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Determination of the frequency of excess electron trap to trap migration in saturated hydrocarbons by an OD ESR technique
Chemical Physics 144 ( 1990) 241-248 North-Holland
DETERMINATION OF THE FREQUENCY OF EXCESS ELECI’RON TRAP TO TRAP MIGRATION IN SATURATED HYDROCARBO NS BY AN OD ESRTECHNIQUE S.N. SMIRNOV, A.Yu. PUSEP, O.A. ANISIMOV and Yu.N. MOLIN Institute of Chemical Kinetics and Combustion, Novosibirsk, 630090, USSR Received 20 November 1989; in final form 12 February 1990
The temperature dependence of the ESR linewidth of excess electrons in squalane and 3-methylpentane has been measured by the OD ESR technique. When the temperature dependence of the linewidth is fit by a model which inchules a temperaturedependent jump frequency, Ef, it is found to be smaller than the activation energy, Er. for the electron mobility. For 3methylpentane the activation difkrence is small and the jump length calculated from the Einstein formula, 1s 10 ,k, therefore has weak dependence on the temperature. In squalane the difference between Er and Ef is greater, and the calculated large values of at hi temperatures indicate a wide depth-distribution of trap energies.
1. Introduction
The excited molecules and radical ion pairs which appear in a high-energy particle track as the initial step of any radiation-chemical process define the behavior of further chemical reactions. One of the main primary particles, the excess electron, is mobile and highly reactive. Its high mobility determines both the time scale of electron/radical-cation recombination and the rate of most chemical processes in the track. The character of the excess electron motion depends on the nature of the matrix. In polar matrices the thermalized (i.e. in the thermal equilibrium with the medium) excess electron is rapidly localized in a potential well, resulting from the polarization interaction between the electron and dipolar molecules. Since orientation polarization is not essential for electron localization in nonpolar liquids, it is not clear yet whether the excess electron state is localized [ 11. For most nonpolar molecular liquids, the equilibrium electron is considered to be neither fully localized nor quasi-free but, alternatively, in one or the other state. For hydrocarbon liquids in which the electron mobility is below 1 cmZ/V s, electron motion is usually described by one or two phenomenological models (trapping and hopping). In the trapping model [ 2-61, electrons are thought 0301-0104/90/$03.50 (North-Holland)
8 Elsevier Science Publishers B.V.
to be trapped for a time r,. Thermal activation may deliver the electron from the trap to the conduction band where it moves for time rr with high mobility & until capture. The measured mobility & is
(1) Here, A, the trapped electron mobility, is, in hydrocarbons, probably equal to the ionic mobility. Assuming rfh w r,~ we obtain: &
=Ptrf/%
*
(2)
For the simplest case of identical traps: z,=qexp(E,/kT)
(3)
and p(T)=Cbexp(-E/W),
(4)
where ~4,has replaced fi to indicate the preexponential for any temperature-activated mobility. A comparison of eq. (4 ) with the experimental dependence gives approximately the same value, cb = 100 cm2/V s, for different hydrocarbons [ 31. For a linear increase in the trap density with trap energy from 0 to E [4]: p( T)=b(
1 +E/kT)
exp( -E/kT)
.
(5)
Eq. (2 ) shows that, in the trap model, the temper-
242
S.N. Smirnov et al. /Excess electron migration in saturated hydrocarbon
ature dependence of the mobility is mostly determined by that of the residence time of electrons in the traps, or, in slightly different terms, by the temperature dependence of the frequency of changing localization sites. In this case the jump length 1, mean full path as seen from the Einstein relation: (6)
has a weak temperature dependence. Here f. is the frequency and L the jump length. In the hopping model [ 1,5,7-g] the electron is assumed to transfer directly from one trap to the other bypassing the conduction band, i.e. by tunneling. Thus the activated character of the mobility (4) results not only from the temperature dependence of the probability of electron escape from a trap (3) but also by that for creation of a neighboring trap [ 11. As in the trapping model, the electron executes a random walk and eq. (6) applies. Now however, 1 is the mean tunneling distance and f ; ’ the mean time for tunneling. In both models, the mobility is determined by two parameters, f, and 1. Which of these has a greater effect on the mobility temperature dependence a priori is not clear. To answer the question the temperature dependence must be determined separately. The most direct determination of the hopping frequency is obtained from the exchange ESR line narrowing of the excess electron. This method has long and fruitfully been used to study the spin exchange frequency for stable radicals and radical-ions [ 10,111. The excess electron, however, cannot be studied using conventional ESR techniques due to its high reactivity at temperatures when its motion is substantially unfrozen. The OD ESR technique is very useful for this purpose [ 12-141. Earlier we have used this method to investigate the spectra of excess electrons over a wide temperature range [ 141. This paper is concerned with the experimental study of the temperature dependence of the OD ESR linewidth of the excess electron in squalane and 3-methylpentane. Its relation to the hopping frequency was verified theoretically. The depth of the excess electron traps has been shown to have a substantial dispersion in these hydrocarbons; the dispersion for the more branched
squalane is larger than that for 3-methylpentane.
2. Theory Nonexponential decay kinetics is a characteristic feature of pair recombination in OD ESR experiments. As already mentioned [ 15 1, this may lead to a different relationship between the parameters of the exchange-narrowed line and the exchange frequency than that in conventional ESR experiments. Consequently, theoretical studies have been undertaken in this work to determine this relationship which is then used in interpreting experimental OD ESR data. The OD ESR spectrum lineshape is known to be described by the function: Ccl G(w-oo)
=G(Ao) =I
(7)
0
where p&A, t) is the population of the radical pair singlet state; AU is the difference between the microwave field frequency w and that for electron spin precession w. in a constant field; F(t) describes the radical pair lifetime distribution. The angular brackets in (7) denote averaging over the ESR frequency shift, A, defined by the hyperfime interaction of electron spins: 00 0ss(A, t) > = I Z&A, MA)
do >
(8)
-Go
where p(A) is a function of the frequency shift distribution, describing the static ESR spectrum shape. We now treat the pairs consisting of the electron and radicalation whose spectra do not overlap. Therefore the lineshape can be described [ 15 ] by the expression: 00 G(Am)=t
1
[l+(W,O)lJ’(Odt,
(9)
0
where n(A, t) is the difference of the populations of the electron Zeeman levels at time t. At the initial moment one takes the condition n (A, 0) = 1. In terms of the OD ESR technique the difference Z(Ao) =G(oo) -G(Ao) is then:
243
S.N. Stiirnov et al. /Excess electron migration in saturated hydrocarbons wfi-l=
Q)
Z(Ao)=t
j [l-(n(A,t))]F(t)dt.
