J. Phys. Chem. 1994,98, 13249-13261
13249
FDMR in Low-Temperature Solids. 1. Theory? I. A. Shkrob" and A. D. Trifunac Chemistry Division, Argonne National Laboratory, Argonne, Illinois 60439 Received: November 23, 1993; In Final Form: July 6, 1994@
Various mechanisms of geminate recombination in spin-correlated radical ion pairs produced by radiolysis of organic solids under cryogenic conditions are considered and compared with experimental data. Single-step electron tunneling and slow resonant hole transfer are examined from the viewpoint of their manifestation in pulsed time-resolved fluorescence-detected magnetic resonance (FDMR). Two theoretical models treating the phenomenon are developed, and their predictions are compared with the experimental results. These models are found to be consistent with many observed features: t-I decay kinetics, unusual evolution of the FDMR spectra with time, narrow radial distribution of the FDMR-active pairs, and efficient spin memory transfer from the primary pairs. However, with very few exceptions the tunneling theories fail to reproduce the exact scale of the effect and its dependence on solute concentration and thermalization distance of electrons. This discrepancy suggests that the negative charge is more mobile than predicted by the single-step tunneling model.
1. Introduction Despite extensive study over the last 30 years, electron transfer (ET) and charge mobility in primary radiolytic events are still the subject of considerable attention and controversy. While in liquids two key mechanisms of charge migration, ionic diffusion, and resonance transfer provide a consistent rationalization for the majority of experimental results, neither of these appears to dominate in frozen organic solids.' For lowtemperature solid media, the mobility of charged particles has been usually assigned to quantum-mechanical electron tunneling (Figure 1).2-6 The manifestations of this tunneling, however, may be very different depending on the time scale and conditions of a given experiment. The accepted viewpoint is that the tunneling in rigid matrices is a single-step process occurring over a time scale of 14 orders of magnitude and the range 1-200 A.2,3,6However, for some media, such as amorphous AszSe3 (e.g., ref 7), polyvinylcarbazole films (e.g., ref 8), and few soft organic glasses at 77 K? more sophisticated types of mobility involving site-to-site tunneling of the charges has been found. In this context, observations of steady-stateloand pulsed timeresolved fluorescence-detected magnetic resonance (FDMR)' from a number of matrices at 4-80 K are quite intriguing. In the presence of an aromatic scintillator (A) an ionization of a solvent molecule (S) with energetic electrons or X-rays
'
S-''S**--s++efollows a familiar sequence of ET reactions: e- +A-'A-
's+ + 'A-S+
+
-
(1.2)
S + 'g3A*
+A s + -c
+ 'Ae- +
-A+
e-
-c
2
~
+
A + 1,3A*
-c
'p3A*
- -sH-*+ S*
(1.1)
(1.3) (1.4)
(15 )
(1.6)
H*
(1.7) leading to delayed fluorescence from the excited singlet 2 ~ +
* Author to whom correspondence should be addressed. +Work performed under the auspices of the Office of Basic Energy Sciences, Division of Chemical Science, US-DOE, under contract number W-31-109-ENG-38. Abstract published in Advance ACS Abstracts, November 15, 1994. @
scintillator:
'A* - A
+ hv
(1.8)
FDMR is the observation of the fluorescence yield as a function of the external magnetic field under microwave @w) resonance conditions.'* The technique is very discriminative toward the reactions of geminate radical ion pairs generated in steps 1.2-1.5. The method exploits spin selection rules in ET reactions and singlet correlation in radiolytically formed geminate pairs. The yield of the excited singlet 'A* depends on the number of preceding singlet pairs, which, in turn, depends on the efficiency of intersystem crossing in the pairs. Under resonance conditions the microwave field accelerates spin transitions between the split triplet sublevels of the geminate pairs. Only one of these triplet states is energetically close to the singlet state. Consequently, pw pumping mixes the spin states more uniformly and speeds up intersystem crossing, increasing the fraction of the triplet pairs and decreasing the fluorescence yield. Thus, the recombination has to compete with the fast transitions between the spin states to provide the FDMR effect. It is far from evident why in rigid glasses (or, generally, in solids at 4-20 K), where any diffusion or displacement of the species is inhibited, the recombination is sufficiently rapid to compete with these fast transitions (lo7109 s-l). Note that in liquids it is diffusion (driven by the Coulombic attraction) and the re-encounters of partners which cause FDMR.12 Diffusion does have the necessary timing (0.01-1 ps) to interplay with the spin d y n a m i ~ s . ' ~ Electron J~ tunneling is most routinely observed on the time scale of seconds and and, at least at first glance, it seems too slow a process to provide any significant FDMR. Perhaps, it was this sort of reasoning which led to the revitalization of alternative mechanisms of charge mobility in solids, such as resonance charge transfer,15
's+(s> + SCSJ
s(%)+ 2S+cOn)
(1.9)
(here a, and ,& are the spin orientations of a given nucleus in the hole), and thermal detrapping of the electIons,7-9,ii as possible origins of the observed FDMR. Both these mechanisms (in application to organic glasses) were the subject of decades-
0022-3654/94/2098-13249$04.50/0 0 1994 American Chemical Society
13250 J. Phys. Chem., Vol. 98, No. 50, I994 1. Single-Step Tunneling
Shkrob and Trifunac 2. Resonance charge transfer
13. Multi-site diffusion of electrons
I
II Figure 1. Mechanisms of charge transport in low-temperature dielectric solids.
