Electron Spin Dynamics as a Controlling Factor for Spin-Selective

Jul 23, 2010 - Hopping versus Tunneling Mechanism for Long-Range Electron Transfer in Porphyrin Oligomer Bridged Donor–Acceptor Systems...
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Electron Spin Dynamics as a Controlling Factor for Spin-Selective Charge Recombination in Donor-Bridge-Acceptor Molecules† Tomoaki Miura, Amy M. Scott, and Michael R. Wasielewski* Department of Chemistry and Argonne-Northwestern Solar Energy Research (ANSER) Center, Northwestern UniVersity, EVanston, Illinois 60208-3113 ReceiVed: April 16, 2010; ReVised Manuscript ReceiVed: June 30, 2010

Photoinitiated charge separation and thermal charge recombination (CR) in a covalent donor-bridge-acceptor (D-B-A) system consisting of a perylene-3,4:9,10-bis(dicarboximide) (PDI) acceptor, 2,7-oligofluorene bridge (FLn), and phenothiazine donor (PTZ) (PTZ-FLn-PDI) have been shown to transition from superexchange to charge hopping mechanisms as the D-A distance increases. In work presented here, the spin-selective multiple CR pathways in PTZ-FLn-PDI are studied by a detailed analysis of the magnetic field effect (MFE) on the radical ion pair (RP) lifetime and triplet yield. A kinetic analysis of the MFE gives the spin-selective CR rates and the RP singlet-triplet (S-T) relaxation rates for n ) 2-4. When n ) 2 and 3, where the S-T splitting (2J) of the RP is large, slow S-T relaxation results in a kinetic bottleneck slowing the observed total CR rate at zero magnetic field. These results show that spin state mixing is an important controlling factor for CR reaction rates in these systems. The CR rate constant for the triplet RP (kCRT) obtained by MFE analysis is about 10 times faster than the corresponding rate for the singlet RP (kCRS) when n ) 2-4, indicating that kCRT occurs near the maximum of the Marcus rate vs free energy dependence, whereas kCRS is deep in the inverted region. The distance dependence of both kCRS and kCRT is explained by the crossover from superexchange (n ) 1 and 2) to distant independent thermal hopping (n ) 3 and 4). A possible mechanism of the S-T relaxation is proposed based on S-T dephasing, which may be induced by fluctuations of 2J resulting from bridge torsional dynamics. Introduction Photoinduced electron transfer reactions are important in numerous biological processes, such as photosynthesis1-4 and light-induced DNA repair,5,6 as well as in photofunctional materials for artificial photosynthesis,7-12 photovoltaics,13-15 and optoelectronic devices.16,17 In these systems, charge separation takes place from an excited state or exciton to generate charge carriers (electrons and holes) that are converted into electrical or chemical potential. To optimize the efficiency of photon-toenergy conversion, it is well-known that charge recombination (CR) reactions must be suppressed. CR reactions are also important in organic-LEDs where photons are emitted from excitons, which are generated by the recombination of electrically generated electron-hole pairs.18-20 The detailed mechanism of CR reactions in covalently linked donor-bridge-acceptor (D-B-A) systems has been studied extensively by analyzing electron transfer rate constants obtained by transient absorption and/or time-resolved fluorescence spectroscopy.10,21-26 Two distinct mechanisms of charge transfer in these D-B-A systems have been proposed, namely, coherent superexchange27-29 and thermally induced incoherent hopping.30,31 It is well-known that superexchange depends exponentially on donor-acceptor distance29 (rDA), but hopping is only weakly distance dependent (1/rDA).32 Thus the latter mechanism is sometimes referred to as “wire-like” transport, which is ideal for long distance charge transport. These two mechanisms can be distinguished by a detailed analysis of charge transfer rates as a function of rDA and temperature.21,22,33,34 We have recently demonstrated that a

CHART 1: Structure of PTZ-FLn-PDI (where n ) 1-4 and R ) 3,5-di-tert-butylphenyl)

covalently linked D-B-A system consisting of a perylene-3,4: 9,10-bis(dicarboximide) (PDI) acceptor, a 2,7-oligofluorene bridge (FLn) and a phenothiazine donor (PTZ), (PTZ-FLn-PDI, Chart 1), transitions from superexchange to charge hopping mechanisms at relatively short D-A distances.21,33 The electron spin dynamics of radical ion pairs (RPs) is a powerful probe of electron transfer reaction mechanisms, especially CR. The singlet-triplet (S-T) energy gap of RPs, 2J, is mainly determined by the donor-acceptor electronic coupling interaction,VDA as given by