(10)
WI512
-iwl@1,2-a,>-fc(~-(ii>),
=iU-Aw)
~712 -h/2
-_&(A2
-
0512
> 1
0 -P12/
Thus, to calculate the OD ESR spectrum, < n(A, t) > should be determined from the equation for the density matrix describing the evolution of the electron spin states. If migration over the static spectrum contour defined by the processes of electron transfer is allowed, the equations for the density matrix are of the form (in a coordinate system rotating with frequency 0):
T2
Later we will restrict ourselves to consideration of the spectrum at small wI (neglecting saturation effects ) . Hence eqs. ( 15 ) may be solved by taking fi for 0, = 0 as follows: ii=(
w-1.
-h/2W+f,
=iU-Aw)
dt
-~12lT2
~12 -iwn/2-_Lh
3
- (~12 > )
(A2
W+f,+Ty’+i(Ao-A)
“‘=
4h2
-
(16)
Substitution of eq. ( 16) into the second equation of ( 15 ) yields:
dn - =-iwl(Plz-Pzl)-fc(n-
(15)
.
>
(17)
*
Averaging p,2 over the spectrum bp(A) gives: (11)
where pr2 is the off-diagonal element of the density matrix, f, is the spectral migration frequency; T2 is the time of transverse relaxations and o1 = yZZ,where y is the magnetogyric ratio and Hi is the microwave amplitudes. In order to determine the lineshape Z(b), it is convenient to expand the lifetime-distribution function in powers of the exponents:
where ca Z=
I
[ W+f,+Ty’+i(Ao-A)]-‘(p(A)
&I.
-03
(19) Using the same integration of the first equation of ( 15 ) and taking into account ( 18 ), we obtain:
co P(W) exp( - Wt) dW,
F(t)=!
(12)
(fi)=W-zRe{Zl(l-LZ)},
(20)
0
where P( W) is a weighting function. Thus the spectral lineshape may be written as:
where the real part of the complex expression is designated by Re. Thus substituting (20) into ( 13) we find that the spectral lineshape is:
Z(A++(W)(W-I-(A))dW,
Z(Aw) = to: ca
(13)
0
x
t
m I
P(W) W-‘Re{Z/(l-f,Z)}dW.
(21)
0
where ii=
I
n(A,t)exp(-Wt)dt.
(14)
0
In order to calculate Z(Aw) the Laplace transformation is used for the set of equations ( 11). Using the initial conditions n(A, 0) = 1, pL2(A,0) =O, yields:
Eq. ( 2 1) together with ( 19) gives a general solution. Within the range accessible to measurements, the lifetime-distribution function of the electron-cation pairs was shown [ 16-181 to be F(t)at-” where (Y decreases from 1.5 to 1 as temperature decreases. The function F(t) at- 3/2 typical of molecular-ion recombination at high temperatures, was used [ 15 ] to calculate the linewidth of the exchange-narrowed OD
244
S.N. Smirnov et al. /Excess electron migration in saturated hydrocarbons
ESR spectrum. Since in our case 1~ cr < 1.5 we have employed a distribution function of a more general type:
where d is the dispersion of the static contour. In this case formula ( 19 ) reduces to (at W= 0 ) : Z= (lc/2d)“*
F(t)=A(t+G)-*exp(
-/It)
,
where A is a normalization constant and the positive values 6 and B are introduced to provide integration convergence with time. In the final expression the 6 andpvalues will approach zero to exclude their effect on the lineshape. The Laplace transformation of eq. (22) is: P(W)=
A(W-w-’ r(a)
ew[-S(W-B)lT
(23)
(24)
~(Aw)aRe{Z/(l-f,Z)}l,=~.
Thus the non-exponential recombination kinetics examined above fail to have any effect on the lineshape described for this case by expression (24) as is common for ESR spectra when account is taken of migration over the inhomogeneous contour. Hence in what follows, the conventional theory of migration over the inhomogeneous contour [ 191 may be used. Note that this conclusion holds true only for small wI. The reason is that calculating Z(b) as/?-+0 leads to a saturated spectrum if o, is constant. To avoid saturation, o, must be decreased as 0 so that the probability of the microwave induced transitions is always less than /3.It is this calculation procedure that corresponds to result ( 24 ) . The OD ESR lineshape was detected experimentally as the first derivative, hence: aRe{ (1 -AZ)-‘dZ/dAw}
The static spectrum Gaussian,
erfc(S/,/%)
,
(27)
e&(v)=
2w Ic exp(-x2) JI Y
dw,
and S=f,+T,‘+iAw.
where r( (Y) is the gamma function, and the integration in ( 12) must be performed for /I< WC co. Substituting (23) into (2 1) we see that if a < 2 (i.e. for reasonable microwave power values), the effect of the recombination lineshape is exclusively due to the exponential cofactors in the distribution function (22). Indeed, as p-0, and for (r<2, the singularity in the integral (2 1) for W= 0 allows writing the intensity as:
$
exp(S2/2d)
(22)
,
Thus from (25 ), (27 ) we obtain &aIm{(l-Sz)/(l-LZ)*}.
(28)
Further calculations were carried out numerically based on this expression. Fig. 1 presents the calculated dependence of the linewidth (between the points of the maximum slope) on the migration rate over the inhomogeneous contour. In the region of high frequencies cf B $) , the linewidth is seen to be determined only by the transverse relaxation time T2 (the lineshape in this case being Lorentzian). In the region of low frequencies cf, c5=,/&, the linewidth is somewhat greater than that of the static contour because of contributions from transverse relaxation.
4.
3cl
(25)
shape was assumed to be
Q(A) = (27rd)-‘I* exp( -A2/2d2)
,
(26)
Fig. 1. The dependence of the contour ESR linewidth on the frequency of spectral exchangef.. The static contour width = 3.3 G: (a) l/T,=0.6G, (b) l/T, (G) =0.5+5X10-“J (Hz).
