (Mozumder, 2002). These values increase

with mobility, while, in the high-mobility case, L is approximately a few tens of Å and the inelas-

tic events outnumber the elastic ones by a factor of ∼2–4.

Comparing the hopping and two-state models, it can be said that the hopping model is appli-

cable only to low-mobility liquids (vide supra). The jump lengths are usually large and vary errati-

cally with the nature of the liquid and with temperature. The two-state models have a wider range

of applicability, but they require additional parameters. Consistency between free-ion yield and

mobility has been achieved only in the quasi-ballistic model. In either type of model, it is hard to

reconcile the activation energy of mobility with the peak of the optical absorption spectrum, where

observable. Sometimes it has been attributed to two kinds of trapped electrons, shallowly trapped

and deeply trapped. However, no consistent workable theory has yet been proposed along this line.

9.5 ConCluding remarks

In this chapter, we have considered electron localization and trapping in hydrocarbon liquids.

We have proposed and developed the application of the Anderson model for initial localization.

However, the transformation from the initially localized or delocalized doorway state to eventual

trapping, or promotion to the quasi-free state, has not yet been achieved theoretically. Comparing

with experiments, it seems that the two-state model has wider applicability with reasonable

values for quasi-free mobility, trap concentration, and binding energy in the trap. Consistency

between free-ion yield and mobility is possible within the context of the quasi-ballistic version of

the two-state model.

232 Charged Particle and Photon Interactions with Matter

In both hopping and two-state models, as applied to liquid hydrocarbons, it is a reasonable

assumption that the traps are preexisting. The situation is quite different in polar media, or when

the mobility is very high. In polar media and in dense cold gases, such as He, H

, and Ne, there is

evidence

of self-trapping.

aCknowledgments

The authors are thankful to Dr. G. Ferraudi for numerous discussions on molecular and liq-

uid structures. This paper is Document No. NDRL-4808 from the Notre Dame Radiation

Laboratory, which is supported by the Office of Basic Energy Sciences of the U.S. Department

of Energy.

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237

Reactivity of Radical Cations in

Nonpolar

Condensed Matter

Ortwin Brede

University of Leipzig

Leipzig,

Germany

Sergej Naumov

Leibniz Institute of Surface Modication

Leipzig,

Germany

10.1 introduCtion

As a consequence of the radiation-induced ionization of matter, electrons and parent radical

cations are primarily formed. Whereas in polar media the parent ions are extremely unstable,

in nonpolar systems (alkanes, alkyl chlorides, freons, etc.) such radical cations may persist for

longer times, depending on the temperature and viscosity. The lifetime of radical cations in neat

systems is determined by neutralization, deprotonation, and fragmentation. This is known since

decades (see, e.g., Mehnert, 1991; Tabata etal., 1991), and will be briey reviewed in this chapter.

In the presence of solutes having lower ionization potential (IP) than the matrix (liquid solvents,

solid-state glasses, etc.), electron transfer (ET) from the solute to the solvent parent radical cat-

ions happens. This is often applied for the generation and characterization of radical cations of

various solutes. These studies were preferably performed with the solid-state matrix isolation tech-

nique (Hamill, 1968; Shida, 1988) as well as electron pulse radiolysis (Baxendale and Busi, 1982;

Fartahaziz and Rodgers, 1987). The detailed investigation of the ET from substituted aromatics to

Contents

10.1 Introduction ..........................................................................................................................237

10.2 Ionization

of Nonpolar Systems ...........................................................................................238

10.3

Reaction

of Parent Solvent Cations with Added Scavengers................................................240

10.4

Free

Electron Transfer ..........................................................................................................243

10.4.1

The

Phenomenon ......................................................................................................243

10.4.2

Molecule Dynamics..................................................................................................246

10.4.3 Energetics..................................................................................................................247

10.4.4 Generalization

of the FET Phenomenon ..................................................................249

10.4.4.1

Benzylsilanes .............................................................................................249

10.4.4.2 Trityl-Hetero-Substituted

Aromatics ......................................................... 251

10.4.5

Driving Force of FET ............................................................................................... 253

10.4.6 Relation

between Molecule Dynamics and Reaction Kinetics.................................255

10.4.7

Comparison

of ET Theories .....................................................................................256

10.5

Transformation

of Donor Radical Cations in Freon Matrices..............................................257

10.5.1

Reaction

Principles...................................................................................................257

10.5.2

Intramolecular Transformations............................................................................... 257

References...................................................................................................................................... 261

238 Charged Particle and Photon Interactions with Matter

solvent parent cations implies mechanistic peculiarities insofar that in the liquid phase, molecule

dynamics governs the product formation. Some of the ET phenomena, such as free electron trans-

fer (FET) in liquids (Brede and Naumov, 2006) and special transformations of radical cations in

low-temperature glasses (Naumov etal., 2003a,b,c,d, 2004a,b,c, 2005), will be described here.

10.2 ionization oF nonpolar systems

Because of its high energy excess, ionizing radiation does not act on matter very selectively. Two

pathways are generally passed, namely, ionization and excitation of the material. Concerned with

nonpolar substances represented here by alkanes (RH) and alkyl halides (RX), it implies that in the

course of gamma or electron irradiation, besides the direct ionization products, fragmentation and

dissociation

products originating from excitation of matter also appear (cf. Equation 10.1):

RX R H Neutral fragments X H Cl etc

solv

→ → RX e

i i+ −

+ + + =

; , , , . (10.1)

We will focus on the ionization products given in boldface. The radical cations derived from pure

σ-bonded saturated molecules represent structures where the spin density is distributed over the

whole molecule (Brede and Naumov, 2006). This can be seen in Figure 10.1 where spin densities

and

charge distributions of some RX

•+

are demonstrated.