2J ) ∆ES - ∆ET )

[

∑ n

|〈ψRP |VRP-n |ψn〉| 2 ERP - En - λ

[

∑ n

]

-

S

|〈ψRP |VRP-n |ψn〉| 2 ERP - En - λ

]

(1)

T

†

Part of the “Mark A. Ratner Festschrift”. * To whom correspondence should be addressed, m-wasielewski@ northwestern.edu.

where the indicated matrix elements couple the singlet and triplet RP states to states n, ERP and En are energies of these states,

10.1021/jp103441n  2010 American Chemical Society Published on Web 07/23/2010

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CHART 2: Structure of DMJAn-Phn-NI (where n ) 1-5 and R ) n-C8H17 when n ) 1-3 and R ) 2,5-di-tert-butylphenyl, when n ) 4-5)

respectively, and λ is the total nuclear reorganization energy of the CR reaction.35,36 Thus direct measurements of 2J by timeresolved EPR (TREPR) or reaction yield detected magnetic field effects (MFEs, discussed later) give us more direct quantitative information on VDA compared to that estimated from the semiclassical Marcus equation.21,22,25,36-38 Competitive spin-state-dependent CR reactions to the singlet ground state and to the local neutral triplet state are frequently complex, especially in the case of RPs generated from singlet excited states, where the RP energy is higher than that of the local neutral triplet state.38-41 The interconversion between the singlet and triplet RP states is usually driven by very small electron-nuclear hyperfine interactions ( aeff (middle column), and 2J e aeff (right column) at zero magnetic field (top row), resonance (middle row), and very high magnetic fields (bottom row).

The nanosecond transient absorption (TA) detected MFE apparatus has been described elsewhere.38 All the measurements were carried out in room temperature toluene. PTZ-FLn-PDI and DMJAn-Phn-NI, were excited with 7 ns 532 and 416 nm laser pulses, respectively. The total instrumental response function was ∼7 ns, which is largely determined by the laser pulse duration. The magnetic field was applied by an electromagnet whose magnitude was measured by a Hall effect probe. For the magnetic field vs rate or triplet yield plot (magnetically affected reaction yield (MARY) spectroscopy), reference kinetics at zero field were collected during the experiment in fouror five-step increments to compensate for possible sample degradation. Results and Discussion Mechanism of MFEs on RP Recombination Dynamics. As has already been mentioned in the introduction, a RP born in the singlet state frequently recombines by spin-selective pathways to the singlet ground state and local triplet excited state. The rates at which the RP decays and these products are formed (kobs) depend on two major considerations: (1) Spin dynamics controls the interconversion between the singlet and triplet RPs and determines population flow between the singlet and triplet manifolds. (2) The CR rates within the singlet and triplet manifolds depend on the electronic coupling matrix elements, VCRS and VCRT, as well as the Franck-Condon weighted density of states, which involves the reaction free energy and the total reorganization energy for CR. It should be noted that (1) is relatively easily controlled by applied magnetic fields42 or resonant microwave irradiation,49-51 whereas (2) is never field dependent because the interactions involved are much larger (.10 cm-1) than the Zeeman interaction between electron spins and the typical applied magnetic field ( 0 or 2J < 0, respectively) becomes isoenergetic with S when the field corresponds to

gµBB0 ) 2|J| p

(2)

where g, µB, and B0 are the averaged g value of the RP, the Bohr magneton, and the magnetic field strength, respectively. For the D-B-A molecules considered here, 2J is positive as confirmed by TREPR spectroscopy.54 At resonance, coherent S-T+1 mixing occurs in about 10 ns induced by the hyperfine interaction, which opens the CR channel to the neutral triplet state. If this mixing is fast compared to kCRS and kCRT, the spin state population quickly reaches a steady state, where S and T+1 are equally populated, and the observed decay rate is approximately described as

kobs(2J) ) kCRS /2 + kCRT /2

(3)

At higher fields, S-T mixing becomes inefficient again due to further splitting of the triplet sublevels. Thus the CR rate observed at high fields returns to kobs (B0 . 2J) ) kCRS. Consequently, the magnetic field dependence of the RP reaction yield or kobs (MARY spectra) shows a resonance at B0 ) 2J. This mechanism has been sometimes referred to as the levelcrossing mechanism.55,56 The sign of the MFE on kobs is determined by the relative magnitudes of kCRS and kCRT, which is positive (increase of kobs at resonance) for kCRS < kCRT and negative for kCRS > kCRT, whereas the triplet yield shows an increase at resonance regardless of the relative CR rates. In the second case 2J e aeff (Figure 1, right column), which corresponds to the so-called hyperfine mechanism.57,58 In this case, all four spin sublevels of the RP are nearly degenerate at zero magnetic field and are well mixed by the hyperfine interaction. The steady-state approximation gives the decay rate of the RP as

kobs0 ) kCRS /4 + 3kCRT /4

(4)

