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OF STANDARDS

LIBRARY

JR. 7 m

Drift Mobilities and Conduction

fieri Band Energies of Excess Electrons

in Dielectric Liquids

U.S. DEPARTMENT OF COMMERCE, Elliot L. Richardson, Secretary

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

~ NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

ill

Issued May 1976

A. O. Allen

Department of Chemistry

Brookhaven National Laboratory

Upton, N.Y. 11973
Library of Congress Cataloging in Publication Data

Allen, Augustine 0.

Drift mobilities and conduction band energies of excess elec- trons in dielectric liquids.

(Nat. stand, ref. data ser. ; NSRDS-NBS 58)

“CODEN: NSRDAP.”

“Prepared for, and in cooperation with, the Radiation Chemistry Data Center of the Radiation Laboratory, University of Notre Dame.”

Supt. of Docs. No.: 03.48:58

1. Dielectrics. 2. Liquids—Electric properties. 3. Excess elec- trons. 4. Drift mobility. 5. Conduction band. I. Notre Dame, Ind. University. Radiation Chemistry Data Center. II. Title. III. Series: United States. National Bureau of Standards. National stand- ard reference data series; NSRDS-NBS 58. QC100.U573 No. 58 [QC585] 602'.ls [537.2'4] 75-44460

NSRDS-NBS 58

Nat. Stand. Ref. Data Ser., Nat. Bur. Stand. (U.S.), 58,23 pages (May 1976)

CODEN: NSRDAP

1976 by the Secretary of Commerce on Behalf of the United States Government

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1976

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD Catalog No. Cl 3.48:58). Price 70 cents. (Add 25 percent additional for other than U.S. mailing).
Foreword

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Ill
Preface

This report is one of a series of data publications on radiation chemistry; the aim of the series

is to compile, evaluate, and present the numerical results on processes occurring in systems which

have been subjected to ionizing radiation. Various kinds of data are important in radiation chemistry. The quantities which were measured first were the observed radiation yields or G values (molecules formed or destroyed per 100 eV). Various indirect methods based on G values have been used to determine yields of transient species and relative rates of reactions. The spectral properties (optical,

electron spin resonance) of transients have provided a direct method for their identification, and rates of the very fast reactions of transients which occur in irradiated systems have been measured directly

by spectroscopic methods. Conductivity and luminescence methods have also provided a means of

measuring properties of transients and their kinetics. Some reactions which occur in irradiated systems

have also been studied by other methods, such as photochemistry, electric discharge, ultrasonics, chemical initiation, electron impact, etc. The emphasis in these publications is on the data of radiation

chemistry, but where other pertinent data exist, they are included.

The data of radiation chemistry are voluminous; thousands of systems have been investigated.

As a result there are certain collections, e.g. rate constants of particular types of reactions or certain

properties of transients, for which tabulations of the data are considered essential, but for which

critical assessment of each value is impossible. On the other hand, certain systems and properties

have been studied so extensively that critical examination of these data is desirable and timely.

Authors of this series of data publications have been asked to evaluate the extent to which the data can be critically assessed, to describe their criteria for evaluation, and to designate preferred values

whenever possible.

IV
Drift Mobilities and Conduction Band Energies of

Excess Electrons in Dielectric Liquids *

A. O. Allen

Department of Chemistry

Brookhaven National Laboratory

Upton, N.Y. 11973

The properties of electrons in liquids which are covered in this review include drift mobilities and the energy level (V 0 ) at the base of the conduction band. Tables are given

of zero—field electron mobilities for several liquefied diatomic and inert gases, a number of aliphatic and aromatic hvdrocarcarbons, and polar solvents. V0 values are given for liquid

helium and argon, a number of hydrocarbons, tetramethylsilane and tetramethyltin.

Theories attempting to explain these results are described in a qualitative way.

Keywords: Conduction band; drift mobility; energy levels; excess electrons; hydrocarbons;

ion mobility; Uquids; mobility; quasifree electrons; solvated electrons.

1. Introduction

Modern views on the nature of matter arose from experiments on electrical conduction in gases,

which led to the discovery of the electron as a constituent particle of all substances. Application of the

electron concepts to electrical conduction in solids then led to such advances that the majority of all physicists now devote themselves to solid state studies. Though such great results have followed from

studies of electrons in gases and solids, little work on electrons in liquids has been done until very

recent years. Experiments at the University of Chicago on electrons in liquid helium and argon, and

their remarkably successful theoretical interpretation, have in the last seven or eight years stimulated

work at other laboratories on a variety of other liquids. Much of this work has been motivated by desire

to obtain better basic understanding of effects of radiation on materials and biological organisms.

The most significant, readily measurable properties of the electron in a medium are its drift mobility, and the energy level of the conduction electrons. This publication summarizes data on these

topics.

2. Definitions

An electron or ion, accelerated in a fluid medium by an electric field E, quickly attains a terminal

velocity v (when the frictional force balances the electrical force) which is proportional to the field, if the

field is not too large:

v = tiE (1) The mobility /x is always given in cm/s per volt/cm, or cm2 V‘1 s'1

. Values of fj. are characteristic of the

charged species and the medium. The mobility of an electron or singly— charged ion is related to the

diffusion coefficient D by the Nernst equation

fj. = De/kT (2)

* This is a data review prepared for. and in cooperation with, the Radiation Chemistry Data Center of the Radiation Laboratory. University of Notre Dame. Notre Dame,

Indiana 46556. The Laboratory is operated under contract with the Energy Research and Development Administration. The work of the Center is supported in part by the National Bureau of Standards. Office of Standard Reference Data.