245
S.N. Smimov et al. /Excess electmn migration in saturated hydrocarbons
3. Experimental
OD ESR spectra were detected with an apparatus described earlier [ 12-15,201. The temperature was measured to an accuracy of f 0.5 ’ C by a thermocouple placed inside the specimen. The specimen was cooled by nitrogen gas passing through the cavity. The solvents, 3-methylpentane, squalane (hexamethyltetracosane), and 3-methylheptane were purified as described in refs. [ 13,141 by passing through activated silica gel. The purity was checked by the UV. To observe the excess electron signal one or another of the positive charge acceptors, N,N,N’,N’-tetramethylparaphenylene diamine (TMPD), tetramethylbenzene (durene), or triethylamine (TEA) were added to the solution at a concentration of about 10T2 M. Each substance has a negative electron affmity and a large fluorescence quantum yield. Solutions were degassed using a few freeze-pump-thaw cycles. OD ESR spectra were detected as first derivatives. Widths were estimated as the distance between the points of maximum slope in the low microwave-field amplitudes limit.
4. Results and discussion Shown in fig. 2 is the temperature dependence of the OD ESR linewidth of the excess electron in squal-
ane. The width is observed to decrease smoothly from 4.5 to 0.6 G with increasing temperature. This dependence is the same for both hole acceptors, durene and TMPD. A similar dependence is detected for the excess electron linewidth in 3-MP. A slight increase in the width with increasing temperature from 125 to 145 K is, probably, due to the temperature dependence of the transverse relaxation time T2 (see fig. 3 ). Note that the linewidth observed in the low-temperature limit is practically the same for the OD ESR and conventional ESR spectra (see table 1). This suggests that in the OD ESR technique (as in the case of ESR), the linewidth is determined by the hf interaction of the excess electron with the magnetic nuclei of the solvent trap. The lineshape is diffkult to analyze due to the distortions on the wings caused by the flipflop transitions [ 201. However, at small microwave power, it is closer to Gaussian than to Lorentzian. As shown previously [ 201, matrix rearrangement is negligible over the temperature range studied. Therefore the decrease in the linewidth with increasing temperature is caused by the averaging of the hf interaction due to electron hopping. As mentioned in section 2, the temperature dependence of the electron migration frequency between the traps (i.e. hop ping frequency) may be determined from the temperature dependence of the OD ESR linewidth
I
I
4 -
3 ,i3
z2
1 1.I.I.I.
4.5
4
4.5
5
5.i 1000/T
6
6.5
Fig 2. The temperature dependence of the OD ESR linewidth of excess electrons in squalane. Hole acceptor LO-*M durene (0 ) , IO-*MTMPD (0). (a)and(b):calculationinvolvingparameters listed in table 2; (a) corresponds to Ef=E,,=4250 K and (b ) fit to experimental data with Ef= 1800 K.
O
I.I.I.I
7
8
9
10
11 ‘is”
1000/T Fig. 3. The temperature dependence of the OD ESR linewidth of excess electrons in 3-methylpentane. Hole acceptor: lo-* M TEA. (a) and (b): calculation involving parameters listed in table 2; (a) corresponds to Ef= E,,= 4600 K and (b ) fit to experimental datawithE~=31OOK.
246
S.N. Smirnov et al. /Excess eiectron migration in saturated hydrocarbons
Table 1 ESR linewidth of excess electron Solvent
squalane 3-methyl~ntane 3-methylheptane
Our measurements (OD ESR)
Literature data (standard ESR method)
AH(G)
T(R)
AH tG)
T(R)
4.5kO.l 3.8kO.l 4.2f0.2
< 100 77 17
4.0 3.5 + 0.2 4.0+ 0.2
77 [ 18) 77 [2l] 77 [21]
without additional information about the recombination kinetics. For simplicity, the hopping frequency is assumed to be of the form
f,(T)=.&(T/1000) exp(-G/T)
procedure. Curves (a) conform to the activation energies equal to the E,,values of the escaped electron mobility [ 2 1,22 ] ; curves (b) are best fits and yield, the parameters listed in table 2. Within the temperature range under study, as seen from figs. 2 and 3 and table 2, the observed activation energies of the hopping frequency for squalane and 3-MP are different from the literature values for the activation energy of the electron mobility EN in the same solvents. On the one hand, this may be attributed to a difference in the activation energies of the mobility and of the hopping frequency (which may be explained in terms of the hopping mechanism). On the other hand, the disagreement may imply inadequacy of the model. Using the known values for&T) [22-241, a jump distances can be calculated by eq. (6) for different temperatures. For 3-MP it increases from d= 6 A at T=95Kto1=17AatT=110K.Asimilarvalueof the jump distance ( 10 A) has been obtained for amorphous 3-MP [ 221. According to Yakovlev et al. [ 221, the small value of the jump distance and, moreover, the difference between the optical absorption and bleaching spectra testify to the hopping mechanism of the excess electron mobility in amorphous 3-MP.