At higher fields than aeff, the S and T0 states can still mix, but T+1 and T-1 are isolated from the S-T0 mixed states. Yet, the population in the mixed S-T0 states can still flow into the T(1 states by incoherent spin relaxation, typically occurring in microseconds and usually slowing at higher fields.59 In the high

field limit, where relaxation is much slower than kCRS and kCRT, one can ignore the relaxation processes and the observed decay rate is approximated as

kobs(high field) ) kCRS /2 + kCRT /2

(5)

As a consequence, the observed rate monotonically increases with applied field if kCRS > kCRT and decreases if kCRS < kCRT, whereas the triplet yield monotonically decreases. This field dependence is opposite to that of the level crossing mechanism. A small minimum is sometimes observed at the magnetic field corresponding to 2J but is usually difficult to observe because the S-T mixing efficiency is not drastically changed by magnetic fields comparable to the hyperfine interaction. For the intermediate case, where 2J > aeff (Figure 1, middle column), spin dynamics and CR kinetics are complex. Since 2J is not very large relative to aeff, spin relaxation between the S and T0 states (described in detail later) followed by triplet CR should be taken into account, even if coherent S and T0 interconversion by the hyperfine interaction is almost negligible. MARY spectra for kobs in this case become mixtures of the two cases described above, namely, a peak at B0 ) 2J and a monotonic decrease at higher field for kCRS < kCRT, and vice versa. It should be noted that these three cases cannot be differentiated by simply considering the absolute magnitude of 2J. Which regime dominates depends on the relative relaxation and recombination rates. The experimentally obtained magnetic field dependence of kobs is the best diagnostic tool for this classification. Likewise, the “high field limit” condition should also be confirmed experimentally; namely, the magnetic field vs kobs plot should show a clear plateau at high fields. Kinetic Analysis of MFE for PTZ-FLn-PDI. The distance dependencies of the CR rate (kobs0) and 2J for PTZ-FLn-PDI have already been reported.21 The value of kobs0 decreases from n ) 1 to 2 but increases again at n ) 3 to 4 (Figure 2a). This behavior has been explained qualitatively by the crossover from the coherent superexchange mechanism to the thermally induced incoherent hopping mechanism,22,30 but the increase in the rate from n ) 3 to 4 has not been explained adequately in terms of conventional electron transfer mechanisms. In order to separate the spin dynamics and CR mechanisms within the singlet and triplet manifolds, we have analyzed the CR kinetics using the MFE theory explained above. We have observed MFEs for n ) 2-4, but not for n ) 1, which may be due to a very large 2J value that could exceed our experimental limit (1.2 T). Another possibility is that fast SO-induced direct recombination of the singlet CT state (SOCT) to the triplet state could significantly reduce the MFE on the RP lifetime and triplet yield. The spin polarization pattern of the time-resolved EPR spectra of 3*PDI resulting from RP recombination of PTZ+•-FL1-PDI-• in frozen toluene is indicative of SOCT.54 For n ) 4, the MARY spectra do not have a distinguishable resonance feature (Figure 3c), so that n ) 4 corresponds to the case (2J e aeff). The observed rates at high field and zero field give kCRS and kCRT from eqs 4 and 5, which are plotted in Figure 2a. On the other hand, n ) 2 and 3 seem to correspond to the intermediate case (2J > aeff) because their MARY spectra show clear peaks at the resonance fields and further increases in the magnetic field result in changes opposite to the direction of the resonance peak; this behavior is understood as a mixture of the level crossing and hyperfine/ relaxation mechanisms. Triplet formation at out-of-resonance fields also clearly indicates a contribution from S-T0 relaxation.

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Figure 2. (a) Distance dependence of spin selective CR rates (kCRS and kCRT, for the singlet and triplet channels, respectively) and S-T relaxation rate obtained by kinetic analysis of MFE. kobs0 is the CR rate observed at 0 mT. (b) The distance dependence of 2J.