1
where e is the electronic charge, k Boltzmann's constant and T the absolute temperature.

Under steady—state conditions with a constant applied field, the amount of positive charge reaching

the cathode per unit time must equal the amount of negative charge reaching the anode. Then the total current density in unit field (in units of ions per second per cm2

) is

i\E = l ca zn

/Ji n + 1c zp

/x

p (3)

n p

v K K where the subscripts refer to negative and positive charge carriers, c is concentration in ions per cm3

, z

is the number of electron charges on each ion (unity in all cases considered here) and the two sums are equal. The current density 1 in amperes per cm2 is related to i by i = IN/F = / • 6.24 x 10 18 where N is Avogadro's number and F is the Faraday constant. The ratio I/E is the conductivity cr; if E is in

volts/cm, cr is given in ohm-1 cm_1

. However, under different conditions, the physical systems governed by this equation may be quite

different. In electrolyte solutions, the applied field is always relatively small, and the concentration of

ions so large (c > 10 15

) that a small fractional excess of ions of one sign would lead to an internal field

large compared to the applied field, so this excess charge would neutralize itself by diffusion. Hence in

the physical chemistry of electrolyte solutions, the maintenance of electrical neutrality in all macroscopic volumes is assumed. The conductivity of the solution per g—equivalent of total electrolyte

per cm3 (extrapolated to zero concentration) is called Ac

: A0 = o-N/lcz = Fi/EZcz (4)

where Xcz is the total number of unit charges of either sign per cm3. A0 is the sum of equivalent

conductances of each individual ion present, weighted by their respective relative concentrations:

Ac = 2Xn cn /2cz + 2Ap c /Xcz (5) n p Comparing eqs (4) and (5) with eq (3), we see that

\ — F/jl = 96500 ^ (6)

if as usual A. is given in cm2 ohm_1 g—equiv-1 and /z in cn^V^s-1

. This equation is used in determining

the mobility of solvated electrons.

In an ionization chamber, as used to determine the dose rate from ambient y— or X—rays, the

situation is quite different. Here the field is kept high so that ions are swept out of the chamber as fast as formed. The positive and negative ions have concentration gradients in opposite directions, and the

mean concentration of each ion varies inversely with its mobility; so with parallel plates electrical

neutrality exists only at one plane in the chamber. It is still true under constant irradiation, that at the

steady state half the current is carried by positive charges and half by negative; but the time required to attain this state after any change in conditions is the time for the slowest ion to traverse the entire width

of the chamber. Drift time determinations for highly mobile electrons are made under conditions

resembling those of the ion chamber.

The states available to an electron in a liquid resemble those in a solid, and fall into two classes:

localized and delocalized. The delocalized state is that of a conduction electron in a metal or semiconductor, or an electron moving through a dilute gas; such electrons are often called quasifree.

Only an electron in a vacuum is really free; the motion of the quasifree electron is limited by scattering

from gas molecules, or (in crystals) from lattice imperfections or irregularities due to thermal motions

(phonon scattering). That existence of delocalized states in condensed matter does not depend on longrange crystalline order is proved by the simple fact that liquid mercury is a good conductor. In most

non—metallic liquids, electrons introduced by ionization (or by photodetachment from immersed metal

surfaces) enter the delocalized conduction state and will remain there if the electrical potential field

within the liquid is reasonably regular. Monatomic liquids (argon) and those polyatomics having

molecules of nearly spherical symmetry (neopentane) provide the most regularly varying potentials.

2
Anisotropy of the molecular polarizability and the presence of permanent dipoles in the liquid create

stronger irregularities which break up the electron wave and cause the electrons to become localized at the points of lowest potential. In a solid crystal or glass, these electrons are said to be trapped, and do

not contribute to the conductivity. In a liquid, the electrons are said to be solvated, and the whole

complex of electron and trapping molecules can move in an electric field and exhibit conductivity; but

its mobility is orders of magnitude less that that of a quasifree electron, and is similar to that of ordinary

molecular ions. The mobility of the quasifree electron normally decreases with increasing temperature,

because of increased phonon scattering; but the mobility of a solvated electron like that of a molecular

ion depends largely on the fluidity of the medium and increases with increasing temperature.

The energy level of a quasifree electron in a liquid, relative to a free electron in the vacuum, is called VG , and is always given in electron volts. For a given liquid, V0 increases with density and

therefore decreases with rising temperature. The exchange forces tend to repel the electron from the

molecules of the liquid, but the electron also induces dipoles in the molecules by polarization, which

tend to attract the electron. The greater the attraction compared to the repulsion, the more negative is V0

. In liquid helium, the polarizability is low and V„ is distinctly positive. An electron formed by

ionization in this liquid repels the atoms of the liquid and forces about itself a region of reduced density,

the behavior of which can be closely approximated by assuming the electron to be enclosed by a bubble

with a sharp boundary, containing no helium. The mobility of such a “bubble” is very close to that of a molecular ion, like that of a solvated electron; but the nature of these three negatively charged entities'

is thought to be entirely different.

Liquid hydrocarbons exhibit a wide range of electron mobilities lying between those of quasifree and

solvated electrons. It is thought that the traps in these liquids hold electrons less firmly than those in

polar liquids, so that diffusion of electrons from trap to trap is possible. Two mechanisms are proposed

for this passage. In one model, the electrons are in shallow traps having energies not far below the

conduction band, so that a thermodynamic equilibrium exists between solvated and quasifree states. On

this model, the observed mobility is given by the quasifree mobility, multiplied by the quasifree

equilbrium fraction, which increases with temperature. For the other model, we note that quantum

mechanics allows an electron, retained in a certain energy state by a potential barrier, to leak through

the barrier, provided a state of matching energy exists for the electron on the other side of the barrier. In general, a potential trap in a liquid, adjacent to a solvated or trapped electron, is not likely to be of

the right size to provide the energy match that will make leakage possible; but thermal agitation

continually alters the shape of the traps, so that the higher the temperature the more frequently the

energy match will occur. This model is called phonon—assisted—hopping

,

or simply hopping.