,
where&, Efare variable parameters. The third variable parameter is the phase relaxation time T2.However, due to a strong dependence offe on Tand of the width on f, these parameters are determined practically independently. The temperature dependence of the phase relaxation time Tz may also be taken into account in the calculations. It manifests itself, however, only in the high-temperature region where the width sometimes increases slightly with increasing temperature and has a minor effect on the deteimination off, ( T) . Note thatfc ( T) may be estimated using this method only in the temperature region where the exchange narrowed linewidth may bc found, i.e. where 1/T2< AH< 2& For other temperatures, only the upper and the lower limits of the hopping frequencies may be estimated. The solid curves in figs. 2 and 3 correspond to the theoretical dependences calculated using the above
Table 2 Parameters h and E,, for calculating mobility by eq. (4) and parameters used to calculate the AH(T) dependence in figs. 2 and 3 by formulas (26)-(29) Jb (cm2/V s)
4
fo
Ef
1/T,
(RI
(s-l)
(RI
(G)
2@ fG)
squalane
1.4x
4250 [2l]
3-MP
1.6x 10”
2.8x 6.4x 5.9x 3.4x
4250 1800 4600 3100
0.5 0.5 cl d)
4.25 4.25 3.5 3.5
Solvent
105
4600 [22]
*) Curve (a) in fip. 2 or 3. b, Curve (b) in fig. 2 or 3. d)0.5+8.6~ lO’Texp( -3100/T). ‘) 0.5+7X 10sTexp( -4600/T),
10”‘) 10” ‘) 1026” 1020b’
S.N. Smirnov et al. /Excess electronmigration in suturated hydrocarbons
More dramatic results have been obtained for squalane. The jump distances calculated from the Einstein formula (6 ) , increase from Iz= 12 A at 115 K to 92 A at 225 K. As for 3-MP the strong dependence of Aon temperature could be accounted for in terms of the hopping mechanism of electron mobility. This however, disagrees with the large absolute values of L at high temperatures. Moreover, the small values of the free ion yield for squalane at high temperatures [ 241 testify to much lower values. In this connection let us consider our model in more detail. It assumes that all traps are identical. This suggests (a) the same form of the static contour for all the traps; (b ) a Poisson time-distribution of electron escape from traps (each with the same trap depth). The former is a satisfactory approximation since the ESR spectral width for electrons in hydrocarbons is constant over the temperature range 774.2 K [ 211, even though at 4.2 K electrons may be captured by shallow traps with high probability. The presence of a distribution of trap depths has more serious consequence. Electrons hop out of traps of differing depth with rates that decrease strongly with depth. In this case the ESR spectrum is composed of the contours for electrons exchanging at different rates. The width observed will be inconsistent with the mean hopping frequency and will be defined mainly by high frequency contributions, An essential feature of the OD ESR technique is that the contribution to the OD ESR signal is made by the electron-radical-cation pairs recombining within a definite time range: 0,’
(30)
For the rapidly recombining pairs ( t < coi ’ ) the microwave field has no effect on the spin state in the pair. The pairs that recombine slowly (t > TI ) lose their spin correlation due to interaction with the lattice and therefore also fail to change the ratio between the singlet and triplet recombination products under microwave pumping, i.e. make no contribution to the OD ESR signal. Initially the distance between electron and radical cation is only a few tens of angstroms. Therefore, prior to recombination, the electron visits a limited number of traps. Thus the magnitude of the contribution to the OD ESR signal depends on the traps visited by the electron before recombination. At high tempera-
247
tures, most of electrons recombine rapidly, and the largest contribution on the OD ESR signal is made by electrons in deep traps. Consequently, the frequency of electron jumps measured from the OD ESR spectrum, will be an underestimate with respect to that for the majority of electrons. At low temperatures, electrons move towards the counter ion slowly and the recombination time for most of them is greater than T,. Then only the electrons that visit shallow traps (their recombination being fast) contribute to the OD ESR signal. This leads to overestimation of the electron migration frequency in fitting the OD ESR spectrum. Thus in the presence of a wide depth-distribution, the temperature dependence of the CD ESR spectrum width is inconsistent with that of the mean frequency of the electron jumps among the traps. A squalane molecule is very long and branched and possesses rich possibilities for conformational variations and so it has some properties analogous to polymeric liquids. For example, diffusion coefficient of solute molecules in squalane is significantly higher than calculated from viscosity [ 251. On the other hand the statistical model of the excess electron trap [ 26,271 shows that the greater the number of possible conformations for the hydrocarbon matrix molecules, the greater are the possibilities for constructing electron traps of various geometries and various depths. Thus, in squalane the trap depth-distribution function must be wider than in 3-MP. This is reflected in the temperature dependence of the electron OD ESR spectrum width. However, it is quite possible that a trap depth-distribution is also required for a complete account of the excess electron spectrum in 3-MP.
5. Conclusion The temperature dependence of the electron OD ESR spectral width in 3-MP may be described satisfactory by a model in which the traps have equal depth. In this case the frequency of the electron jumps the dependence among obeys the traps f,(T)=3.4x101’T exp(-3100/T) s-l. A similar valueforsqualanef,(T)-64xlO*Texp(-1800/T) s- ! Je&ls to ,mgscmablv large values of the iumo distance at high temperatures; 1% 100 A. This indi-
248
S.N. Smirnov et al. /Excess electron migration in saturated hydrocarbons
cates that the assumption of equal trap depths is obviously inadequate, and testifies to a wider trap depthdistribution for squalane than that for 3-methylpentane. It can be explained by differences in molecule structure of above compounds. A more rigorous justification of hypothesis of wide trap depth-distribution requires additional parameters, and accordingly, additional experimental study of such factors as lineshape and the dependence of signal intensity on temperature and microwave power.
Acknowledgement
The authors are grateful to C.L. Braun, A.B. Doktorov, B.S. Yakovlev and A.M. Raitsimring for helpful discussions.
References [ I] B.S. Yakovlev, Usp. Khim. 48 ( 1979) 1153. [2] W.F. Schmidt and A.O. Allen, J. Chem. Phys. 52 (1970) 4788. [ 3 ] K Fuery, D.F. Feng and L.A. Kevan, Chem. Phys. Letters 13 (1972) 616. [4]A.O. Allen and R.A. Holroyd, J. Phys. Chem. 78 (1974) 796. [5] J.P. Dodelet and G.R. Freeman, Can. J. Chem. 53 (1975) 1263. [ 61 J.P. Dodelet, K. Shinsaka and G.R. Freeman, J. Chem. Phys. 63 (1975) 2765. [ 71 Y. Waruyama and K. Funabashi, J. Chem. Phys. 56 ( 1972) 2342.