long discussion and were gradually abandoned in favor of the more verified concept of long-range single-step tunneling. Undoubtedly, sufficiently fast resonance charge transfer would lead to a higher hole m0bi1ity.l~ The process may be easily fingerprinted with a variety of spin resonance techniques since the hopping mixes nuclear subensembles of the radical cations and causes a pronounced spectral diffu~ion.'~*'~ In a number of works, this mechanism was proposed to explain the higherthan-diffusional hole mobilities in liquid cyclohexane and its derivative^.'^ However, later work showed that even in this case the high mobility was most likely due to the proton transfer in cyclohexyl cations.16 Contemporary ET theories predict that the resonance charge transfer is unlikely to be a source of any significant charge mobility at 4-20 K. Indeed, the transfer requires the excess energy of -0.1-0.5 eV for medium reorganization.2d The barrier manifoldly exceeds the thermal activation energy under cryogenic conditions. This implies that the transfer must be a very slow process. In accordance with this conclusion, for most examined systems, no changes in the spectral lines from the solvent holes, exceeding the expected variation of the inhomogeneous broadening, have been found in both steady-statelh and pulsedllb solid-state FDMR. Moreover, experiments on hole scavenging showed independence of C50 (the concentration of scavenger at which the FDMR signal from the hole decreases twice) on the concentration of the hole precursor.lh These findings are hardly compatible with the paradigm of resonance transfer, or at least with a concept of a short-range electron-hopping reaction.( 1.9). Nevertheless, in the later work on steady-state FDMR from glassy 3-methylpentane (at 77 K), Anisimov et al.*Obonce more invoked resonance charge transfer to explain spectral variations observed with increase in the field modulation frequency (0.212.5 kHz)and the pw power. A theoretical simulation of these effects yielded a hopping time of -0.65 ps. This time is considerably longer than the typical period of observation in our fast time-resolved experiments (100-200 ns). Therefore, in the pulsed experiments the spectral effects due to the hole hopping would have been less significant and easily obscured by the pulse and power distortions. Even on the longer time scale, the occurrence of reaction 1.9 is not firmly established. Similar effects may be caused by
a hindered rotation in the methyl and methylene fragments of the radical cations. Though the authorslobfound this improbable, at the present moment it is not clear how such an effect would interplay with the anisotropic hyperfine interaction (hfi), inhomogeneous broadening, electron spin exchange, and dipoledipole interaction, which were ignored in the simulations performed. Furthermore, the reactivity of the solvent hole may depend on the conformation of the radical cation. On the other hand, reaction 1.9 was assumed to have a single first-order rate constant, which implies that the transfer occurs between neighboring species. These very short and slow hops could not provide the observed charge migration over -20-50 solvent molecules within less than 100 ns. It was estimated in ref 10b that the observed spectral effects occur after -150 hole jumps over -80 ps. In sum, resonance charge transfer between the neighboring solvent molecules is too slow to account for the charge mobilities observed on the sub-microsecond scale in the solid. Of course, this does not contradict the fact that electron hopping strongly affects the millisecond-resolved FDMR and, as will be shown in the second part of this series, in some systems (such as n-alkanes) operates even on the microsecond scale. An alternative approach to the excessive hole mobility recalls the fact that within the first picosecond the solvent holes are vibrationally and electronically excited and the ion-molecular transfer between the excited species and solvent molecules is energetically favorable. According to ref 17, the excited holes could migrate over -lo00 solvent molecules (although many results reported in ref 17 may be easily reinterpreted in terms of single-step tunneling%). If the holes were so exceptionally mobile, the cation-electron distribution would be nearly random. This would result in the independence of FDMR on the thermalization distance of the electrons. Furthermore, the migration over such large distances would destroy the spin correlation in the pairs. Our experimental results are inconsistent with this picture. On the same basis we should omit a similar approach suggesting an involvement of extremely mobile and reactive pretrapped ("dry") electrons.'*'* If not due to reaction 1.9 the mobility could have been caused by electron tunneling. In fact, the FDMR results discussed above were simulated within this assumption: the hopping was considered as a side process which accompanies the tunneling but does not add to the charge mobility.'("' The ET rate constant
FDMR in Low-Temperature Solids
J. Phys. Chem., Vol. 98, No. 50, 1994 13251
K ( r ) = exp(-r/A) is known to decrease nearly exponentially with the separation distance r between the radical ions.2,6 Assuming the decrease parameter A 1 A, for fast charge transfers the constant K(r) is -106-107 s-l at r 20 and -103-104 s-l at r 30 A. Thus, only spin-selective transfers from pairs with 20 A < r < 30 8,contribute to the signal. These separations are 3-4 times smaller than the thermalization distances in nonpolar hydrocarbon liquids.l9 Seemingly, our estimates indicate that the mobilities of ionic species in the media are higher than the values provided by single-step tunneling models. In a previous paper from this laboratory,llb we proposed that these higher-than-expected mobilities are due to the so-called phonon-assisted detrapping of charge^.^,^ The conclusion was based on the -trl time dependence of the observed FDMR signal. Such time dependences were previously obtained in the photoconductometric experiments with some solid materials.’~~ However, heretofore it has never been proved to be the case with rigid glasses at liquid helium temperature^.^.^ For this case, the single-step tunneling is a much more plausible explanation of the result^.^ Although the tunneling mechanism seems to account for many observations, most importantly for the results on electron and hole scavenging,2 it is nevertheless must be supplemented with other concepts to account for the observed data quantitatively or, sometimes, even q u a l i t a t i ~ e l y .In~ many ~ ~ ~ ~cases ~ ~ the trap depth of electrons is time-dependent, slowly increasing with time.2b In contrast to thermal detrapping this process does not require phonon-assisted activation. The resulting site-to-site electron migration is frequently referred to as a multisite diffusion (parallel to the Brownian diffusion with which it shares some common features) (see ref 20b for a review). According to some theoretical work, this mechanism might also cause a t-a dependence of the effective reaction rates.8*20A “trap deepening”, which is the final result of this diffusion, decreases exothermicity of the ET reactions and, consequently, reduces scavenging rates. This progressive reduction of the reaction constants causes a flattening of the electron decay curves in many systems involving both very weak and very strong electron acceptors.2bld “Deepening of traps” constitutes only a part of the problem. In many cases an excess of energy released with ET is sufficient to excite the radical anions electronically.2d Since by probing the fluorescence we observe the final result of ET reactions, an involvement of these excited intermediates may, and probably does, affect the formation of FDMR. These examples demonstrate that single-step tunneling is inevitably accompanied by a variety of side processes and should not be envisaged as a universal clue to all the secrets of solid-state chemistry. One of the main obstacles in application of FDMR and the related techniques to the analysis of low-temperature kinetics is a lack of theoretical background. The knowledge of how the results “should” look within this or that model permits an elimination of false concepts and the development better ones. With modem FDMR spectroscopy this problem becomes rather serious since so far the theoretical apparatus for this technique was targeted on the diffusion-controlled reactions in liquid13J4 or ET reactions in photosynthetic centers.21 Neither of those resembles a solid at cryogenic temperatures. In this work we consider the solid-state FDMR formation within the assumption of single-step tunneling as the sole mechanism of the charge mobility. We are aware that, perhaps, our treatment is quite mechanistic, and we will attempt to apply several corrections later.22 However, this single-step tunneling forms the keystone of our framework.
-
-
-
2. General Framework An overview of the low-temperature data reveals an important feature which illustrates the difference between solid-state FDMR spectroscopy and the liquid-state analog. While with the latter technique the resonance signals from the [e-. .2A+] and [2A+..2A+]pairs in the systems involving the most common electron donor-acceptor scintillators, namely, aromatic hydrocarbons, were readily observed even at quite low concentrations M),l2sZ3in the solid only the signals from the [%+. .2A-] pairs were detected.’0s11*22This qualitative difference is a consequence of the inefficiency of ET over large distances by which randomly distributed solute molecules are separated. Indeed, statistical weights of the favorable arrangements when the trapped electron et-, the scintillator molecule A, and the solvent hole 2S+ are sufficiently close in space are already so small that an addition of another solute molecule (which has to be in the vicinity of the hole for reaction 1.4 to occur) gives a minor contribution to the overall FDMR effect. As for reaction 1.6, which is statistically allowed, an absence of the FDMR signals from the corresponding pairs is probably due to a strong state-locking (see below) in the radical ions under the pw conditions we use. Consequently, we will limit our consideration to the analysis of spin and reaction dynamics in the [W..2A-] pairs. For the systems involving strong electron donors D (e.g., aromatic amines) the fluorescence was formed exclusively in the [e-.?D+] pairs since in this case reaction 1.2 either does not occur or is too slow.2,22Nevertheless, very distinct signals from the solvent holes were found even in these electron donor systems, indicating a substantial spin mixing in the primary [et-. pairs prior to the solvent hole scavenging.22 We may suppose that to some extent the same process occurs in the systems involving scintillators of the acceptor type (as was observed for octafluoronaphthalene in methylcyclohexane22) and even those of the donor-acceptor type. Indeed, in the frozen solid the lifetime of primary pairs may be rather long, providing that the solute molecule is remote or ET to this molecule is a slow reaction. Although the transfer of spin memory developed in the primary pairs has not been observed in radiolytic systems previously, a fairly analogous transfer from the [*A-. .2S+] pairs to the [*A-. .2A+] pairs is documented.23b All these radical ion pairs are influenced by miscellaneous magnetic forces, varying in their effective times and magnitudes. Those of prime importance are the Zeeman interaction of the electrons with the static field Bo of the electromagnet, the Zeeman interaction with the pw field B1, the hyperfine interaction of the electrons with nuclei, and the anisotropic dipoledipole interaction. It is convenient to consider all these interactions in a frame rotating with the right-hand component of the plane-polarized pw radiation (the so-called rotating frame). Provided that the magnetic field of the spectrometer (or Bo field) is sufficiently high and B1 = const, the spin-Hamiltonians H1,2 of the pairs in the rotating frame are time-independent. Therefore, we introduce density matrices Q1,2 = @1,2(r1,81;r2,82;~;t) of the primary [%+. .et-] and secondary [%+. .*A-] (or [et-. .2D+]) pairs, respectively, in the rotating frame, where 71 is the distance between et- and 2S+, r2 is the distance between 2Sf and 2A(or e,- and 2D+), 8 1 , ~is the angle between the Bo field and the radius-vectors connecting the correspondent ionic centers, and 4 is the angle between the planes defined by these radius-vectors and Bo (Figure 2a). The density matrices e 1 . 2 obey the master Liouville equation12-14
13252 J. Phys. Chem., Vol. 98, No. 50, I994
t
Shkrob and Trifunac mixed behavior
Bo
mixed behavior cofigurations favorable for FDMR from the secondary pairs only
*S+
I (b)
configurations favorable for spin memry transfer
(c)
Figure 2. (a) Spatial arrangement, reaction parameters, and constants of an ensemble including a trapped electron et-, a solvent hole 2S+,and an electron donor molecule D (or the foregoing radical cation 2D+). (b) Pictorial representation of the areas in which ET reactions of D/ZD+ with 2S+/e,- have the optimum rate constants K(rmin) < K122 < K(rm) for FDMR formation and spin memory transfer from the primary pairs andor FDMR in the secondary pairs. Since the correspondent spherical layers do not intersect, the given configuration yields no FDMR. (c) Configurations of *S+, et-, and DPD+ favorable for observations of FDMR from the secondary pairs (vide supra) and primary pairs (vide infra).