The bubble model, hopping model and passage through the conduction band lead to similar

phenomenological predictions. Perhaps the future will provide a more comprehensive theory for cases in

which the electron is neither firmly solvated nor entirely “quasifree”.

3. Methods of Measurement

3.1. Electron Mobilities in Non—Polar Liquids

Measurements of drift times in work reported here have involved electron lifetimes of at least

several microseconds. Extremely high purity of the liquid is then required with respect to electronattaching impurities. If the rate constant for the electron—impurity reaction is 10 13 M-1 s

_1 (not unusually

high) and a mean electron lifetime of 10_5 s is required for the measurement, the concentration of

reactive impurity should be kept below 10-8 M.

Given sufficient purity, the double—shutter method is the most direct and accurate [660394]. 1 Electrons formed at a cathode, usually by ionization, drift to a collecting anode, while retarding

potentials are periodically applied to a pair of double grids, acting as gates, located in the drift space.

The current reaches a maximum when the time to drift between the grids is a multiple of the period

between openings of the gates. By this method, electron mobilities in liquid argon have been measured

with an accuracy of ±1.5% [719353].

. Numbers in brackets indicate the references at the end of this paper. 3
Alternatively a single grid near the cathode can be pulsed; or, if the electrons are generated by

illuminating the cathode with ultraviolet light, the period is obtained by winking the light on and off with

a shutter (single shutter method) [729022].

A simpler method (thin layer method), used for tetramethylsilane [700140] is simply to turn on X—rays, which pass through the solution only in the vicinity of the cathodes, and note the time required

for the current to reach its maximum, which is the drift time across the cell. In the “thick layer

method” [731036] the entire cell is irradiated with a very short pulse of X—rays. As the ions are pulled

out by the field, the current decreases linearly with time, reaching zero at the drift time.

The Hudson method [460003] has proved useful for measuring electron mobilities in hydrocarbons

[700140]. When X—rays are turned on, irradiating uniformly the entire volume between cathode and

anode, the current rises parabolically with time to a maximum value, the time to maximum being the

drift time of electrons across the cell. The rise of the electron current is followed by the much slower

rise of the positive ion current. Impurities distort the electron rise curve and tend to give too high

mobility values; a method of correcting for the impurities has been devised [741036].

All the above methods require the use of extremely small electron currents in order to avoid space

charge effects, with corresponding care to avoid spurious readings in the electrical measurements.

Judging by consistency of the results when applied to hydrocarbons, the thin layer, thick layer and

Hudson methods are subject to errors of about ±10%. However, the shutter methods do not seem to have given much better results with hexane, and the difficulties in obtaining higher accuracy with

hydrocarbons may lie more in impurity effects than in the physical methods used.

A less direct method, which entails less strict purity requirements, is used at laboratories which

have pulse radiolysis setups with nanosecond resolution times. Conductivity is measured after a known

pulse of radiation, much more intense than used in the drift time methods, and the decay of the current

is usually largely due to ion recombination. In liquids such as hydrocarbons, where the electron mobility

is hundreds of times greater than that of the positive ions, the current is fixed essentially by the product

of the electron concentration and mobility; if the electron yield is known by other measurements, and

the absolute dose has been accurately determined, extrapolation of the current back in time to the end

of the radiation pulse allows the mobility to be calculated. Uncertainties as great as 30% in the value of

the free—ion yields make this method less accurate as a measure of mobility than the drift—time

determinations.

3.2. Mobilities of Solvated Electrons

The pulse conductivity method has been used to determine the mobility of solvated electrons in

water. The concentration of the electron is determined optically, simultaneously with the conductivity.

As the electron disappears, it is replaced in the solution by the OH- ion; the change in conductivity measures the difference in mobility of e~ and OH-

, which turns out to be surprisingly small. Similar

methods have been used for other polar liquids [720153]; but the mobility of other ions in these solvents

is less well known than in water, so these results are presumably less accurate.

Ammonia dissolves alkali metals such as Na to form stable solutions. The extrapolated conductivity

Ac = A.(c~) + MNa+

), and a transference number experiment has given the mobility of e~ in this solvent.

The stability of the electron thus allows the mobility to be determined by purely 19th-century methods.

3.3. Conduction Band Energies

Conduction band energies are measured by noting the difference in work function of a metal in

vacuum and immersed in the liquid. The logarithm of the photoelectric current emitted from a metal

illuminated at different wavelengths in the ultraviolet and visible is plotted against the quantum energy

of the light, showing a curve (Fowler function) which rises rapidly when the energy exceeds a certain

value, the work function, characteristic of the metal. Liquid is distilled onto the metal, and the

experiment repeated; the same shape curve is found, but is shifted along the energy scale to give a

different value of the work function. The electrons initially go into the conduction band of the liquid, so the change in the work function is equal to Vc

. After the liquid measurement, the liquid is distilled out

4
and the vacuum measurement repeated to make sure no change has occurred in the metal surface to affect the work function; clean hydrocarbons do not affect the surface.