[8] L. Kevan, J. Phys. Chem. 76 (1972) 3830. [ 91 T. Huand and L. Kevan, J. Chem. Phys. 6 1 ( 1974) 4660. [lo] J.E. Wertz and J.R. Bolton, Electron Spin Resonance (McGraw-Hill, New York, 1972). [ 111Yu.N. Molin, K.M. Sahkhov and K-1. Zamaraev, Spin Exchange (Springer, Berlin, 1980). I 121 O.A. Anisimov, V.M. Gtigoryants, V.K. Molchanov and YuN. Molin, Chem. Phys. Letters 66 ( 1979) 265. 13lYu.N. Molin, O.A. Anisimov, V.I. Melekhov and S.N. Smirnov, Faraday Discussions Chem. Sot. 78 (1984) 289. r14]O.A.Anisimov,Y~.N.Molin,S.N.SmimovandV.A.Rogov, Rad. Phys. Chem. 23 (1984) 727. [ 151 V.O. Saik, N.N. Lukzen, V.M. Grigoryants, O.A. Anisimov, A.B. Doktorov and Yu.N. Molin, Chem. Phys. 84 (1984) 421. [ 161 J. Klein and R. Voltz, in: Proc. 6th International Congress of Radiation Research, ed. S. Okada, p. 257. [ 171 A.I. Fiksel. V.N. Parmon and K.I. Zamaraev, Khim. Vys. Ener. 18 (1984) 232. [ 181 A.I. Fiksel, Doctoral Thesis, Novosibirsk, USSR ( 1984). [ 191 A. Abragam, The Principles of Nuclear Magnetism (Oxford Univ. Press, Oxford, 196 1) . [ 201 S.N. Smimov, O.A. Anisimov and Yu.N. Molin, Chem. Phys. 124 (1988) 81. [2l]L.Kevan,Advan.Rad.Chem.4(1974) 181. [22] B.S. Yakovlev, Doctoral Thesis, Chemogolovka, USSR (1979) p. 339. [23] G.F. Novikov and B.S. Yakovlev, Intern. J. Rad. Phys. Chem. 8 (1976) 517. [ 24 ] L.V. L&in, A.P. Tchalov and B.S. Yakovlev, Khim. Fiz. 7 (1988) 1250. [ 25 ] V.O. Saik, Ph.D. Thesis, Novosibirsk, USSR ( 1985). [26] M. Sopek, M. Bartczak and J. Kroh, Bull. Pol. Acad. Sci. Chem. 34 (1986) 41. [ 271 W.M. Bartczak, M. Hilczer and J. Kroh, J. Phys. Chem. 91 (1987) 3934.
DETERMINATION OF THE FREQUENCY OF EXCESS ELECI’RON TRAP TO TRAP MIGRATION IN SATURATED HYDROCARBO NS BY AN OD ESRTECHNIQUE S.N. SMIRNOV, A.Yu. PUSEP, O.A. ANISIMOV and Yu.N. MOLIN Institute of Chemical Kinetics and Combustion, Novosibirsk, 630090, USSR Received 20 November 1989; in final form 12 February 1990
The temperature dependence of the ESR linewidth of excess electrons in squalane and 3-methylpentane has been measured by the OD ESR technique. When the temperature dependence of the linewidth is fit by a model which inchules a temperaturedependent jump frequency, Ef, it is found to be smaller than the activation energy, Er. for the electron mobility. For 3methylpentane the activation difkrence is small and the jump length calculated from the Einstein formula, 1s 10 ,k, therefore has weak dependence on the temperature. In squalane the difference between Er and Ef is greater, and the calculated large values of at hi temperatures indicate a wide depth-distribution of trap energies.
1. Introduction
The excited molecules and radical ion pairs which appear in a high-energy particle track as the initial step of any radiation-chemical process define the behavior of further chemical reactions. One of the main primary particles, the excess electron, is mobile and highly reactive. Its high mobility determines both the time scale of electron/radical-cation recombination and the rate of most chemical processes in the track. The character of the excess electron motion depends on the nature of the matrix. In polar matrices the thermalized (i.e. in the thermal equilibrium with the medium) excess electron is rapidly localized in a potential well, resulting from the polarization interaction between the electron and dipolar molecules. Since orientation polarization is not essential for electron localization in nonpolar liquids, it is not clear yet whether the excess electron state is localized [ 11. For most nonpolar molecular liquids, the equilibrium electron is considered to be neither fully localized nor quasi-free but, alternatively, in one or the other state. For hydrocarbon liquids in which the electron mobility is below 1 cmZ/V s, electron motion is usually described by one or two phenomenological models (trapping and hopping). In the trapping model [ 2-61, electrons are thought 0301-0104/90/$03.50 (North-Holland)
8 Elsevier Science Publishers B.V.
to be trapped for a time r,. Thermal activation may deliver the electron from the trap to the conduction band where it moves for time rr with high mobility & until capture. The measured mobility & is
(1) Here, A, the trapped electron mobility, is, in hydrocarbons, probably equal to the ionic mobility. Assuming rfh w r,~ we obtain: &
=Ptrf/%
*
(2)
For the simplest case of identical traps: z,=qexp(E,/kT)
(3)
and p(T)=Cbexp(-E/W),
(4)
where ~4,has replaced fi to indicate the preexponential for any temperature-activated mobility. A comparison of eq. (4 ) with the experimental dependence gives approximately the same value, cb = 100 cm2/V s, for different hydrocarbons [ 31. For a linear increase in the trap density with trap energy from 0 to E [4]: p( T)=b(
1 +E/kT)
exp( -E/kT)
.
(5)
Eq. (2 ) shows that, in the trap model, the temper-
242
S.N. Smirnov et al. /Excess electron migration in saturated hydrocarbon
ature dependence of the mobility is mostly determined by that of the residence time of electrons in the traps, or, in slightly different terms, by the temperature dependence of the frequency of changing localization sites. In this case the jump length 1, mean full path as seen from the Einstein relation: (6)
has a weak temperature dependence. Here f. is the frequency and L the jump length. In the hopping model [ 1,5,7-g] the electron is assumed to transfer directly from one trap to the other bypassing the conduction band, i.e. by tunneling. Thus the activated character of the mobility (4) results not only from the temperature dependence of the probability of electron escape from a trap (3) but also by that for creation of a neighboring trap [ 11. As in the trapping model, the electron executes a random walk and eq. (6) applies. Now however, 1 is the mean tunneling distance and f ; ’ the mean time for tunneling. In both models, the mobility is determined by two parameters, f, and 1. Which of these has a greater effect on the mobility temperature dependence a priori is not clear. To answer the question the temperature dependence must be determined separately. The most direct determination of the hopping frequency is obtained from the exchange ESR line narrowing of the excess electron. This method has long and fruitfully been used to study the spin exchange frequency for stable radicals and radical-ions [ 10,111. The excess electron, however, cannot be studied using conventional ESR techniques due to its high reactivity at temperatures when its motion is substantially unfrozen. The OD ESR technique is very useful for this purpose [ 12-141. Earlier we have used this method to investigate the spectra of excess electrons over a wide temperature range [ 141. This paper is concerned with the experimental study of the temperature dependence of the OD ESR linewidth of the excess electron in squalane and 3-methylpentane. Its relation to the hopping frequency was verified theoretically. The depth of the excess electron traps has been shown to have a substantial dispersion in these hydrocarbons; the dispersion for the more branched
squalane is larger than that for 3-methylpentane.