In the above equations the quantities q5s and & are the relative yields of the singlet and triplet reaction products. Recent studies indicate that for a typical recombination in the radiolytic systems q5s t$t.24 Although nothing is known about the temperature dependence of q5s and 4, we may hypothesize that their ratio is approximately the same even in the cryogenic region," and for simplicity we will assume that Ks = Kt = K. It will also be accepted that the decrease parameter iZ is the same for both the singlet and triplet pathways. The assumption is important since, as will be shown, most FDMR-active transfers occur at substantial separation between the radical ions (>20 A), and even a small difference in the decrease parameters may result in an order of magnitude difference in the ET rate constants at these separations. We will also neglect the singlet-singlet excitation energy transfer from the excited solvent molecules (reaction 1.7) to the scintillator due to the rapid dissociation of these solvent states.25 By definition, the FDMR signal S(t) is the difference between the fluorescence traces with and without the applied pw field,
-
where the first term involves spin transitions in the pair (here [ 3- and [ ]+ are the commutator and anticommutator, respectively). The following terms describe a spin-selective decay of the correspondent pairs via the singlet and triplet routes, respectively (in our notations Qs= Is>(Sl is the projector on the singlet states of the pair, Qt = &dlT9)(TdI is the projector on the triplet states, q, q' = -1, 0, +l); K1,2;s,tand K&) are the first-order rate constants of ET in the correspondent pairs. These ET reactions will be treated as single-step electron tunneling; that is, it will be assumed that the radial dependence of the constants K(r) is given by a single-exponential function,
S(t) = I(t;B,=O) - I ( @ , ) where A e ( f i 2 / 8 ~ ) - l nis the decrease parameter close to the Bohr radius (-1 - 1.2 A (ref 2)), B is the binding energy at the electron donor (-1 - 1.6 eV), me is the electron mass, and r is the minimum approach between the partners involved in the ET. The last term describes a non-spin-selective transformation of the primary pairs into the secondary ones with a constant K12, where r12 is the distance between e- and A (or 2S+ and D, respectively).
(2.3)
The fluorescence trace I(t) is a convolution of (i) the exponential function describing radiative decay of the singlet product with the rate constant of kf,(ii) a probability P(rl,Ol;r2,&;~$;J)for the secondary pair to undergo ET from the singlet state at the moment J provided that the pair has been born at a given separation r1.2 and orientation 01.2 at t = 0 and (iii) distribution pe(r1) of the trapping distances
FDMR in Low-Temperature Solids Z(t) = L ' d t
J. Phys. Chem., Val. 98, No. 50, 1994 13253 are the Laplace-transformed density matrices obeying the equations
kf exp[-k&-
t')]f2nr12 dr, f2nr:
dr2 JndO1 sin(8,) JZdO2 sin(8,) x
(2.5) This distribution p,(r) may be simulated with $-exponential function,26
or $-Gaussian function,
pe(r)= (4n)-l[*] A b :
I');(
exp[-
(2.6b)
(where re = 3 b = ~ 2n-ll2b~is the thermalization distance). The last term in eq 2.5 represents a probability to find a scintillator molecule (evenly distributed with a bulk concentration C) at a given separation i-12 between A and e,- (or D and 2S+) and no other scintillator molecules in between.& Thus, Z(r) is a rather complex quantity reflecting the formation of the primary and secondary pairs, their spin dynamics, reactive decay, and the radiative depletion of the singlet product. Computationally, solving eqs 2.1-2.5 is impossible without simplifying assumptions. Here we examine two such approaches. Within the first one, called here exponential windowing, the experimental boxcar integration of the fluorescence traces (which may be regarded as a convolution of the signal with a rectangular window function) is simulated by a convolution of the signal with an exponential window function F(t) = In other words, a quantity
Z, = h m d rZ(r) exp(-tlz)
(2.7)
will be considered as an approximation to the boxcar-integrated signal. Note that the first derivative (-z dZJdz) would give a result of convolution of Z(t) and the t-exponential window function F(t) = (t/z)exp(-tlz), which has a maximum at t = z and therefore simulates the experimental rectangular integration somewhat better. This derivative may be found numerically. We will also assume that pw pulses affect the system in the continuous-wave fashion (B1 = const during the lifetime of the pairs or at least the sampling period). This condition may be easily met experimentally. Note that the quantity I, (or S, = Z,(off) - &(on)) is, in fact, a Laplace transformation of Z(t) (or, respectively, the signal S(t)). Using the fundamental property of this transformation, we may separate all three contributions to Z(t):
exp(zq where
In the strong magnetic field of the spectrometer, projections of nuclear spins on the Bo field are constants of motion, and the pairs with different nuclear spin configurations may be treated separately. For a given spin configuration the spinHamiltonian H(r,8) (we dropped indices 1,2) may be divided into four terms,
where the term (i) includes Zeeman interactions of the radical ions with the Bo field and the local field B1, of nuclei, (2.12) ga,b are the g-factors of the correspondent partners (marked a and b, respectively), p~ is the Bohr magneton, o is the pw frequency, and Bloc;a,b
= zAk%,k
(2.13)
k
Ak and mZ,k are the hfi constant and spin projection of a kth nucleus, respectively. The next three terms in eq 2.11 involve, respectively, (ii) Zeeman interaction of electrons with the pw field (where Wla,b = ga,WB& are Rabi frequencies of the radical ions), (iii) electron spin exchange, and (iv) the dipole-dipole interaction. The exchange potential J(r) may be well simulated with the exponential dependence14a J(r) = Jo exp[-r/AJ]
-
(2.14)
-
with AJ i1/2 and JO (2-10) x lo9 r a d / ~ . l ~ ~Our ,l~ calculations clearly demonstrate that in the time interval z 0.1-1 ps the spin exchange does not affect spin dynamics significantly. Indeed, as may be shown, the FDMR signal reaches a maximum when one of the constants Kt or K2 is close to z-l. At K1,2 1010-1014 s-l, it gives r 12-20 A. For these r the exchange constant J(r) is much less than the hyperfine coupling constants and B1 field and hence may be neglected. Note that contrary to the viscous solutions the adiabatic ST- transfers (a common manifestation of the spin exchange effects in the confined pairs12a)cease to exist in the solid, where the interaction is not tuned by a relative motion of the partners. As for the dipole-dipole interaction, taking into account the high-field condition, only the secular part of it may be considered,
-
-
D(r) = (gag@,2)tir-3(3Sa&
-
(2.15)
For the sake of simplicity other relaxation channels were neglected in our model. The relaxation should not be significant at t < 1 ps. At longer times the TI process may be included in the model by changing z to z TI-'. This process is,very slow at 4-20 K (TI> 100 ps). The spectral diffusion due to various dephasing interactions may be formally considered as
+
13254 J. Phys. Chem., Vol. 98,No. 50, 1994
Shkrob and Trifunac
analogous to the mixing of spin states on resonant charge transfer and will be considered later. To find IT in eq 2.7 the following procedure was used. First, the distributions of the local magnetic fields Blocwere generated. Then, in order to accelerate computations, the graining of these distributions was made more coarse so that 15-30 bars represented variations in hfi.14a For each of these nuclear configurations and for each of the orientations 81,2 and separations r1,2 the spin-Hamiltonians (eq 2.11) were calculated and the system 2.10 of 32 linear equations was solved. Finally, the obtained QZ, were angle and configuration averaged with proper weight functions. The procedure may be simplified if the spin mixing in the primary pairs is neglected. The numerical integration may be further accelerated by rewriting eq 2.8 in the prolate spheroidal coordinates. Finally, one obtains