The kinetics observed at zero field, at resonance, and at high field are analyzed by numerical simulation of the kinetics shown in the middle column of Figure 1.38 At zero magnetic field, S-T mixing is likely governed by incoherent spin relaxation considering the fact that the energy gap is much larger than aeff. The detailed mechanism of such zero quantum relaxation with large 2J has not been reported thus far, but it is reasonable to assume that the S-T+1, S-T0, and S-T-1 relaxation rates are all the same because the quantization axis does not exist at zero field. At resonance, S-T+1 mixing occurs efficiently by the hyperfine interaction, and this process is treated approximately as an incoherent kinetic process with a rate constant of khfcc ) 1 × 108 s-1. This approximation is valid if other processes are slow compared to coherent mixing.60 S-T0 relaxation as well as single quantum T0-T+1 and T0-T-1 relaxation is also taken into account. The relaxation rates are all assumed to be equal for simplicity. This assumption may not be strictly valid considering the different spin multiplicity but does not cause a serious problem for fitting the data because the main population flow at resonance is due to fast hyperfineinduced S-T+1 mixing. In the high field limit, the T+1 and T-1 states are well separated from the S and T0 states, while the S and T0 states are still mixed by the relaxation process. The relaxation rate (krlx) is assumed to be independent of magnetic field because the S-T0 energy gap is determined solely by 2J. The kinetic equations at zero field, at resonance, and at high field are solved numerically to calculate the time evolution of the RP population in each spin sublevel. The transient optical absorption signal of the RP depends linearly on the sum of populations in the four RP sublevels. It is also possible to calculate the time evolution of the population of the neutral triplet excited state generated by CR from the triplet RP states. The calculated time evolution of the RP and/or triplet state is

Figure 3. MARY spectra obtained by plotting the observed CR rate (kobs, blue lines) and the relative triplet yield at late times (T(B)/T(0), red lines) as a function of applied magnetic field for n ) 2 (a), n ) 3 (b), and n ) 4 (c).

used to fit the transient absorption kinetics at the three field conditions. The fitted kinetic traces of the triplet signal for n ) 2 and 3 are shown in Figure 4. The magnetic field dependent rise time, which reflects the decay lifetime of the RP, and final triplet yield are nicely simulated by this model, and kCRS, kCRT, and krlx obtained in this manner are plotted in Figure 2a. Importance of Spin Dynamics on Charge Recombination Kinetics. The kinetic analysis shows that kCRT is about 10 times faster than kCRS for n ) 2-4, both of which show similar rDA dependence, namely, an increase from 2 to 3 and almost the same rate for 3 and 4. The rDA dependence for multiple CR pathways with regard to electron transfer mechanisms is discussed in detail in the next section. Here we focus on the

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Figure 4. Transient absorption kinetics of the PDI triplet signal (monitored at λ ) 690 nm) at zero field (blue), resonance (green), and high field (red) associated with the simulated kinetics (black) at the corresponding magnetic field conditions for n ) 2 (a) and n ) 3 (b).

Figure 5. (a) Distance dependence of spin selective CR rates (kCRS and kCRT) and S-T relaxation rates for DMJAn-Phn-NI. kobs0 is the CR rates observed at zero field. kCRS is equal to kobs0 for n ) 1 due to large 2J (see ref 38). (b) Distance dependence of 2J for DMJAn-Phn-NI.

relationship between the spin dynamics and the CR kinetics at zero field taking into account the spin-selective recombination pathways. The observed CR rates at zero field increase from n ) 3 to 4, although the intrinsic CR rates for both the singlet and triplet channels are nearly constant. This fact clearly indicates the importance of RP spin dynamics on the CR kinetics even at zero magnetic field. For n ) 2 and 3, as is observed in the MARY spectra (Figure 3), 2J is large enough compared to aeff, so that S-T conversion at zero field is governed by the relatively slow relaxation process. For n ) 4, on the other hand, coherent S-T interconversion occurs at 108 s-1 induced by the hyperfine interaction because 2J is smaller than the hyperfine interaction. In this particular D-B-A system, kCRS is about 10 times slower than kCRT and the fit gives krlx in between the two recombination rates for both n ) 2 and 3. In terms of the population flow among RP sublevels, this means that the RP avoids fast recombination from the triplet sublevel as a result of slow S-T interconversion; namely, the nonequilibrated spin dynamics presents a bottleneck to the overall RP decay. Population flow for n ) 4 is totally different from that of n ) 2 and 3 because S-T conversion is fast compared to the CR rates and the spin system quickly reaches a steady state where the singlet state and degenerate triplet states are equally populated. In this case, the RP cannot avoid fast recombination from the triplet sublevel as described by eq 4. This difference in spin dynamics gives rise to an increase of kobs0 from n ) 3 to 4, although the intrinsic CR rates are almost the same. The increase of krlx from n ) 2 to 3 looks reasonable because 2J is smaller for longer rDA as shown in Figure 2b. This increase in krlx is also observed for the DMJAn-Phn-NI system (Figure 5a). The picture obtained here indicates that spin dynamics can be a controlling factor for CR, if the RP has a sufficiently large