2. Theory Nonexponential decay kinetics is a characteristic feature of pair recombination in OD ESR experiments. As already mentioned [ 15 1, this may lead to a different relationship between the parameters of the exchange-narrowed line and the exchange frequency than that in conventional ESR experiments. Consequently, theoretical studies have been undertaken in this work to determine this relationship which is then used in interpreting experimental OD ESR data. The OD ESR spectrum lineshape is known to be described by the function: Ccl G(w-oo)
=G(Ao) =I
(7)
0
where p&A, t) is the population of the radical pair singlet state; AU is the difference between the microwave field frequency w and that for electron spin precession w. in a constant field; F(t) describes the radical pair lifetime distribution. The angular brackets in (7) denote averaging over the ESR frequency shift, A, defined by the hyperfime interaction of electron spins: 00 0ss(A, t) > = I Z&A, MA)
do >
(8)
-Go
where p(A) is a function of the frequency shift distribution, describing the static ESR spectrum shape. We now treat the pairs consisting of the electron and radicalation whose spectra do not overlap. Therefore the lineshape can be described [ 15 ] by the expression: 00 G(Am)=t
1
[l+(W,O)lJ’(Odt,
(9)
0
where n(A, t) is the difference of the populations of the electron Zeeman levels at time t. At the initial moment one takes the condition n (A, 0) = 1. In terms of the OD ESR technique the difference Z(Ao) =G(oo) -G(Ao) is then:
243
S.N. Stiirnov et al. /Excess electron migration in saturated hydrocarbons wfi-l=
Q)
Z(Ao)=t
j [l-(n(A,t))]F(t)dt.
(10)
WI512
-iwl@1,2-a,>-fc(~-(ii>),
=iU-Aw)
~712 -h/2
-_&(A2
-
0512
> 1
0 -P12/
Thus, to calculate the OD ESR spectrum, < n(A, t) > should be determined from the equation for the density matrix describing the evolution of the electron spin states. If migration over the static spectrum contour defined by the processes of electron transfer is allowed, the equations for the density matrix are of the form (in a coordinate system rotating with frequency 0):
T2
Later we will restrict ourselves to consideration of the spectrum at small wI (neglecting saturation effects ) . Hence eqs. ( 15 ) may be solved by taking fi for 0, = 0 as follows: ii=(
w-1.
-h/2W+f,
=iU-Aw)
dt
-~12lT2
~12 -iwn/2-_Lh
3
- (~12 > )
(A2
W+f,+Ty’+i(Ao-A)
“‘=
4h2
-
(16)
Substitution of eq. ( 16) into the second equation of ( 15 ) yields:
dn - =-iwl(Plz-Pzl)-fc(n-
(15)
.
>
(17)
*
Averaging p,2 over the spectrum bp(A) gives: (11)
where pr2 is the off-diagonal element of the density matrix, f, is the spectral migration frequency; T2 is the time of transverse relaxations and o1 = yZZ,where y is the magnetogyric ratio and Hi is the microwave amplitudes. In order to determine the lineshape Z(b), it is convenient to expand the lifetime-distribution function in powers of the exponents:
where ca Z=
I
[ W+f,+Ty’+i(Ao-A)]-‘(p(A)
&I.
-03
(19) Using the same integration of the first equation of ( 15 ) and taking into account ( 18 ), we obtain:
co P(W) exp( - Wt) dW,
F(t)=!
(12)
(fi)=W-zRe{Zl(l-LZ)},
(20)
0
where P( W) is a weighting function. Thus the spectral lineshape may be written as:
where the real part of the complex expression is designated by Re. Thus substituting (20) into ( 13) we find that the spectral lineshape is:
Z(A++(W)(W-I-(A))dW,
Z(Aw) = to: ca
(13)
0
x
t
m I
P(W) W-‘Re{Z/(l-f,Z)}dW.
(21)
0
where ii=
I
n(A,t)exp(-Wt)dt.
(14)
0
In order to calculate Z(Aw) the Laplace transformation is used for the set of equations ( 11). Using the initial conditions n(A, 0) = 1, pL2(A,0) =O, yields:
Eq. ( 2 1) together with ( 19) gives a general solution. Within the range accessible to measurements, the lifetime-distribution function of the electron-cation pairs was shown [ 16-181 to be F(t)at-” where (Y decreases from 1.5 to 1 as temperature decreases. The function F(t) at- 3/2 typical of molecular-ion recombination at high temperatures, was used [ 15 ] to calculate the linewidth of the exchange-narrowed OD
244
S.N. Smirnov et al. /Excess electron migration in saturated hydrocarbons
ESR spectrum. Since in our case 1~ cr < 1.5 we have employed a distribution function of a more general type:
where d is the dispersion of the static contour. In this case formula ( 19 ) reduces to (at W= 0 ) : Z= (lc/2d)“*
F(t)=A(t+G)-*exp(
-/It)
,
where A is a normalization constant and the positive values 6 and B are introduced to provide integration convergence with time. In the final expression the 6 andpvalues will approach zero to exclude their effect on the lineshape. The Laplace transformation of eq. (22) is: P(W)=
A(W-w-’ r(a)
ew[-S(W-B)lT
(23)
(24)
~(Aw)aRe{Z/(l-f,Z)}l,=~.