}he(r,)C exp( -4n;”c) where
rl
+
= r ~ ( < q)/2,
r12
(2.17)
= r2(< - 7)/2, and
,,
5 de2 ~in(Q&(rJ Tr[Q8L(r2~Q-’Qsl Pt(r2) = (‘)JZ
(2.18)
Formulas 2.8-2.22 provide a basis for simulations of FDMR in the exponential windowing model. Unforhmately, the validity of this approach is somewhat questionable since, first, the reallife measurements are not performed with the exponential or t-exponential windowing (so the results are semiquantitative, while rather bulky computations are to be performed) and, secondly, most of the measurements were obtained in the pulsed pw fashion in order to increase the signal-to-noise ratio and control the state-locking effect^.'^^^^^ Our second model addresses these two facets of the timeresolved FDMR. Though the approach (which will be called “pulsed FDMR’) is more explicit, perhaps, it is less applicable to the systems in question. Within the second framework we will (i) neglect the distance-dependent and anisotropic interactions, including spin exchange and dipole-dipole interaction, (ii) artificially segregate spin and reaction dynamics (which is possible only in the case of very slow reaction kinetics), (iii) consider the formation of [2S+. .2A-] pairs as an instantaneous process, and (iv) assume that resonant charge transfer does not occur. Under these assumptions the spin-Hamiltonian of eq 2.10 may be rewritten ~ s ~ ~ ~ J ~ ~
+
~ ( r ~= ; ei [~~ )~ ( r ~ , @ ~ ) , . .[z-l+ . ] - K2(r2)]1 (2.19) The resulting quantities P,(r) (eq 2.18) represent the recombination probability for a hypothetical secondary pair whose spin and reaction dynamics compete with a spin-independent decay occumng with the rate constant of 2-l. In other words, the quantity approximates a radial distribution of the secondary pairs participating in the FDMR formation over a certain period z (we shall call them “FDMR-active” pairs). Correspondingly, the function pT(r)in eq 2.17 is the cation-anion distribution developed over the period of 2. Formulas 2.8-2.19 may be easily modified to take into account finite volumes of the radical ions. So far we neglected relatively slow reactions (1.9) which mix nuclear spin states of the solvent hole. As will be shown below, this mixing significantly alters the spectral patterns at longer delay times. Following ref lob, we will consider this exchange as a pseudo-fist-order reaction (with a rate constant kh) which gives no additional mobility to the solvent holes. Thus, we will assume that the charge transfer involves the neighboring species, and the number of subsequent reactions (1.9) over the observation period is small. In this case eq 2.18 must be modified. Let us introduce new quantities L(ialjb), W(ia), and W(jb) which are, respectively, the Liouville matrix 2.19 and the weights of nuclear spin configurations { ia} and G b } in the corresponding radical ions (let us also assume that the radical ion “b” is the solvent hole). At a given rz the master equation 2.10 for the integrated density matrices @Zr(ia$b) should be changed Qs
= L(iagb) @2~(~;&)+ kh{@2r(ia&)- x w ( % ) @2r(ia;kb)} b
(2.20) @2T
= cw(ia)cw&) a
b
where na,b
= (Wa,dQa,b;o;w1a,dQa.b) 2
‘a,b
= [@a,b + Wla,b
2 112
1
(2.24) (2.25)
and the signal S(t) is given by S(t) = Jtdt’
kfexp[-k&t’)]AP&’)ll(t’)
(2.26)
AP,(t) = P,2(/;B1=O) - P,2(/)
(2.27)
P,(r) = (SIT exp{ih‘dt‘ H(B,=B,(t’))jIS)
(2.28)
where
is the projection of the electron spin wave function on the singlet state.
l l ( t ) = f d r 4n&I(t;r) p(r)
(2.29)
ll(t;r)= K2(r)exp[-K2(r)tl
(2.30)
is the probability for the singlet pair to react at a time t, and p(r) is the cation-anion distribution. The calculated signal S(t) may be treated exactly as the experimental one, which gives an opportunity to estimate distortions of the signals due to the finite duration of pw pulses, boxcar sampling, and the power effects. The projection P,(t) may be calculated with the formula
(2.21) exp[iQt(nS)] = cos(Qt/2)
+ 2i(nS) sin(Qt/2)
(2.31)
FDMR in Low-Temperature Solids
J, Phys. Chem., Vol. 98, No. 50, 1994 13255
The final result of the calculation for the pair probed with a rectangular pw pulse with the delay time of T, duration of A, and the peak intensity of B1 is given in the Appendix.
3. Modeling of the Solid-state FDMR 3.1. Kinetics of FDMR as Simulated with the Pulsed FDMR Approach. It is instructive to calculate the quantities APs(t) and n(t)separately. At B1 hfi the pw field induces TO T+1 and To T-1 transitions between the triplet states T+1, To, and T-1 decoupled in the high BO field of the spectrometer (here the subscripts denote their spin projections on the Bo field), causing a faster depletion of the TOstate. Since this state is energetically close and Mi-coupled with the singlet S state, the application of the pw field results in more rapid decay of the singlet pairs via the population transfer to the energetically remote upper T+1 and lower T-1 states. An excessive population of these triplet sublevels facilitates the triplet route of the ET followed by formation of the nonobservable (on our time scale) triplets 3A*. Consequently, the yield of fluorescence decreases (in our notation FDMR > 0). The effect, however, may be reversed by application of stronger pw pulses due to the so-called “state-locking”. The phenomenon was considered in detail in ref 14a,b. State-locking is an isolation of the singlet and triplet states in the strong pw field. We would like to point out a source of the signal decrease specific to the short-pw-pulse-induced FDMR which may be observed when the delay time T of the pulse is smaller than a characteristic time of the hfi-induced STo transitions (‘thfi sz 2dA, where A is the mean hfi constant). Since the pw field does not mix the singlet and triplet states, its application cannot affect the singlet term unless preceded by these hfi-stimulated STo transitions. This mechanism is responsible for a substantial decrease in the peak amplitude of S(t) when the pw pulses are applied at T I1/22a (Figure 3a, trace (i)). This decrease may have been observed experimentally but was interpreted in terms of the decreased Q-factor of the microwave cavity due to the electron beam interference,23awhich is known to occur from pulsed EPR experiments. If the number of nuclear spins is sufficiently great and the spin degeneracy is moderate, the dependences AP,(t) rapidly achieve a plateau. The signal magnitude S, in this range is a function of the pulse duration A and w1 but is independent of the pulse delay T provided that T > ‘ta(Figure 3a). The conclusion is in excellent agreement with the liquid-state FDMR experiments:23awhile an increase in the delay time T of 30-11swide pw pulses resulted in the pronounced changes in S(t) traces immediately after the pulse, over a period of 50-100 ns the traces converged to a single curve. Indeed, as soon as AP,(t) achieves the plateau (Figure 3a), S(t) follows the time dependence n ( t ) . The calculated power dependences are remarkably different when the flip angle 8, = o l A of the pulse is small and when 8, > 50-60’. In the former case, the signal is proportional to :8 at B1/A < 1% and linearly proportional to 8, at the higher pw power (Figure 3b). When the flip angle is close to or exceeds 90°, the curves are considerably distorted (Figure 3c). With a further increase in the flip angle S, decreases and finally the signal becomes negative at longer times (“B~-beats”~~). As illustrated in Figure 3c, at such high 8, the desired increase in the peak signal is accompanied by a considerable decrease in the period over which the signal is to be sampled. At their maxima the signal intensities are proportional to 8, unless B1/A > 0.1, when the state-locking is strong. An interesting feature of the spectra is its nonuniform response to the pw power. It has already been shown that the central lines of FDMR spectra
-
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0
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!