2J and spin-selective CR pathways with different rates. This is interesting because 2J in such long-range electron transfer systems originates from the charge transfer interaction between donor and acceptor. Under certain conditions, where kCRS , krlx , kCRT, stronger D-A coupling results in slower CR as a result of the large S-T energy gap. As a consequence, it is possible to control the lifetimes of charge-separated states by careful tuning the D-A coupling to take advantage of the spin dynamics of the RP. The effect of spin dynamics on the CR kinetics is less obvious in the case of the previously studied DMJAn-Phn-NI system given that kobs0 almost coincides with kCRT for n ) 1-4 (Figure 5a). However, this is due to coincidentally similar kCRS and kCRT values for the shorter bridges (n ) 1 and 2), where S-T mixing is inefficient due to the large 2J. For the shorter bridges, triplet RP recombination is not active at zero field, even though kCRT is as large as kCRS. This is obvious from the very small triplet yield at zero field. For longer bridges (n ) 3 and 4), kCRS is much smaller than kCRT,, but S-T interconversion is efficient because 2J is smaller. In this case, the total rate is governed largely by kCRT thanks to efficient S-T mixing (see eq 4), and a large triplet yield is observed at zero field. Thus kobs0 is similar to kCRT for all bridges, but the actual population flow and yield of recombination products are still gated by spin dynamics (singlet recombination for n ) 1 and 2, and triplet recombination for n ) 3 and 4). The results shown here indicate that the observed CR rates at zero field (kobs0) do not always reflect the intrinsic spinselective CR rates. It is possible that kobs0 is determined by the spin relaxation bottleneck, especially when 2J is in the intermediate region. This is obviously the case for n ) 2 because krlx, which is on the same order as kobs0, is between the wellseparated values of kCRS and kCRT. The rate-determining step

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for n ) 2 is clearly S-T relaxation, which is not an intrinsic CR rate that can be used to discuss electron transfer mechanisms. Using conventional chemical kinetics, disregarding the spin dynamics, and assuming that CR occurs from a single RP site, it is possible to separate kCRT starting with

d[RP] ) -(kCRS + kCRT)[RP] dt

(6)

If the initial yield of the RP ([RP0]) and the final yield of the triplet state ([Tf]) are measured by transient absorption spectroscopy, one would simply calculate kCRS and kCRT from the observed decay rate of the RP (kobs0) as

kobs0 ) kCRS + kCRT kCRS )

[RP0] - [Tf] kobs0 [RP0]

kCRT )

(7)

[Tf] k [RP0] obs0

However, when krlx , kCRT as for n ) 2, the kinetic equations, which take into account the spin dynamics are simplified using the steady-state approximation

d[RPS] ) -(kCRS + krlx)[RPS] dt d[RPT] ) krlx[RPS] - (kCRT + krlx)[RPT] dt = krlx[RPS] - kCRT[RPT] ) 0 (8) d[T] ) kCRT[RPT] ) krlx[RPS] dt These equations can be easily solved to give kobs0 and [Tf] as