Relative values of Vc for different liquids are also obtained by determining photoionization

thresholds in different liquids, using solutions of a readily ionized solute such as TMPD

(tetramethylphenylenediamine). Photoionization shifts from one liquid to another were found to be equal

to photoelectric work—function differences between the same liquids [747297]. These determinations are not sensitive to the presence of electron—attaching impurities, even at concentrations as high as 10

~3 M.

4. Data on Electron Mobilities

4.1. Elements and Diatomic Gases

Electron or negative ion mobilities /z_ in some liquefied monatomic and diatomic gases, and in

liquid sulfur are shown in table 1. All these data were obtained by some version of the double— or single— shutter methods. Some data on solids are included to show the behavior of /x_ on melting.

The bubble model for the electron in liquid helium [658008, 658009] is now generally accepted (see

for example [730316]). Above the A. point (2.2 K) the mobility varies approximately inversely with the

viscosity; but at lower temperatures, though the viscosity vanishes there is no discontinuity in the

mobility; it rises (fig. 1) from 0.033 at 2.2 K to 220 at 0.5 K, and at lower temperatures is proportional to T~3

; /aT3 = 27.64 K

3 cm2 V

-1 s_1 (T < 0.5 K), according to [729251] which presumably supersedes an

earlier publication [680854] by the same author. These results have led to some new insights on

superfluid behavior.

In liquid neon, as in helium, the electron mobility is close to that of the positive ion. In solid neon,

however, the electron mobility is many orders of magnitude larger, and is close to that of solid argon;

but in argon, krypton and xenon the mobility falls only by a factor of 2 on melting. The difference is that

electrons injected into the solid are expected to remain in the conduction band, even though the

Table 1. Zero—field mobility of electrons (or negative ions) in condensed

phases of elements and diatomic gases

Substance State M- cm2 V

_1 s_1 T, K Temp, effect, equation

Temp, range, K

Reference

He liq. 0.020 4.2 See fig. 1 0.274-5.18 620159,

600073, 729251 Ne solid 600. 25. — 720839

liq. 0.0016 25.5 = 3.0 X 10'4 r+ 0.0070 26-31 720839

liq. 0.0008 25.5 In

(

M_r> = 2.03 - (151 ID 25.5-43 729075 Ar solid 1000 82 689006

liq. 475 85 — — 689006

liq. 470 90.1 See fig. 2 90-160 660394,

719353 Kr solid 3700 113 — 689006

liq. 1800 117 — — 689006

liq. 1300 120 See fig. 5 120-180 660394

Xe solid 4500 157 — — 689006

liq. 2200

1080

163

167 See fig. 6 167-315

689006

741134

N2

‘ solid 0.0017 63 — 720840

liq. 0.00065 63 = 3.57 x 10~s r-0.00160 63-77 720840

02

‘ liq. 0.00045 62 /a_ = 3.70 x 10 _s r-0.00185 62-89 720840

CO* solid 0.003 68 — — 720840

liq. 0.00087 69 /x_ = 4.78 x 10

-5 r-0.00242 69-78 720840

h2 * liq. 0.0086 21 — — 638002

Sulfur* liq. 0.000100 393 = 0.15 ± 0.02 eV 393-433 687280

‘Interpretation of these data as electronic rather than ionic mobilities is quite doubtful; see text. 5
repulsive terms lead to a highly positive V„; but in the liquid, the atoms are free to move away from the

electron and form the low—mobility “bubble”. The bubble will be stable when V0 is larger than the sum

of energy terms required to form the bubble (pressure, surface tension and electrostatic forces). Before

the mobility measurements were made, Miyakawa and Dexter [699050] calculated VD for neon by two

methods, and concluded that the electron in a bubble should be only slightly more stable than in the

conduction band. If this is true, application of a few hundred atmospheres pressure should collapse the

bubble and cause a dramatic increase in the mobility; but this experiment has not been tried.

The electron mobility in liquid argon has been examined in great detail both experimentally and

theoretically (figs. 2—4). Lekner [670446] calculated the drift velocity of electrons in liquid argon for a temperature near the melting point (84 K) using methods developed for explaining the conductivity of

semiconductors, but calculating the potentials within the liquid from the atomic polarizability and

electron scattering data for dilute argon gas. The final result gave the zero— field mobility of the liquid to within 5% of the experimental value, and also agrees extraordinarily well with the high—field electron

velocity data, much of it obtained after the calculation was done (fig. 3). The levelling off of the velocity

at the very highest fields studied was explained in a theory of Spear and Le Comber [690202]. Moreover,

the decrease in mobility between 84 and 110 K follows the expected 7^2/3 law. Thus it may be said that

the electron mobility in liquid argon up to 110 K is basically understood at least as well as any property

of the liquid. At higher temperatures the mobility rises to a high maximum, with a sharp fall appearing

just below the critical temperature (151 K) (fig. 2). The cause probably lies in changes of the electron's

environment caused by microscopic density fluctuations that appear in the liquid as the critical

FIGURE 1. Mobility of electrons in liquid helium at pressures below 1 atm, plotted against recriprocal absolute temperature: open circles ’[729251]; closed circles [6000073]: open triangles [620159].

700

660

620

580

540

500

<SI

'<? 460 “ 420

380

340

FIGURE 2. Comparison of apparent electron mobilities at —100

V/cm on a 50-atm isobar in liquid argon. Data from

the isothermal experiments were obtained by taking

the value at 50 atm from plots of apparent mobility versus pressure at constant temperature. Refer to [660394] for data obtained using a Po210 alpha-ray

source. The figure shows the agreement between data using different sources and between data obtained through different thermodynamic routes. Figure

from [719353],

1

1

1 1 r E= -100 V/cm

° ISOBARIC EXPERIMENTS

(50.0 £Q5 atm)

^ ISOTHERMAL EXPERIMENTS

(DATA AT 50.0 ±0.5 atm

OF ISOTHERM)

6
Figure 3. The field dependence of the electron drift velocity

in liquid Ar at 85 K. Curve S, calculated from Shockley theory: curve C, theory of Cohen and Lekner;

curve B is corrected for multiple scattering and also indicates the linear field dependence. Curve A: typical

results from a speciment with short electron lifetime (— 200 ns). The sound velocity in the liquid is de- noted by uL. Figure from [689006].