0.2
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IAV (1)
0
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I
a
i
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111
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tlw-
Ill
Figure 3. Evolution of the yw-field effect on the singlet-state population, M&),for the radical ion pair of anthracene-dd-/cisdecalin’+ (with no reaction depletion of the pair) as a function of pulse
delay time, duration, and power. On-resonance conditions: BO = w/gpB; J = D = 0. (a) B1 = 0.25 G, A = 30 ns, T = 0 ns (i), 50 ns (ii), and 150 ns (iii); (b) A = 30 ns, T = 50 ns, and B1 = 0.25 G (i),
0.50 G (ii), 0.75 G (iii), and 1.0 G (iv); (c) the same Tand B1 fields as in b, but A = 300 ns. High-frequency oscillations are STOquantum beats due to Mi in the cis-decalin radical cation. are state-locked at smaller pw p ~ ~ e and r ~shorter ~ , delay ~ ~ , ~ ~ ~ times (t - 2714bthan the side lines. The simulations show that at S(t) = max the opposite is true: due to a consecutive interference of STo quantum beats (at ( t - 27 ZM), the centerto-the-wings ratio may be 1.5-2 times greater than at the longer delays. In the plateau region ( t - T > (2-3)‘t~) the spectral shape does not evolve (in the absence of transverse relaxation in the radical ions). This brief analysis shows that if the flip angle 8, is sufficiently small, 8, 5-30’, and t IT (2-3)%, the tail of S(t) reflects the kinetics of ET without possible complications from the simultaneous spin dynamics. Both these conditions were met in the previous study on the low-temperature FDMR when S(t) = t-’ behavior has been observed.lla The kinetics was interpreted in terms of a site-to-site diffusion. Thus, it is important to demonstrate that the proposed single-step tunneling does provide the required t-’ behavior at longer times. Indeed, according to eq 2.24 the individual probabilities n ( r ; t ) reach a maximum at K2(rm)t 1 when n(r,;t) Ut. Since &(r) depends on the separation distance r exponentially, big changes in t correspond to relatively small variations in rm and within, say, an order of magnitude in the time t the gross quantity n(t)is proportional to p(rm)lt. These estimates may be readily confirmed by a numerical simulation of the kinetics, which demonstrates the t-’ behavior at any given time scale (though different slopes of the ll vs t-’ curves when the time scales are very different, Figure 4). It appears that the t-’ behavior is an inherent feature of single-step tunneling; so the
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13256 J. Phys. Chem., Vol. 98, No. 50, 1994
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Shkrob and Trifunac
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Z(O)/Z(r) = 1
+ const x t
20
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.
10
15
20
25
rIA-
(3.1)
behavior were repeatedly reported in the isothermal luminescence (ITL) experiments performed in various organic solids and p~lymers.~ The major difference between the recombination kinetics as recorded by means of FDMR and ITLis in the time scale of these techniques. Indeed, to observe pulse FDMR, the recombination kinetics has to occur on the appropriate submicrosecond time scale. On the contrary, the ITL signal is observed on the scale of minutes and hours3 It is fascinating that despite the dramatic difference in the probing period, both these experiments may be rationalized by assuming a singleexponential dependence of K(r)! It is noteworthy that for some solvents (e.g., for 3-methylpentane) the t-' behavior was observed only at 4 K, while a t-2 behavior of ITL was found at 77 K.3b The feature is likely to result from the thermal detrapping of electronsg or tunneling of the electrons between the trapping sites? Independent evidence that these process do occur in the system at temperatures > 35 K is given by photoconductivity experiments.gaWe may neglect the process when dealing with the solid-state FDMR at liquid helium temperatures since, fist, this site-to-siterandom walk occurs on a much longer time scale (the sites are separated by 100 A3,9,20) and, second, no such process has ever been found at 4 K. 3.2. FDMR As Simulated with the Exponential Windowing Model. The results obtained with this model are costly in terms of computation time, and therefore we will concentrate on the qualitative conclusions considering a hypothetical radical ion pair having a single nuclear spin as one of the partners and neglect any spin mixing in the primary pairs and reaction 1.9. Let us f i s t examine the radial distribution AP,(r) of the FDMR efficiency. Figure 5a shows a family of distribution curves obtained at different widths z of the exponential windows. These curves are bell-shaped and centered at r = r, (at which K2(rm)z 1; we will drop the index further). The left margin of the bell (r-) is given by an approximate formula K(rm)rw 1, which simply reflects the fact that at t < zm the ,uw field does not produce spin mixing. At bigger separations ( r > rm) FDMR progressively decreases and rapidly (an exponential dependence!) the kinetics becomes so slow that no FDMR signal can be formed over the period of z (K(rmax) z&z2). The width of the distribution logarithmically increases with z and linearly increases with 1 but is independent of KOprovided that the dipole-dipole interaction is neglected. As Figure 5 indicates, in the model with no dipole-dipole interaction the
-
18
18
4
Figure 4. Simulated recombination kinetics l l ( t ) at to = 1 ps (i) and 0.1 ps (ii); KO = lOI4 s-l, 1 = 1 A.