Thus the non-exponential recombination kinetics examined above fail to have any effect on the lineshape described for this case by expression (24) as is common for ESR spectra when account is taken of migration over the inhomogeneous contour. Hence in what follows, the conventional theory of migration over the inhomogeneous contour [ 191 may be used. Note that this conclusion holds true only for small wI. The reason is that calculating Z(b) as/?-+0 leads to a saturated spectrum if o, is constant. To avoid saturation, o, must be decreased as 0 so that the probability of the microwave induced transitions is always less than /3.It is this calculation procedure that corresponds to result ( 24 ) . The OD ESR lineshape was detected experimentally as the first derivative, hence: aRe{ (1 -AZ)-‘dZ/dAw}
The static spectrum Gaussian,
erfc(S/,/%)
,
(27)
e&(v)=
2w Ic exp(-x2) JI Y
dw,
and S=f,+T,‘+iAw.
where r( (Y) is the gamma function, and the integration in ( 12) must be performed for /I< WC co. Substituting (23) into (2 1) we see that if a < 2 (i.e. for reasonable microwave power values), the effect of the recombination lineshape is exclusively due to the exponential cofactors in the distribution function (22). Indeed, as p-0, and for (r<2, the singularity in the integral (2 1) for W= 0 allows writing the intensity as:
$
exp(S2/2d)
(22)
,
Thus from (25 ), (27 ) we obtain &aIm{(l-Sz)/(l-LZ)*}.
(28)
Further calculations were carried out numerically based on this expression. Fig. 1 presents the calculated dependence of the linewidth (between the points of the maximum slope) on the migration rate over the inhomogeneous contour. In the region of high frequencies cf B $) , the linewidth is seen to be determined only by the transverse relaxation time T2 (the lineshape in this case being Lorentzian). In the region of low frequencies cf, c5=,/&, the linewidth is somewhat greater than that of the static contour because of contributions from transverse relaxation.
4.
3cl
(25)
shape was assumed to be
Q(A) = (27rd)-‘I* exp( -A2/2d2)
,
(26)
Fig. 1. The dependence of the contour ESR linewidth on the frequency of spectral exchangef.. The static contour width = 3.3 G: (a) l/T,=0.6G, (b) l/T, (G) =0.5+5X10-“J (Hz).
245
S.N. Smimov et al. /Excess electmn migration in saturated hydrocarbons
3. Experimental
OD ESR spectra were detected with an apparatus described earlier [ 12-15,201. The temperature was measured to an accuracy of f 0.5 ’ C by a thermocouple placed inside the specimen. The specimen was cooled by nitrogen gas passing through the cavity. The solvents, 3-methylpentane, squalane (hexamethyltetracosane), and 3-methylheptane were purified as described in refs. [ 13,141 by passing through activated silica gel. The purity was checked by the UV. To observe the excess electron signal one or another of the positive charge acceptors, N,N,N’,N’-tetramethylparaphenylene diamine (TMPD), tetramethylbenzene (durene), or triethylamine (TEA) were added to the solution at a concentration of about 10T2 M. Each substance has a negative electron affmity and a large fluorescence quantum yield. Solutions were degassed using a few freeze-pump-thaw cycles. OD ESR spectra were detected as first derivatives. Widths were estimated as the distance between the points of maximum slope in the low microwave-field amplitudes limit.
4. Results and discussion Shown in fig. 2 is the temperature dependence of the OD ESR linewidth of the excess electron in squal-
ane. The width is observed to decrease smoothly from 4.5 to 0.6 G with increasing temperature. This dependence is the same for both hole acceptors, durene and TMPD. A similar dependence is detected for the excess electron linewidth in 3-MP. A slight increase in the width with increasing temperature from 125 to 145 K is, probably, due to the temperature dependence of the transverse relaxation time T2 (see fig. 3 ). Note that the linewidth observed in the low-temperature limit is practically the same for the OD ESR and conventional ESR spectra (see table 1). This suggests that in the OD ESR technique (as in the case of ESR), the linewidth is determined by the hf interaction of the excess electron with the magnetic nuclei of the solvent trap. The lineshape is diffkult to analyze due to the distortions on the wings caused by the flipflop transitions [ 201. However, at small microwave power, it is closer to Gaussian than to Lorentzian. As shown previously [ 201, matrix rearrangement is negligible over the temperature range studied. Therefore the decrease in the linewidth with increasing temperature is caused by the averaging of the hf interaction due to electron hopping. As mentioned in section 2, the temperature dependence of the electron migration frequency between the traps (i.e. hop ping frequency) may be determined from the temperature dependence of the OD ESR linewidth
I
I
4 -
3 ,i3
z2
1 1.I.I.I.
4.5
4
4.5
5
5.i 1000/T
6
6.5
Fig 2. The temperature dependence of the OD ESR linewidth of excess electrons in squalane. Hole acceptor LO-*M durene (0 ) , IO-*MTMPD (0). (a)and(b):calculationinvolvingparameters listed in table 2; (a) corresponds to Ef=E,,=4250 K and (b ) fit to experimental data with Ef= 1800 K.
O
I.I.I.I
7
8
9
10
11 ‘is”
1000/T Fig. 3. The temperature dependence of the OD ESR linewidth of excess electrons in 3-methylpentane. Hole acceptor: lo-* M TEA. (a) and (b): calculation involving parameters listed in table 2; (a) corresponds to Ef= E,,= 4600 K and (b ) fit to experimental datawithE~=31OOK.
246
S.N. Smirnov et al. /Excess eiectron migration in saturated hydrocarbons
Table 1 ESR linewidth of excess electron Solvent
squalane 3-methyl~ntane 3-methylheptane
Our measurements (OD ESR)
Literature data (standard ESR method)
AH(G)
T(R)
AH tG)
T(R)
4.5kO.l 3.8kO.l 4.2f0.2
< 100 77 17
4.0 3.5 + 0.2 4.0+ 0.2
77 [ 18) 77 [2l] 77 [21]
without additional information about the recombination kinetics. For simplicity, the hopping frequency is assumed to be of the form
f,(T)=.&(T/1000) exp(-G/T)
procedure. Curves (a) conform to the activation energies equal to the E,,values of the escaped electron mobility [ 2 1,22 ] ; curves (b) are best fits and yield, the parameters listed in table 2. Within the temperature range under study, as seen from figs. 2 and 3 and table 2, the observed activation energies of the hopping frequency for squalane and 3-MP are different from the literature values for the activation energy of the electron mobility EN in the same solvents. On the one hand, this may be attributed to a difference in the activation energies of the mobility and of the hopping frequency (which may be explained in terms of the hopping mechanism). On the other hand, the disagreement may imply inadequacy of the model. Using the known values for&T) [22-241, a jump distances can be calculated by eq. (6) for different temperatures. For 3-MP it increases from d= 6 A at T=95Kto1=17AatT=110K.Asimilarvalueof the jump distance ( 10 A) has been obtained for amorphous 3-MP [ 221. According to Yakovlev et al. [ 221, the small value of the jump distance and, moreover, the difference between the optical absorption and bleaching spectra testify to the hopping mechanism of the excess electron mobility in amorphous 3-MP.