FIGURE 4. Zero-field electron mobilities as a function of number

density in fluid argon. Figure from [719353].

temperature is approached. The treatment which was so successful at lower temperatures assumed the

potential to be isotropic about the average position of each atom, which predicts that mobility should

depend only on the liquid density. Figure 4, which summarizes the extensive and precise data of Jahnke,

Meyer and Rice, shows that this is indeed true at the higher densities, but that at the higher

temperatures the mobility for a given density is higher, the higher the pressure (and therefore the higher

the temperature). Thus thermal fluctuations of the density lead to an increase in mobility, though it is not obvious that this should be so. Lekner [688043] pointed out that his calculated scattering length or square root of the scattering cross section for the electron in the liquid is positive, corresponding to scattering from repulsive centers, while in argon gas the scattering length is negative. At some

intermediate density then the mean scattering length should be zero, and the mobility should fail to become infinite only because of fluctuations in the scattering process. It was to test the theory derived

from this model that Jahnke carried out his extensive measurements. The results agreed with the theory

in that the maxima at different pressures always occurred at temperatures such that the density was the

same (0.0120 atoms/A3

), and predicted the right shape for the mobility rise from the high—density side,

but failed to explain other features of the data. Later, Jahnke, Holzwarth and Rice [728005] reported that no theoretical calculation of the scattering length could predict a zero at densities above the critical

liquid density (0.0080 atoms/A3

). Thus at present there is no good theory for the maximum in mobility

below the critical temperature.

The mobility maximum also occurs in liquid krypton (fig. 5) and xenon (fig. 6). It is not known

whether it exists in any of the many polyatomic liquids which have been studied, since measurements in

these liquids have not been made sufficiently close to their critical temperatures.

Electron mobility increases with increasing density and polarizability in the series argon, krypton, xenon. The field dependence of the mobility sets in at lower fields in the higher atomic weight liquids,

and the extrapolation to zero field is somewhat uncertain for xenon. In fact, Robinson and Freeman

[741012] who looked at electron velocities in xenon by the maximum pulse current method, found that

the velocity was proportional to the square root of the field, so extrapolation of /z to zero field would be

impossible. However, their velocity figure was obtained by dividing the maximum conductivity by the

free—ion yield, the value of which is perhaps somewhat dubious and is also field— dependent. More

recently, Kimura and Freeman [741134] have published numbers obtained at fields of only 16 V/cm.

Neither theory nor experiment has been applied as thoroughly to krypton and xenon as to argon. Since

the theory of these cases, in which self—trapping is essentially absent, is basic to the understanding of

7
Apparent

Electron

Mobility

(cm'

FIGURE 5. Apparent electron mobility in liquid krypton as a function of temperature. All points are measured at a constant electric field strength of —100 V/cm.

Figure from [660394].

FIGURE 6. Apparent mobility of electrons in liquid xenon as a function of temperature. The pressure on the sample was the vapor pressure. Applied field strength = 16 V/cm. Triple point, t.p. = 161.3 K; critical temperature Tc — 289.7 K. Figure from [741134].

Figure 7. Vo in electron volts vs. mole fraction hexane for neopentane-n-hexane mixtures; open points are from work function shifts of Zn; filled points are from TMPD photoionization threshold shifts; , bubble model prediction, [720242]; , ef- fective medium prediction, [738029]. Figure from

[740079],

8
electrons in polyatomic liquids, in which widespread interest is now being shown, additional study of

krypton and xenon would be desirable.

The other substances in table 1 all show much lower mobilities. Hydrogen was predicted by

Miyakawa and Dexter [699050] to form “bubbles” about electrons, and the measured mobility, which lies between those of helium and neon, is consistent with this prediction. Oxygen gas (but not N2 or CO) at reasonably high pressures reacts with electrons to form the ion 02 , and the negative species in liquid

oxygen would be expected to exist in this form.

Mobile positive and negative species were seen in solid N2 and CO, with /li 1—3 x 10-3 cm2 V

-1 s-1

, and in the low-temperature form of solid 02 (36-44 K) but not in the high-temperature form (44-54 K). In liquid N2 and CO, n_ increased from about 0.8 x 10-3 to 1.3 x 10-3 cm2 V

_l s_l in the range from the

melting point to 10° above it, in parallel with /x+ and with the decrease in viscosity of the liquids. In

sulfur similar phenomena were found, with /x_ falling by a factor of 25 to 1 x 10-4 on melting, then

increasing at higher temperatures. At 160° C sulfur suddenly begins to polymerize and the viscosity

increases by more than two orders of magnitude over a range of 4°; but in this range the mobilities /jl_ and fi + showed no significant change. The: experimenters, in the belief that ionic mobilities (but not

electronic) should vary inversely with viscosity, ascribed the mobilities in sulfur to electron hopping, but

in N2 , CO and 02 to ionic drift (presumably of impurity negative ions in N2 and CO). However,

Drickamer and coworkers [539005, 539006] showed that the rate of self-diffusion in liquid sulfur changes

little over the range where viscosity increases, and the effect of pressure on self—diffusion showed the

activation volume to be only 1 to 6 atomic volumes. Apparently diffusion of small molecules may occur

with little hindrance within the network of polymer molecules, while viscous flow requires motion of the

macromolecules. Ionic drift should be more closely related to self—diffusion than to viscosity, so it seems possible that in all these liquid elements the mechanism of charge transfer is ionic. However, this can hardly be true for the solid crystals, where electron hopping would seem to be the only plausible

mechanism; and the fairly small change of mobility and its temperature coefficient on melting argues in

favor of the same mechanism holding in both phases. The data on mobility of negative charge carriers in

these liquids is therefore included in table 1, with the notation that interpretation of these numbers as electron mobilities is doubtful.