models are consistent with experimental kinetics of FDMR. The simulations also demonstrate that the shape of n(t)is largely independent of the parameters KO and 1,so an analysis of S(t) would gain very little insight into the mechanism of ET besides the fact that ET reaction constants rapidly decrease with r. Observations of the approximate t-' or, more correctly,
14
a*
Q
-
-
Figure 5. (a) Distribution A,P(r) of FDMR-active pairs as a function of the time window t (in the exponential model) for a hypothetical one-nucleus pair, A = 10 G , B1 = 0.25 G, g = 2 , K2.0 = IOl4 s-l, 1 = 1 A, RO (molecular/ionic radius) = 3 A; on-resonance conditions t = 100 ns (i), 1 ,us (ii), 10 ps (E),0.1 ms (iv), and 1 ms (v). (b) Effects of dipole-dipole interaction on the distribution and the spectral shape as a function of ET parameters and t. The distribution AP(r) at t = 1ps and KO = loL1s-l (i) and KO= lOI4 s-l (ii): solid lines, no dipole-
dipole interaction; broken line, with dipole-dipole interaction. (c) Normalized FDMR spectra of the pair at (i) z = 1 ms and KO= lOI4 s-l, (ii) z = 1 ps and KO = IOl4 s-I, and (iii)z = 1 ps and KO = 10" s-l(& = 20 A). As KOand t decrease, the dipole broadening becomes more pronounced. increase in KO causes a mere shift of the whole curve (solid lines) toward the higher separations. However, when the dipole-dipole interaction was included in the computation, a remarkable change in the distribution curves was obtained (Figure 5, broken lines). This interaction causes a characteristic nonuniform broadening in the spectral lines accompanied by a substantial decrease in the signal intensity (Figure 5b,c). Since the magnitude of dipole-dipole interaction contribution is roughly proportional to r,,,-6, the decrease and the broadening are much smaller at higher KO(and A) or bigger sampling periods z (Figure 5c). In the steadystate mode (z 1 ms) these distortions are relatively small. At KO 1O'O s-' and z 1 ,us the lines are so broad that the signal is 42%of the signal without dipole-dipole interaction. However, when KO = YOFC (where FC stands for the FranckCondon factor, which varies within 10-5-1296) is close to its maximum values (YO 1014-1015 s-l (ref 2)), the decrease in the signal at z 1 ,us is only 30-40% and the line broadening is -1 G (Figure 5c). Thus for FDMR to be observed on the sub-microsecond scale without significant dipolar broadening the constant KO of ET has to be close to its maximum values. Dependence of the FDMR signal on KO has an important implication since the model predicts a strong correlation between the ET parameters, the scale of the FDMR effects, and the spectral broadening.
- -
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FDMR in Low-Temperature Solids Without dipole-dipole interactions, variations in the FranckCondon factors might only change r, while the signal intensity would be roughly the same, being slightly corrected by a factor Qr(rm) rm2. Another significant conclusion we can make is that the signal is formed from recombination reactions in a relatively narrow spherical shell around the solvent hole (or et-). Since it is unrealistic to expect any sharp features in the cation-anion distribution on the scale of 5-6 A (which is a width of the AP,(r) distribution), we may conclude that (i) the spectral shape must not be affected by the changes in the distribution despite the fact that distance-dependence dipole-dipole interactions strongly affect the spectra and (ii) the magnitude of the FDMR signal is roughly proportional to pz(rm). Experimentally, this implies that with a narrowed cationanion distribution (e.g., in the confined radical ion pairs) one may predictably change the spectral intensity, but it is not possible to modify the spectral shape, which is a function of r, and only r,. This prediction of the tunneling model is supported by the results of our recent study on the solid-state FDMR from fluorescing polyoxyethylene detergents:28with increase in the length of the oxyethylene tail the signal passed through a maximum (which corresponded to the optimum cation-anion distance r, 20 A) without significant changes in the spectral shape. This would not be the case if FDMR were formed in the pairs whose separation distances varied broadly. In this case the spectral shape would inevitably show more dipolar broadening for the shorter detergents. This key experiment further justifies our scheme of the FDMR formation. 3.3. Cation-Electron and Cation- Anion Distributions. So far we have avoided any speculation on the cation-anion distribution in frozen solids. However, this distribution is crucial for the models presented. Indeed, as was shown above, if no spin mixing in the primary pairs occurs, the signal is proportional to the number of radical anionskations in a rather narrow spherical layer around a given solvent hole or trapped electron (Figure 2b). Since experimentally the signal may be observed M, any plausible model has to even at C as low as 5 x explain why this very unfavorable statistic does not eliminate the FDMR effect. To illustrate the point, let us estimate the weight of the FDMR-active pairs assuming that all the scintillator molecules at C lop3M are converted into radical anions. This calculation demonstrates that at most -2-3% of the pairs would participate in the formation of the FDMR signal. On first glance, this estimate (which by itself does not rule out the models used since the observed signals are an order of magnitude smaller than the corresponding signals in liquids) must be further decreased by 2-3 orders of magnitude when one takes into account that in solids only a small fraction of et- may be scavenged by the scintillator on the time scale of our experiments. In the context of these estimates, it should be stressed that little is known about the spatial distribution of the trapped electrons in solids. Moreover, even for the most studied systems, namely, liquid alkanes, this distribution is a subject of continuing controversy since, depending on the ionization density and the type of experiment, some data are better rationalized with the exponential distribution26awhile others were better fitted with the ?-exponential26d or $-Gaussian curves.26b As for crystalline solids, polymer films, and amorphous media, the only distribution applied to the interpretation of photoconductivity was the d3-function, whose use is more a matter of convenience rather than a reflection of the underlying physical reality.29 There is no doubt that electron trapping in low-temperature solids is dissimilar to the thermalization process in the liquid. 0~
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J. Pkys. Ckem., Vol. 98, No. 50, I994 13257 In solids the trapping occurs in the preformed sites of different depths, such as the lattice defects or specifically oriented clusters of molecules in amorphous matrices.30 Concentration of these sites varies strongly with the nature of the solid. The only data on the temperature dependence of bG in organic solids available from the literature31 indicate various trends. For example, in cyclohexane a 15% decrease relative to the room temperature value was found at -50 "C (from re = 60 to 50 A), whereas for isooctane a 50% increase at -150 "C was observed. The temperature interval explored in the cited study, however, is considerably above the temperature region at which electron tunneling plays the central role, so the applicability of these results to the 4-20 K range is problematic. Conductivity experiments indicate that the electron mobility in solids comprised of spherical molecules is 2-3 times higher than the mobility in the equivalent liquids, while for the lowmobility liquids (e.g., cyclohexane, methyltetrahydrofuran, 3-methylpentane) the mobility is less in the solid than in the liquid.' For the latter compounds the mobility was found to be fairly constant below 0 = 35 K (the Debye temperature), slowly increasing with temperature at 35-90 K due to phonon-assisted electron d e t r a ~ p i n g . ~ ~ Another technique used to probe cation-electron distributions in frozen solids was ITL deconvolution.3c Although instructive in that the results point out that the distribution is not a d3function, the obtained distributionswere entirely dependent upon the parametrization of ITL traces, assumed distribution of the trap depths, and left too much room for speculation. Hence, having nothing better to choose, we will assume that re in the rigid solids is close or somehow related to the thermalization distance in the correspondent liquids, although we are aware that this approach may be misleading. These distances have a remarkable feature: the exponential factors bE of the solvents in which the FDMR signals at 4-20 K are relatively strong (see below) are very close to the optimal r, calculated with KO = YO and 1 = 1 A. Experimentally, an increase in the solvent polarity (a decrease in b E ) or using the solvents constituted by spherically shaped