usually much lower than the precursor triplet state.61-63 But in this case again, the T-S relaxation can be a bottleneck for CR, if 2J is sufficiently large. Thus it is very important to determine the magnitude of 2J by MFE experiments or other techniques such as TREPR or low-field CIDNP.64 Spin dynamics is less important if CR is much faster than S-T mixing and takes place from the same spin multiplicity as that of the precursor excited state. This is the case for a series of previously reported D-B-A molecules,30 which charge separate from the singlet excited state and quickly recombine to the singlet ground state by rate constants of 109-1010 s-1. These recombination rates are much faster than the spin dynamics (∼108 s-1), and kobs0 can be safely used as an intrinsic kCRS. In other words, spin dynamics is extremely important for long-lived RPs, which frequently give large MFEs. Charge Transfer Mechanisms for Multiple CR Pathways. Now that the intrinsic CR rates for the singlet and triplet pathways are separated from the spin dynamics by the MFE analysis, we can discuss the CR mechanisms for each process based on electron transfer theory. First, we found that for n ) 2-4 kCRT is about an order of magnitude larger than kCRS, independent of n. This tendency can be explained qualitatively by Marcus theory using the energy levels of the RPs.65 Considering the fact that the solvent reorganization energy λs ∼ 0 in toluene, the total reorganization energy (λ) is determined exclusively by the internal reorganization energy, which is λV ∼ 0.6 eV.21 This means that triplet CR (∆G ∼ -1 eV) occurs close to the maximum in the Marcus rate vs free energy profile, whereas singlet recombination (∆G ∼ -2 eV) is deep in the Marcus inverted region. It is impossible to separate kCRS and kCRT for n ) 1 because a MFE is not observed. It is natural to assume, however, that both kCRS and kCRT are larger than the corresponding rates for n ) 2 because kobs0, which is a linear combination of kCRS and kCRT, is much larger for n ) 1 than for n ) 2. Thus kCRS and kCRT decrease from n ) 1 to 2, increase from n ) 2 to 3, and remain constant for n ) 3 to 4. These changes in CR rates can be explained by crossover of the dominant electron transfer mechanism from superexchange to thermal hopping as indicated previously. It is well-known that electron transfer by the superexchange mechanism is exponentially distance dependent as expressed by

kobs0 ) kCRS + krlx

k ) k0e-β(r-r0)

[Tf] )

[RP0]krlx kCRS + krlx

(9)

krlx )

[Tf] k [RP0] obs0

(10)

which yields

where k0 is the rate constant at the van der Waals contact distance r0 and β is the exponential decay factor for superexchange mediated charge transfer. This distance dependence derives mainly from the distance dependence of the effective electronic coupling between the donor and the acceptor (VDA), which governs the electron transfer rate as

k) This means that if we ignore the spin dynamics, we would misinterpret krlx as kCRT because generation of the triplet state is gated by slow spin relaxation. If 2J is very small, the one site model works well qualitatively because of fast interconversion among all the RP states. However, it should be noted that the observed rate is not a simple sum of kCRS and kCRT but is weighted by the spin multiplicity as described in eq 4. In triplet born RP systems with a few exceptions,60 the RPs recombine from only the singlet state because the RP energy is

(11)

2π |V | 2(FCWD) p DA

(12)

where (FCWD) is Franck-Condon weighted density of states.65 Adopting McConnell’s tight binding model, the electronic coupling for N equivalent bridge sites is expressed as

VDA )

( )

VDBVBA VBB ∆EDB ∆EDB

N-1

(13)

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where VDB, VBB, and VDB are the electronic coupling between donor-bridge, bridge-bridge, and donor-bridge and ∆EDB is charge injection barrier to the bridge site.29 Thus the β value is related to the charge injection barrier and VBB by

β)

( )

∆EDB 2 ln r VBB

(14)

where r is the length of one bridge segment. As has been demonstrated by numerous MFE and TREPR studies, the distance dependence of the superexchange component of CR can be directly measured using 2J and eq 1. Figure 2b shows the 2J value obtained from MARY spectra (Figure 3) plotted as a function of rDA. The log plot of 2J is fitted by the equation

2J ) 2J0e-R(rDA-r0)

(15)