4.2. Hydrocarbons

Electron mobilities in hydrocarbons and tetramethylsilane (TMS) are shown in table 2. The aliphatic

hydrocarbons, saturated and unsaturated, are listed first, in the order of carbon number, followed by the

aromatic liquids.

Table 2. Zero--field mobility in liquid hydrocarbons

Name F- cm2V

-1 s_1 T, K F . ^acdvn. kcal/mole

Method* Reference

Methane 400 111 Negative, small tl 731036

430 140 Negative, small pc 741012

Ethane 0.8 200 See fig. 8 tl 739187

740317

0.97 200 3 pc 741012

Ethylene 0.0030 170 — pc 741012

Propane 0.55 238 3 pc 741012

0.12 200 See fig. 8 tl 740346

Propylene 0.008 234 4 pc 741012

Cyclopropane 0.0043 234 4 pc 741012 n-Butane 0.073 250 4 pc 741012

0.4 296 H 700140 Butene—

1

0.064 293 pc 731035

cis—Butene—

2

2.2 293 3.7 pc 731035

trans-Butene-2 0.029 293 5.4 pc 731035

Isobutane 5. 294 pc 720034

Isobutene 1.44 293 — pc 731035

9
Table 2. Zero-field mobility in liquid hydrocarbons — Continued

Name F- cm2V"1 s"

1 T, K F . «cdvn. kcal/mole

Method* Reference

n—Pentane 0.16 296 H 700140

0.08 300 — ds 719062

0.15 293 — pc 731035

0.14 296 H 741036

2—Methylbutene—

2

3.6 300 2.6 ds 719062

Neopentane 68 296 0.32 ds 729022

70 295 — tl 739187

70 258-296 0 pc 741111

Tetramethylsilane 90 296 — H 700140

100 223-292 0 H 750003

Cyclopentane 1.1 296 — H 700140 n—Hexane 0.076 300 4.35 ds 729022

0.09 296 5.5 H 700140

0.082 296 — H 741036

0.069 294 H 750003

2,2— Dimethylbutane 10. 296 1.4 H 700140

12. 295 _

tl 739187

2,3— Dimethylbutene—

2

5.8 293 — pc 731035

Cyclohexane 0.35 296 — H 700140

0.45 294 2.9 pc 720269

0.24 296 H 741036

0.22 294 — H 750003

Cyclohexene 1.0 293 3.6 pc 731035

Methylcyclohexane 0.068 296 4.7 H 741036

2,2,4-Trimethylpentane 7. 296 — H 700140

5.3 293 1.16 H 750003

2,2,3,3—Tetramethylpentane 5.2 295 1.5 pc 720269

2,2,4,4—Tetramethylpentane 24. 295 1.4 pc 720269

2,2,5,5-Tetramethylhexane 12 293 1.1 pc 720269 Benzene 0.6 300 — ds 719062

0.11 293 7.4 pc 741110

Toluene 0.54 300 — ds 729022

0.063 293 3.4 pc 741110

o-Xylene 0.018 292 4.4 pc 741110

m-Xylene 0.057 292 4.5 pc 741110

p-Xylene 0.062 293 — pc 741110

1,2,3-Trimethylbenzene 0.022 293 4.8 pc 741110

1,2,4-Trimethylbenzene 0.035 292 4.3 pc 741110

1,3,5—Trimethylbenzene 0.16 293 3.2 pc 741110

1,2,3,4-Tetramethylbenzene 0.040 323 5. pc 741110

‘Abbreviations: ds, double shutter; H, Hudson; pc, pulse conductivity; tl, thick layer.

The most striking feature of this work is the variation in the mobilities observed in the alkanes,

which range over a factor of 1000 in a group of compounds whose other properties are very similar.

There is a general correlation of mobility with other electronic properties — free—ion yields and Vc values — but not with dielectric constants or other commonly listed properties. The most important factor in

fixing fx_ seems to be the symmetry of the molecules. Those of nearly spherical shape — methane,

neopentane, TMS — have the highest mobilities, and Bakale and Schmidt [731036, 739187] have shown

that the apparent electron mobility in these compounds decreases with increasing field, as in liquid

argon. The difference in electron mobilities between liquid ethane and methane is striking, and the field

effects are actually opposite, with the apparent mobility at 200 K increasing in ethane at fields above 70

kV/cm [740317]. Propane behaves similarly to ethane, with the apparent mobility increasing above 100

kV/cm. In other hydrocarbons, the velocity remains proportional to the field up to the highest field

studied. In all cases except methane and TMS, the observed mobilities as a function of temperature

10
Figure 8. Temperature dependence of the electron low field mobility of ethane and propane. Ethane curve:

circles, data from [740317]: squares, data from

[741012]. Propane curve: triangles, data from [740346]: squares, data from [741012].

appear to fit an Arrhenius expression, with the activation energies in general being lower the higher the

room—temperature mobilities.