molecules (e.g., tetramethylsilane and 2,3-dimethylbutane; an increase in b E ) caused a substantial decrease or the complete elimination of the FDMR signal. In terms of the Gaussian parameter bG the solid-state FDMR signals from [2S+..2A-] were not found for the compounds with bG (lis) less than 45 A or more than 70 We may assume that this rough correlation between bG and FDMR efficiency is not a coincidence but a reflection of the fact that FDMR is probing a relative narrow band of the cation-anion distribution and some matching between r, and bE is important for the signal to be formed. If this approximate equality takes place, a simple explanation of relatively high efficiency of FDMR in the solid is available. It is intuitively clear that if the electron is distributed -20 A around the given hole and it is quenched by a scintillator molecule within the same 20 8, (-r, at z 1 ps), the resulting anion will be distributed similarly to the electron or even more favorably since the scavenging would occur within a sphere whose radius is comparable with the radius of the electron trapping. To illustrate the point, we used eq 2.16 to calculate the weights 4z$p,(r) of the [2S+. .2A-] pairs at the most typical (for liquid hydrocarbons) thermalization distance re = bG 60 A ( b E 20 A). For simplicity we assumed that both reactions 1.2 and 1.7 occur with the same rate constant, K1 = K12 = K. For KO = 1014s-l, 1 = 1 A, and x = pus (which gives r, = 24 A) this calculation gives the weight 4zrm2p,(rm) of -0.005 at [A] = lov3 M, -0.04 at [A] = M, and -0.2 for the decimolar solutions. Taking into account that AP,(r) is -5-6 A, the statistical weights are -2-3% at millimolar and -20%
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13258 J. Phys. Chem., Vol. 98, No. 50, 1994
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0
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--C
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bEIh
Figure 6. Simulated C- and h-dependences of statistical weights of FDMR-active pairs (normalized to FDMR(rm))at K I Z= K I = Kz,z = 1 ,us, KO= 1014s-l, 1 = 1 A, R = 3 A. (a) C-dependences at LQ = 20 8, (i) and 40 8, (ii); (b) bpdependence at C = M (i) and M
(ii). at centimolar concentrations. Upon further increase in the scintillator concentration the weight achieves the maximum and then slowly decreases. Theory predicts a linear growth of FDMR signals with C up to a certain concentration when V,C 1, where V, = 4nrm3/3is the reaction volume (Figure 6a). Saturation and decrease of FDMR signals at [A] lo-' M have been observed experimentally.22 Figure 6b shows the calculated dependence of the weight 4nrm2pr(rm)on h. The curves exhibit sharp maxima at certain h;this critical h gets closer to rm/2 as C increases. This simply reflects the fact that at high C the distribution p,(r) is close to pe(r),which has an extreme at r = 2h (in the 3-exponential form) or r = bG (?-Gaussian form). According to this calculation, FDMR is optimal in the media with very short trapping distances of 20-30 A. At longer distances the weight is falling nearly exponentially, giving, for example, -10% of the peak value at bE/rm 2. This rapid decrease rationalizes a rough correlation between bG(1iq) and the signal intensity. Although the model developed here qualitatively accounts for the experiment, the quantitative agreement is poor. The actual C-dependence for aromatic hydrocarbons are linear only at C < M and approximately logarithmic with C at intermediate concentrations of solute, 10-4-2 x lov2M.22Only for two radiolytic systems we observed a linear C-dependence in the second concentration range: one such system is benzonitrile; another is N,N,N',N'-tetramethyl-p-ethylenediamine (and the related aromatic amines). Remarkably, these solutes represent, respectively, a pure acceptor and a pure donor, and both yield FDMR indicating the spin memory transfer. Robably, the former circumstance is more significant since, for example, octafluoronaphthalene while showing the transfer exhibits a logarithmic C-dependence. Furthermore, with these two solutes we obtained a much better correlation between the FDMR signals and bG(1iq) than with the donor-acceptor scintillators. However, for all these solutes the estimated optimal b~ values were 2-3 times smaller than the experimental ones.22 A striking feature of the experimental C-dependence is their similarity in straight, branched, and cycloalkanes for a particular
-
-
-
solute. The identical results were obtained in the polycrystalline and vitreous solids of the same chemical composition.22 Another peculiarity is that the FDMR signals could be observed upon addition of molar (!) quantities of the electron acceptors, such as ketones, nitrobenzene, halogenated alkanes, and hexafluorobenzene. The FDMR intensities for the solutes of donor-acceptor type vary even for the solvents with fairly similar bc(liq).2 This difference may be as great as 10-50 times. It is unlikely that the difference originates in the chemical stability of the involved radical ions. According to the tunneling scheme, these variations are due to a difference in the statistical weights of the FDMR-active pairs. These weights are functions of the rate constant KO. However, as was shown above, any dramatic difference in KO and r, has to cause a pronounced dipolar broadening. That did not happen. Since the radical ion pairs in these experiments were not confined (as in our previous study on the fluorescing detergents28), a complete lack of dipolar broadening despite a wide variation in the FDMR efficiency must be interpreted as evidence of large separations between the radical ions. The same message was canied by the C-dependences: the ET occurred over a much longer distance than the single-step mechanism is able to provide. Note that for the donor-acceptor aromatic solutes the C-dependences mainly reflected the reaction radius of electron scavenging. Indeed, nothing indicates a generation of 2A+ in these systems.22 With donor solutes the C-dependences were linear up to a very high concentration of the solute. Moreover, the hole mobility via reaction 1.9 would not drastically change the C-curves for the donor-acceptor systems even if reaction 1.9 were a long-range one. Even in the situation when the mobility of the holes were infinite the FDMR signal would be proportional to the number of solute molecules within a sphere of radius r, around the trapped electron until V,C 1. In accordance with this conclusion we found no changes in the C-dependences obtained in the solvents with different kh. The obtained C-dependences may be interpreted with the assumption that reactions 1.2 and 1.3 are the long-range ones, whereas reaction 4 is the shortrange one. Data on the double scavenging corroborate this conclusion.22 All these findings are hardly compatible with the single-step tunneling model in its original formulation. However, some other aspects of the solid-state FDMR considered below advocate the tunneling scheme. 3.3.1. Isotope Effects. Deuterium substitution changes densities of the vibrational states in molecules and ions. Due to decrease in overlap of the ground vibronic states with the higher lying vibronic states, the FC factors in the corresponding ET reactions decrease upon deuteration. Experimentally, very strong isotope effects on the G-values of trapped electrons were found in many vitreous hydrocarbons at 77 K.5 These effects were interpreted by Willard et al. as evidence of the tunneling mechanism operating in the system. On first glance, FDMR should be as sensitive to the deuteration effects as these G(et-) measurements. The experiment, however, shows no isotope effects upon deuteration. Is that compatible with the tunneling mechanism? Let us assume that the deuteration does not change the b~ and 1 parameters. The only difference is in the rate constants KO and KO* (the asterisk denotes the deuterated solvent). This is not a very realistic assumption since the deuteration would definitely change Mi constants of the solvent hole. This effect may be corrected by a normalization of the experimental ratio of the signals S*/S in the deuterated and protiated solvent to the calculated ratio of the signals in an imaginary situation when KO = KO*. As was shown in section 3.2, in the absence of
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FDMR in Low-Temperature Solids
J. Phys. Chem., Vol. 98, No. 50, 1994 13259
dipolar broadening (which is always the case experimentally), at r = rm the signal is independent of KO,so the ratio of normalized signals is
et-
S*noJsnom
0~
[+
et-
2S+ +
-]%[+
a]
- snom>/snom
bottom spectra: FDMR from *s+ ..... FDMR from *D'
-
-
21 l o g [ ~ ~ ~ * o ~ r m [b ,r m - ~- 2 ~ (3.3) On the other hand, G(et-) may be estimated as a fraction of electrons which escaped prompt recombination over a period T of the steady-state experiment, which is macroscopically long (-lo2 s). Let us introduce a distance m, K(m)T 1. At Q >> bG
..