from which an exponential decay factor of R ) 0.27 Å-1 is obtained. This value indicates that the pure superexchange electron transfer rate is diminished by about a factor of 10 by increasing one FL bridge unit because r ) rDA(n+1) - rDA(n) ) 7.2-7.9 Å. Although kCRS and kCRT cannot be separated for n ) 1, it is likely that the dramatic decrease in CR rate from n ) 1 to 2 is due to the dominance of the superexchange mechanism. This is also consistent with the fact that the charge injection barrier is larger for shorter bridges; namely, thermal hopping is unfavorable. The increase in the CR rates from n ) 3 to 4 can be explained by the dominance of the thermally induced hopping mechanism. From the discussion above, pure superexchange mediated CR for n ) 3 should be 10 times slower than that for n ) 2. Once the charge injection barrier is reduced so that the positive charge thermally transfers to the bridge site, the hopping mechanism dominates over superexchange.30,66 Both kCRS and kCRT stay constant from n ) 3 to 4, strongly indicating that the hopping mechanism is dominant in this region given that the hopping rate is only weakly dependent on rDA. Theory predicts that the hopping rate is approximately proportional to 1/rDA, which means only a ∼20% decrease is expected for increasing n from 3 to 4.32 This decrease is close to the error bars for kCRS and kCRT due to inaccuracies in both the experimental measurements and simulations. A small decrease in the charge injection barrier might increase the hopping rate, which cancels out the decreased distance due to site-to-site intrabridge hopping. Thus, the kCRS and kCRT values obtained by MFE analysis strongly support our earlier conclusion that the superexchange CR dominates for n ) 1 and 2, while thermally activated hopping prevails for n ) 3 and 4. The distance dependence of the CR rates for the PDI-FLn-PTZ series differs from that for the previously studied DMJAn-Phn-NI system.38 In the DMJAn-Phn-NI system, both kCRS and kCRT decrease exponentially with rDA with βCRS ) 0.48 Å-1 and βCRT ) 0.35 Å-1 for the singlet and triplet CR pathways, respectively (Figure 5a). βCRT exactly matches the R value obtained by the distance dependence of 2J (Figure 5b) indicating that the dominant CR mechanism is superexchange for n ) 1-5. The dominance of superexchange vs hopping is explained qualitatively by the high charge injection barrier to both HOMO and LUMO change transfer in the DMJAn-Phn-NI system, whereas the PDI-FLn-PTZ system has a much smaller injection barrier for n ) 3 and 4, which makes the hopping mechanism favorable.

The very similar distance dependence of kCRS and kCRT for PDI-FLn-PTZ indicates that the orbital pathway involved is the same for both the singlet and triplet pathways. It is unlikely that electron transfer occurs via the LUMO of the FL bridges for both superexchange and hopping because reduction of the FL bridges occurs at very negative potentials.21 As has been discussed above, FL has a low oxidation potential, which places its HOMO energy close to that of the donor especially for the longer bridges. Consequently, for both spin selective pathways, charge transfer occurs via the HOMO of the bridge units (hole transfer). Multistep CR occurs via bridge HOMOs for n ) 3 and 4, whereas the bridge HOMOs for n ) 1 and 2 contribute strongly to the virtual bridge states that determine the electronic coupling matrix elements VDB, VBB, and VBA. For both the singlet and triplet pathways, the initial step for the hopping mechanism is PTZ+•-FLn-PDI-• f PTZ-FLn+•-PDI-•, which does not depend on the CR rate for PTZ-FLn+•-PDI-•. The 10-fold difference between the CR rates for the singlet and triplet pathways is attributed to the difference between the hole migration rates from the bridge to PDI acceptor. Previous discussions based on Marcus theory can also be applied to the intermediate PTZ-FLn+•-PDI-• RPs given that the energies of these intermediate states are almost the same as that of the PTZ+•-FLn-PDI-• RP for the long FL bridges.21 One advantage of using spin dynamics as a probe of CR mechanisms is that it is possible to extract the pure superexchange component from the distance dependence of 2J even if it is a minor component.22 The value of R ) 0.27 Å-1 gives the ratio of the bridge coupling to the charge injection barrier as VBB/∆E ∼ 0.37 (eq 13). This value can be compared to the value obtained in the same manner for the DMJAn-Phn-NI system where VBB/∆E ∼ 0.46. These two values are on the same order of magnitude, but considering that PTZ-FLn-PDI has a lower charge injection barrier for both spin selective channels, VBB for the DMJAn-Phn-NI bridge system is likely larger. This is consistent with the fact that the hopping mechanism dominates for the FLn bridges in PTZ-FLn-PDI, whereas superexchange dominates for the Phn bridges in DMJAn-Phn-NI. Mechanism of S-T Relaxation and Simulation of MARY Spectra. It has been established that S-T relaxation plays an important role in the total CR process if 2J is large compared to aeff. However, there have been very few detailed studies on the spin dynamics of RPs with large 2J values. Here we propose a possible mechanism for this relaxation with numerical simulations of the MARY spectra. The MARY spectra reflect the magnetic field dependence of spin state mixing, which involves not only hyperfine-induced coherent mixing but also field-dependent S-T+1 and S-T-1 relaxation processes.59 For n ) 2 and 3, the MARY spectra show moderate changes even at fields much higher than 2J (Figure 3, parts a and b). This feature is explained by the field dependence of spin relaxation, which decreases moderately at higher fields (see eq S4 in the Supporting Information and discussions below). The effect of the relaxation processes can sometimes be obtained using time-resolved MARY spectroscopy, in which the field dependence of the transient absorption signal is measured at each observation time.67,68 The yield changes at high fields are more obvious at later times as shown in Figure 6. This results from the time scale of the relaxation processes being slow (microseconds) and their effect is seen only at later times. The differences between the spectra observed by kobs (Figure 3, blue curves) and by the triplet yield (Figure 3, red curves) are due to the difference in observation time scale.