It is thought that molecular shape influences the mobility through the anisotropy of the polarizability

in unsymmetrical molecules. Orientations of acyclic hydrocarbon molecules in a liquid are mainly

random. If a group of neighboring molecules happens to have their directions of high polarizability

oriented towards a central point, an electron coming to that point would be at a lower potential than at other points, and would tend to be localized there; the less symmetrical the polarizability, the deeper the

traps. Electrons localized in such {raps are solvated, and have the broad absorption spectrum in the

infrared typical of solvated electrons in polar liquids (see, for example, [690400] and [710642]). It is reasonable to suppose that the only difference between the electrons in hydrocarbons and the

permanently solvated, low conductivity electrons in polar solvents such as ammonia is that the traps in the latter are so much deeper that the equilibrium fraction of electrons in the conduction band is 11
negligibly small. These qualitative considerations await confirmation by an actual calculation of the

potentials in liquid hydrocarbons. Such a calculation, even for ethane, would be far more complex and

difficult than for liquid argon.

The effect of the double bond in ethylene and propylene is to decrease the mobility by a large

factor; but in larger molecules this effect is not seen. The remarkable difference in the electron mobility

in the cis and trans forms of butene—2 emphasizes the dependence of this property on the molecular

geometry.

Comparison of the results of different laboratories shows that the absolute accuracy is not always as good as the experimenters believed. The worst discrepancies appear with benzene and toluene; however,

the experimenters who measured drift time reported that only 20% of the current appeared to be carried

by the fast negative carriers, the rest being ascribed to impurity ions; the conductivity method would

then be expected to give lower values. Somewhat discrepant values also appear for cyclohexane and

normal pentane. In the case of neopentane, a number of lower reported values have been excluded from

the table because they have been superseded by more recent values from the same laboratories.

4.3. Polar Solvents

Mobilities of solvated electrons in polar liquids are given in table 3. These measurements are all made by determining that part of the conductivity due to the electron, as outlined above. The results in

water and ammonia are based on rigorous treatment of accurate data on conductivities and transference

numbers, and should be good to a few percent; but the results on the other compounds are based on

small deviations in conductivity decay curves brought about by small additions of anthracene or other

electron acceptors, which are analyzed on the assumption that the mobilities of the acceptor negative ion

and the positive ion are equal. The results are of interest chiefly in showing that the mobilities of

solvated electrons in all these solvents are of comparable magnitude.

Table 3. Mobilities of solvated electrons in polar solvents

Solvent T, K activn. kcal/mole

References

Ammonia 0.0092 239 609010 for transference number;

599008 for conductance,

recalcfl. from

219001 Water 0.00197 298 — 690295

0.00197 295 — 690547

0.00187 298 4.8 700243

0.00162 292 4.2 720404

Methyl ethyl ketone* 0.0015 293 — 720153

Hexamethylphosphoramide 0.00055 293 — 720153 Butylamine 0.0027 293 — 720153

Tributylamine 0.002 293 — 720153 Methyl alcohol 0.00059 293 — 710064

Ethyl alcohol 0.00025 293 — 710064

Isopropyl alcohol 0.00051 293 — 720574

Isobutyl alcohol 0.00068 293 — 720574

rt-Butyl alcohol 0.00075 293 — 720153

0.00073 293 — 720574

Isopentyl alcohol 0.00095 293 — 720153

0.00097 293 — 720574

n-Hexyl alcohol 0.00043 293 — 720574

Ethylene glycol 0.00028 293 — 720574

*May be a molecular anion rather than a solvated electron.

12
5. Conduction Band Energies

Values of V0 for various liquids at selected temperatures are shown in table 4. Temperature effects on Vc are shown in figure 9 for those liquids for which values at more than one temperature were

Table 4. Conduction band energies \ o in various liquids

Liquid T, K V„, eV References

Helium 4.2 +1.05 659054

Argon 84 -0.33 670195

85 -0.18 750045

Xenon 163 -0.65 750045

Methane 109 0.0 740192

95 0.0 749281 Ethane 182 +0.02 740192

Propane 222 -0.07 747297

n—Butane 271 +0.12 747297

cis—Butene—

2

243 -0.16 750009

n—Pentane 295 +0.01 747297

295 -0.01 720439

295 +0.04 719091

295 -0.02 750009

-0.2 727460

Neopentane 295 -0.43 719091

295 -0.35 720439

295 -0.38 747297

Cvclopentane 295 -0.17 720439

295 -0.21 747297

n—Hexane 295 +0.04 719091

295 0.00 720439

295 +0.1 747297

295 +0.16 730414

292 +0.02 750009

2—Methylpentane 295 +0.02 720439

295 +0.01 747297

295 -0.20 730414

3—Methylpentane 295 0.00 720439

295 +0.01 747297

295 -0.14 730414

2,3—Dimethylbutane 295 -0.10 730414

295 -0.15 750009

2,2—Dimethylbutane 295 -0.15 730414

295 -0.26 747297

2,3—Dimethylbutene-2 295 -0.25 727297

Cyclohexane 295 +0.01 727297

Methylcyclohexane 295 +0.08 727297

n-Octane 295 +0.13 727297

2,3,4—Trimethylpentane 295 -0.05 727297

2,2,4—Trimethylpentane 295 -0.18 719091 295 -0.17 747297

2,2,5—Trimethylhexane 295 -0.10 747297

2,2,4,4—Tetramethylpentane 295 -0.33 747297

n—Decane 295 +0.18 747297

1,3-Dimethvladamantane 295 -0.30 747297

n-Tetradecane 295 +0.21 747297 Benzene 295 -0.14 730414

-0.5 727460

Toluene 295 -0.22 730414

Tetramethvlsilane 295 -0.62 719091

295 —0.55 720439

295 -0.59 747297

293 -0.51 750009

Tetramethyltin 293 -0.70 750009

13
Triangles (point down), n-hexane. Circles with cross, n-pentane. Squares, 2,2,5-trimethylhexane. Triangles

(point up), 2,2,4-trimethylpentane. Diamonds, 2,2-di- methylbutane. Circles, 2,2,4,4-tetramethylpentane.