B
-
G*/G = l:dr
pe(r)/l:dr
-- -
(G* - G)/G
pe(r)
t
(3.4) -5
-10
21rdb,' log[KJK0*]
-
-
-
-
-
-
0
5
10
offset field, G
(3.5)
1 A, bG 60 A, and r?. 100 8, At log[K~&*] 1, we obtain G*/G 1.5, which is very close to the experimental values. For instance, in methylcyclohexane at 77 K G = 0.23 and G* = 0.38; G*/G = 1.65.5a In 3-methylpentane at 77 K G*/G = 1.6, and in 3-methylheptane G*/G = 1.3.5b The tunneling model satisfactorily explains the observed scale of the deuteration effect on the electron yield. According to eq 3.2 the isotope effect on FDMR depends on rm. If rm < bG, which is the most reasonable estimate, S* < S: the deuteration decreases the FDMR yield; in the opposite limit the deuteration increases the FDMR yield. We have already shown that the FDMR yield is maximum at rm bG. In this regime there must be no isotope effect at all. The isotope efect on FDMR depends on the solvent bG: if b G is less than the optimum value, the deuteration reduces the signal; if bG exceeds the optimum value, it increases the signal; for b G close to the optimum value there is no isotope efect. 3.4. Corrections to the Model: Spin Memory Transfer. Spin mixing in the primary pairs will improve but will not change the results of section 3.1. It has another meaning for us, since the very observation of the transfer confirms the mechanistic approach we developed. Indeed, as was demonstrated in section 3.2, the lifetime of a radical ion pair is optimum for the FDMR formation when it is comparable with the observation time z. Therefore, for the primary pairs to demonstrate the FDMR probed spin mixing the parameter [Kl(rl) 4- K12(r12)]z should be -1. This could be if rl 2 rm and r12 rm or rl rm and 112 5 rm. if scavenging of the primary pairs is too fast (r12 < rm) or too slow (r12 > rm), the developed spin coherence will be lost, so the latter region is not significant. At r12 rm only the solute molecules with r2 5 r m may participate in the transfer of spin coherence created in reaction 1.7 (Figure 2c, bottom). Note that if the scintillator has an ability to scavenge both the solvent holes and trapped electrons, these configurations are precisely those in which a fast decay of the solvent holes via reaction 1.4 is expected. This should grossly reduce efficiency of the transfer in such systems. If 1-12 < rm, the scavenging of the solvent hole is very fast, and no spin mixing in the primary pairs is possible. In this case only the secondary pairs with r2 rm would give FDMR, the fraction of such configurations being approximately the same as the fraction of the former ones (Figure 2c, top). This brief consideration shows that yields of FDMR formed from spin mixing in the secondary and primary pairs must be approximately equal (provided that the scintillator does not
-
+
pe(rm*)/.e(rm) 1 + (rm* - rm) d[log ~e(rm)l/dr(3.2)
Since K(rm) = K*(rm*) we obtain 1 are not shown). For simplicity we assumed that the partner of the hole is e-. That was the case for the donor systems and a good approximation to the systems involving deuterated acceptors. The solvents differ in two important aspects: (i) hopping in n-alkanes is faster than in cis-decalin; (ii) spectral density at the center in the n-heptane hole is higher than in the cis-decalin hole. We assumed kh IO7 s and 2 x lo6 s for the former and the latter system, respectively. The calculations closely resemble the experimental spectra shown in Figures 2-6, ref 22. The S(M=O)/S(M=&l) ratio systematically decreases with time; this decrease is much faster than the broadening effects in the M = f l lines, which develop only on the microsecond time scale. In the case of cis-decalin, this broadening is practically unobservable within the first 5 ps. The effect of reduction in the S(M=O)/S(M=fl) ratio is power-dependent: the higher the power, the faster the reduction (Figure 8C). At small pw powers the evolution of spectral shape is quite slow. All these features reproduce the experimental behavior.22 In sum, a relatively slow reaction 1.9 accounts for most of the observed features. The obtained estimates of kh agree with the values derived from analysis of spectral multiplets (with the steady-state FDMR at 77 K).lob It should be emphasized that the theory in the present state can not distinguish between short-range and long-range hopping. Knowing the scale of reaction 1.9 is crucial for the simulation of the dilution effects. N
4. Conclusion
We have considered single-step electron tunneling as a mechanism of the solid-state FDMR at very low temperatures. This mechanism successfully reproduces many observed features, particularly the r-' behavior of FDMR narrow spatial distribution of FDMR-active differences between
Figure 8. Simulation of the effects of reaction 1.9 on the FDMR spectra of the pairs comprised by a trapped electron and the holes from (A, C) n-heptane and (B)cis-decalin. For the latter only lines with M = 0, f l are shown; the spectra are normalized for comparison. The curves are calculated with the t-exponential windowing model at t = 0.1 ps (i), 1 ps (ii), and 10 ps (iii Other parameters: kh = 10' s-' (A) and 2 x 106 s-I (B);& = 20 = 1014s-l, 1 = 1 A, R = 6 A, and BI = 2.5 G . (C) Spectra for the [e,-. .n-heptane+]pair calculated at t = 0.1 ps and r = 1 ps with B1 = 0.5 G (i), 2.5 G (ii), and 7.5 G (iii). The higher the pw power, the stronger the broadening/reduction effect in the M = 0 line.
1,
ps- and ms-resolved FDMR spectra, isotope effects, spin
coherence transfer, and the hole hopping.'0*22 However, the tunneling model in its straightforward formulation fails to explain many other results, particularly, a striking absence of dipolar broadening in the spectra, bo- and C-dependences (for donor-acceptor solutes), and, most importantly, the magnitude of the observed FDMR effects.22 The curious ability of the single-step tunneling model to explain the results in general and its failure to reproduce nearly everything concerned with the scale on which the FDMR-active ET reactions operate indicate that this model must be either revised or eliminated. Although further experimentation may lead to the second choice, at the present time no other model can account for so many different findings. A critical assessment of the discrepancies between theory and experiment reveals that in all the cases the reaction radii of ETs 1.2 and 1.3 were grossly undervalued. That may be interpreted both ways: (i) as an indication that the tunneling occurs over much greater distances than was expected or (ii) that the tunneling is not single-step. The fiist assumption implies very low potential barriers for tunneling. The second one is the core postulate of the multisite diffusion:20electrons migrate from the shallower traps to the deeper ones in a jumpwise fashion. However, this picture could not explain the observed Cdependences in the donor-acceptor systems. Indeed, such siteto-site migration is hardly possible with the radical anions, so reaction 1.3 must be a short-range one. Even if the capture of electrons were instantaneous, the FDMR effect would be controlled by a number of solute molecules in a thin spherical layer around the hole. Consequently, the C-dependence would be linear, as in the pure donor and pure acceptor systemsF2 Thus, it is necessary to recall a long-range hole hopping to
FDMR in Low-Temperature Solids
J. Phys. Chem., Vol. 98, No. 50, 1994 13261
account for the speed of recombination 1.3. Furthermore, the multisite diffusion provides no clear explanation for why different classes of the solutes demonstrated different patterns of behavior in the same solvent. Indeed, this diffusion should be directed by a distribution of the traps formed by the solvent molecules. The main problem with the mechanism, however, is its slowness at 4 K. All in all, the multisite diffusion is just a succession of the short-range tunneling steps. As was found, even the single jump is too slow to account for the data if we assume a reasonable density of the location site^.^^^^*^ Both suppositions share a starting point: the electrons are assumed to be very mobile but somehow localized. In terms of the tunneling theory that means a low binding energy of the electron. Thus, we formulate the main conclusion of this work: FDMR is selective to the chemical transfonnations of electrons with low binding energies. This paradigm will be further detailed in the second article of this series.
Acknowledgment. The authors gratefully acknowledge fruitful discussions with Dr. J. R. Miller and Dr. M. C. Sauer, Jr. I.A.S. is indebted to Dr. D. W. Werst for insight-yielding advice, invariably useful criticism, and assistance. Appendix At t < T (or for B1 = 0) P,(t) = cOs(Wabt/2)
At t
(A.1)
=- T + A
+
P,(t) = cos(Wab[t - A]/~)(COS(Q,A/~) cos(QZ,h/2) (oa@dQ2,Qb)x sin(QaA/2) sin(QbA/2)} cos(wab[t -
+
2T - A]/2)(w1,@1b/QaQb) x sin(Qah/2) sin(QbA/2) sin(Wab[t - A]/2){ [wa/Qa] sin(QaA/2) c0s(QbA/2) [ob/Qb] c0s(QaA/2) sin(Qbh/2)} (A.3)
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