,
where&, Efare variable parameters. The third variable parameter is the phase relaxation time T2.However, due to a strong dependence offe on Tand of the width on f, these parameters are determined practically independently. The temperature dependence of the phase relaxation time Tz may also be taken into account in the calculations. It manifests itself, however, only in the high-temperature region where the width sometimes increases slightly with increasing temperature and has a minor effect on the deteimination off, ( T) . Note thatfc ( T) may be estimated using this method only in the temperature region where the exchange narrowed linewidth may bc found, i.e. where 1/T2< AH< 2& For other temperatures, only the upper and the lower limits of the hopping frequencies may be estimated. The solid curves in figs. 2 and 3 correspond to the theoretical dependences calculated using the above
Table 2 Parameters h and E,, for calculating mobility by eq. (4) and parameters used to calculate the AH(T) dependence in figs. 2 and 3 by formulas (26)-(29) Jb (cm2/V s)
4
fo
Ef
1/T,
(RI
(s-l)
(RI
(G)
2@ fG)
squalane
1.4x
4250 [2l]
3-MP
1.6x 10”
2.8x 6.4x 5.9x 3.4x
4250 1800 4600 3100
0.5 0.5 cl d)
4.25 4.25 3.5 3.5
Solvent
105
4600 [22]
*) Curve (a) in fip. 2 or 3. b, Curve (b) in fig. 2 or 3. d)0.5+8.6~ lO’Texp( -3100/T). ‘) 0.5+7X 10sTexp( -4600/T),
10”‘) 10” ‘) 1026” 1020b’
S.N. Smirnov et al. /Excess electronmigration in suturated hydrocarbons
More dramatic results have been obtained for squalane. The jump distances calculated from the Einstein formula (6 ) , increase from Iz= 12 A at 115 K to 92 A at 225 K. As for 3-MP the strong dependence of Aon temperature could be accounted for in terms of the hopping mechanism of electron mobility. This however, disagrees with the large absolute values of L at high temperatures. Moreover, the small values of the free ion yield for squalane at high temperatures [ 241 testify to much lower values. In this connection let us consider our model in more detail. It assumes that all traps are identical. This suggests (a) the same form of the static contour for all the traps; (b ) a Poisson time-distribution of electron escape from traps (each with the same trap depth). The former is a satisfactory approximation since the ESR spectral width for electrons in hydrocarbons is constant over the temperature range 774.2 K [ 211, even though at 4.2 K electrons may be captured by shallow traps with high probability. The presence of a distribution of trap depths has more serious consequence. Electrons hop out of traps of differing depth with rates that decrease strongly with depth. In this case the ESR spectrum is composed of the contours for electrons exchanging at different rates. The width observed will be inconsistent with the mean hopping frequency and will be defined mainly by high frequency contributions, An essential feature of the OD ESR technique is that the contribution to the OD ESR signal is made by the electron-radical-cation pairs recombining within a definite time range: 0,’
(30)
For the rapidly recombining pairs ( t < coi ’ ) the microwave field has no effect on the spin state in the pair. The pairs that recombine slowly (t > TI ) lose their spin correlation due to interaction with the lattice and therefore also fail to change the ratio between the singlet and triplet recombination products under microwave pumping, i.e. make no contribution to the OD ESR signal. Initially the distance between electron and radical cation is only a few tens of angstroms. Therefore, prior to recombination, the electron visits a limited number of traps. Thus the magnitude of the contribution to the OD ESR signal depends on the traps visited by the electron before recombination. At high tempera-
247
tures, most of electrons recombine rapidly, and the largest contribution on the OD ESR signal is made by electrons in deep traps. Consequently, the frequency of electron jumps measured from the OD ESR spectrum, will be an underestimate with respect to that for the majority of electrons. At low temperatures, electrons move towards the counter ion slowly and the recombination time for most of them is greater than T,. Then only the electrons that visit shallow traps (their recombination being fast) contribute to the OD ESR signal. This leads to overestimation of the electron migration frequency in fitting the OD ESR spectrum. Thus in the presence of a wide depth-distribution, the temperature dependence of the CD ESR spectrum width is inconsistent with that of the mean frequency of the electron jumps among the traps. A squalane molecule is very long and branched and possesses rich possibilities for conformational variations and so it has some properties analogous to polymeric liquids. For example, diffusion coefficient of solute molecules in squalane is significantly higher than calculated from viscosity [ 251. On the other hand the statistical model of the excess electron trap [ 26,271 shows that the greater the number of possible conformations for the hydrocarbon matrix molecules, the greater are the possibilities for constructing electron traps of various geometries and various depths. Thus, in squalane the trap depth-distribution function must be wider than in 3-MP. This is reflected in the temperature dependence of the electron OD ESR spectrum width. However, it is quite possible that a trap depth-distribution is also required for a complete account of the excess electron spectrum in 3-MP.
5. Conclusion The temperature dependence of the electron OD ESR spectral width in 3-MP may be described satisfactory by a model in which the traps have equal depth. In this case the frequency of the electron jumps the dependence among obeys the traps f,(T)=3.4x101’T exp(-3100/T) s-l. A similar valueforsqualanef,(T)-64xlO*Texp(-1800/T) s- ! Je&ls to ,mgscmablv large values of the iumo distance at high temperatures; 1% 100 A. This indi-
248
S.N. Smirnov et al. /Excess electron migration in saturated hydrocarbons
cates that the assumption of equal trap depths is obviously inadequate, and testifies to a wider trap depthdistribution for squalane than that for 3-methylpentane. It can be explained by differences in molecule structure of above compounds. A more rigorous justification of hypothesis of wide trap depth-distribution requires additional parameters, and accordingly, additional experimental study of such factors as lineshape and the dependence of signal intensity on temperature and microwave power.
Acknowledgement
The authors are grateful to C.L. Braun, A.B. Doktorov, B.S. Yakovlev and A.M. Raitsimring for helpful discussions.
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