Radical cations derived from n-alkanes are metastable above a particular molecule size (≥C

Below this size, the n-alkane radical cations are less stable and fragmentize in a less specic manner

in olen radical cations or carbenium ions (Mehnert etal., 1984; Le Motais and Jonah, 1989) (e.g.,

see Equation 10.2). This happens also with vibronically excited radical cations existing at a very

early

stage after radiation action.

n n n

n n n n

-C H -C H RH -C H R

6 14 2 2

i i i+ + +

→ + + …;

(10.2)

As a characteristic of radiation chemistry, the ionization products are primarily inhomogeneously

distributed in local entities called spurs, blobs, etc., which have a size around 5–10 Å. In a competi-

tion between neutralization in the spurs and diffusive escape, in nonpolar media, most of the ionic

species decay within the spurs (geminate ions) and only a small part (free ions) results in a homoge-

neous distribution. So, of the primarily formed yield of G ≈ 4…5, only a fraction of G



= 0.05–0.50

results in a homogeneous distribution (Tabata etal., 1991). Here, the G-value represents the yield of

the species per 100eV absorbed energy. The low homogeneous ion yield is a characteristic of the

radiation chemistry of nonpolar media. Table 10.1 gives free ion yields of some nonpolar liquids

(Tabata etal., 1991). It should be taken into account that these yields were determined through

2b1b

2a 3a

Figure 10.1 Spin density distribution (a) and charge distribution (b) of the radical cations derived from

(1) n-heptane,

(2) cyclohexane, and (3) n-butyl chloride.

Reactivity of Radical Cations in Nonpolar Condensed Matter 239

conductivity techniques, which is unspecic, because this sums up parent radical cations as well as

fragment

radical cations and carbenium ions (Tabata etal., 1991).

The

parent radical cations have been observed by matrix insulation experiments in gamma-

irradiated solid samples (glasses, polycrystalline matter) at low temperatures. The detection was

performed by steady state UV–vis spectroscopy (Hamill, 1968; Shida, 1988) or EPR spectros-

copy (Shida etal., 1984; Knight, 1986). From these experiments, structural information originated,

which conrmed the mechanistic postulations of steady state irradiations combined with product

analytics (Földiak, 1981). Although the parent ions of alkenes did not exhibit as remarkable optical

properties as those of the solvated electrons (Baxendale etal., 1973; Allen and Holroyd, 1974), the

observation and kinetic characterization of alkane radical cations in liquids at room temperature,

with electron pulse radiolysis, succeeded in 1977 (Mehnert etal., 1977, 1979a,b). Subsequently,

the Delft, Argonne, and Tokyo pulse radiolysis laboratories also got engaged in this matter. For

example, from recent measurements, Figure 10.2 shows optical absorption spectra of an alkane

and an alkyl chloride radical cation detected by pulse radiolysis at room temperature (Brede and

Naumov, 2006).

The kinetic behavior and reactions of the parent alkane radical cations was intensely studied

(Mehnert, 1991). Similar investigations were performed also with carbon tetrachloride (Mehnert

etal., 1979a,b; Brede etal., 1980) and n-butyl chloride (Arai etal., 1976; Mehnert etal., 1982).

Because of high sensitivity, optical spectroscopy was usually preferred for detection (Baxendale

and Busi, 1982), but conductivity methods (Hummel and Schmidt, 1974; Beck, 1979), the micro-

wave absorption technique (Infelta etal., 1977), as well as electron spin resonance spectroscopy

(Trifunac etal., 1979) were also used for this purpose. The results are presented in detail in some

monographs (Baxendale and Busi, 1982; Fartahaziz and Rodgers, 1987; Lund and Shiotani, 1991).

The fate of alkane radical cations in the liquid phase is determined by neutralization with the

counter charge (solvated electrons or subsequent anions), deprotonation with the solvent, and frag-

mentation to olen radical cations (analogous to reaction 10.2) (see reactions 10.3a through 10.3c):

n n

-C H e or Cl Neutral products

solv.2 2+

+ − −

+ →

(10.3a)

n n

n n n n

-C H RH -C H RH

22 2 2 1+

+ → +

i i

(10.3b)

table 10.1

yields

and m

obilities

of Free i

ons

in n

onpolar

iquids,



as ions per 100 ev, μ in cm

−1

substance G



as ions per 100 ev

μ in cm

−1

Neopentane 0.9–1.0 55

Tetramethylsilane 0.74 90

2,2,4-Trimethylpentane 0.33 7

2,2-Dimethylbutane 0.30 10

n-Butane 0.19 0.4

Isopentane 0.17

Cyclohexene 0.15–0.20

Cyclohexane 0.15–0.19 0.35

n-Butyl

chloride ≈0.20

n-Hexane 0.10–0.18 0.09

Cyclopentane 0.16 1.1

n-Pentane 0.15 0.16

Carbon

tetrachloride 0.05–0.09

Benzene 0.05–0.08