Diamonds with cross, tetramethylsilane.

obtained by work function measurements. The method of photoionization of solutions gives only relative

V0 values for different solutes at any one temperature; the results for room temperature in [747297] and

[727460] were assigned absolute numbers by comparing the results for a few liquids to the VD values

previously determined for the same liquids by the work function method. Temperature effects on VD are

not accurately determined by the photoionization method since the ionization efficiency at any

wavelength may vary with temperature in an unknown manner. Consequently the VD values quoted in

the above references for temperatures other than that of the room are not given here. Determinations by

the work function method made on different days with different preparations reproduce to no better than

±0.05 eV. Determinations made on the same day reproduce to about ±0.02 eV.

Considerable effort has been expended on theoretical estimation of Vc

. Lekner [670446] in the course of calculating the potential distribution in liquid argon for estimation of the mobility, obtained a value of —0.46 eV for VQ , compared with the experimental value of —0.33 eV. Springett, Jortner and

Cohen [688042] developed a more general approximate expression, which in the case of argon gave a value of —0.63 eV. Miyakawa and Dexter [699050] provide an alternative approximate expression, which

leads to similar results. Both methods use the gas—phase scattering length as a basic datum; and this

may not be very accurately established experimentally. They also assume isotropic molecules, so the

theories provide only a very rough guide for polyatomics. K. Fueki [728065] estimated by these methods

V0 for methane to be —0.74 eV, in poor agreement with later measured values 0.0; but he used an

estimated value for the gas—phase scattering length; one hopes that use of an experimental value for this

parameter would result in a better number for VG

. In these theories, V0 is regarded as the sum of a negative polarization energy term and a positive kinetic energy term which results from the electron

being restricted to the free volume. For liquid argon, these terms are respectively —2.50 and 1.87 eV, so

that the sum, V0 , is the difference between two larger numbers and is accordingly inaccurate in absolute

value.

The relation between Vc and /u, e for aliphatic hydrocarbons has been much discussed. Comparison

between the data of tables 2 and 4 shows a close correlation, especially for the linear and branched

alkanes. N. R. Kestner and J. Jortner [738029] suggested an explanation in terms of a “percolation

theory”, in which VG is assumed to be linearly related to an “average electron energy” in the liquid,

which in turn is related to the volume fraction of the liquid in which the electrons are in the quasifree

14
condition. R. Schiller [730414] has developed a theory of the electron in a hydrocarbon based on the

bubble model, assuming that the solvated state can be treated as an electron in a bubble of fluctuating

size. The two theories lead to somewhat different relations between V0 and log /jl c

. For a simple test of

these theories, Holroyd and Tauchert [740079] measured Vc for a series of mixtures of n—hexane and

neopentane, since it had been found (Minday et al., [729022]) that log /u e varied linearly with mole

fraction in these mixtures. It was found that Vc was also linear with mole fraction, and hence with log

/x e , while the theories give a quite different prediction (fig. 7). A similar result was found in mixtures of

hexane and tetramethylsilane. The linear relation found experimentally was unexpected, since many

different factors have been thought to affect the values of Vc and /u e

. Further insights are needed.

Acknowledgements

Figures 2 and 4 are reproduced by permission

of the American Institute of Physics from Physical

Review, Part A, Volume 3, 1971, pp.734—52.

Figures 3 and 5 are reproduced by permission

of the American Institute of Physics from Physical

Review, Volume 166, 1968, pp.871—8 and Volume

150, 1966, pp.122—45, respectively.

Figure 6 is reproduced by permission of the

National Research Council of Canada from the

Canadian Journal of Physics, Volume 52, 1974,-

pp. 2220-2222.

Figure 7 is reproduced by permission of the

American Institute of Physics from the Journal of

Chemical Physics, Volume 60, 1974, p.3715.

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(1972). [728065] Fueki, K., On the mobility of quasifree electrons

in liquid methane, CAN. J. CHEM. 50(20): 3379-80 (1972). [729022] Minday, R.M.; Schmidt, L.D.; Davis, H.T.,

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17
NBS-1 14A (REV. 7-73) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NSRDS-NBS- 58

2. Gov’t Accession

No.

3. Recipient’s Accession No.

4. TITLE AND SUBTITLE

Drift Mobilities and Conduction Band Energies of Excess

Electrons in Dielectric Liquids

5. Publication Date May 1976

6. Performing Organization Code

7. AUTHOR(S) _ A „ A. O. Allen

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

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15. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 75-44460

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

The properties of electrons in liquids which are covered in this review

include drift mobilities and the energy level (V0 ) at the base of the conduction

band. Tables are given of zero-field electron mobilities for several liquefied

diatomic and inert gases, a number of aliphatic and aromatic hydrocarbons,

and polar solvents. VQ values are given for liquid helium and argon, a number

of hydrocarbons, tetram ethylsilane, and tetram ethyltin. Theories attempting

to explain these results are described in a qualitative way.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Conduction band; drift mobility; energy levels; excess electrons; hydrocarbons;

ion mobility; liquids; mobility; quasifree

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electrons; solvated electrons.

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