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∆P(t) ) [RPS] - [RPT] ) cos ω′t +

∆ω2 (1 - cos ω′t) ω′2 (16)

where

ω′ ) √4Vaeff2 + ∆ω2

(17)

Time averaging of eq 16 gives the mixing efficiency as a function of the energy gap as

∆P ) Figure 6. Time-resolved MARY spectra for n ) 2 at 100 ns (blue) and 1 µs (red) associated with simulations using the Liouville equation with the S-T dephasing model (black lines).

The observation of long-lived products generated from the RPs tends to accentuate relatively slow events due to accumulation of a MFE on the products. Thus, MARY spectra for the triplet yield are more likely to be affected by S-T relaxation compared to the RP recombination rates. This difference is small for n ) 4 probably due to the short lifetime of the RP. Conventional spin relaxation is induced by rotational fluctuations of anisotropic interactions in the RP.69 Anisotropic Zeeman interactions are less important considering the low magnetic fields used here. Dipolar relaxation is also considered to be very slow because the interaction itself is very small because rDA is very large. However, relaxations induced by the anisotropic hyperfine interactions of each radical may influence the D-B-A molecules. T1, which is the population relaxation time between the R and β spin states of each radical, works as a single quantum relaxation of the RP between the T(1 states and the S-T0 mixed state (although mixing is inefficient, if 2J is large). On the other hand, T2, which monitors the decoherence of each radical, could induce zero quantum S-T0 relaxation because the zero quantum coherence in the product basis set corresponds to a population difference between the S and T0 states in the S-T basis set. We have attempted a direct spin dynamics simulation for n ) 2 using the Liouville equation taking into account T1 and T2 induced by the anisotropic hyperfine interactions (Supporting Information).67,70 MARY spectra, however, cannot be simulated assuming these conventional spin relaxation mechanisms, indicating that anisotropic hyperfine interactions are not the main source of S-T relaxation. The S-T relaxation can be adequately explained by introducing a phenomenological S-T dephasing term.71 The effect of S-T dephasing on the time evolution of RP populations has been studied in numbers of weakly coupled (2J e Vaeff) RP systems mainly by TREPR.25,37,72-74 It has previously been demonstrated for RPs in SDS micelles that such a dephasing process induces an incoherent population transfer between the S-T0 mixed states and the T(1 states in low magnetic fields comparable to Vaeff.67,70,75 Here, we qualitatively consider the state mixing between two states for simplicity, the singlet (S) and one of the three triplet states (T) in the same manner as the previous studies. The S and T states are separated by energy gap ∆ω and are coherently mixed by the matrix element of the effective hyperfine interaction (Vaeff). Spin state mixing can be calculated by solving the two-state quantum mechanical problem resulting in

∆ω2 4Vaeff + ∆ω2

(18)

Assuming T is the T+1 state and applying a single nucleus approximation, this indicates that the full width at half-maximum (fwhm) of the resonance in the MARY spectra is determined by the hyperfine interactions as

fwhm ) 4Vaeff ) √2aeff

(19)

where

Vaeff ) 〈S, RN |aeffS · I|T+1, βN〉 ) -

aeff 2√2

(20)

if spin state mixing is governed solely by the coherent hyperfine interaction, which is not the case for n ) 2 because the experimental line width (27 mT) is 10 times larger than the theoretical value (2.5 mT). If spin dephasing works on this mixing process, it alters the mixing and eventually induces an incoherent population transfer between the S and T states.67,75 It should be noted here that S-T dephasing is not a direct population relaxation between the S and T states, but the decoherence between the two states, which means the relaxation rate (krlx) is related to, but not the same as, the dephasing rate itself (kSTD). These rates can be analytically correlated in the limit where kSTD is large compared to the time scale of Vaeff, and the mixing is completely incoherent as given by

krlx )

kSTDaeff2 /2 kSTD2 + ∆ω2

(21)

From this equation, kSTD ∼ 109 s-1 is expected, taking into account the corresponding energy gap of ∆ω ) 2J ) 30.8 mT and krlx of ∼106 s-1 obtained from the kinetic analysis. To support this idea, we have simulated the MARY spectra taking account this S-T dephasing term in addition to conventional relaxations. The spectral width of the MARY spectra is very sensitive to the S-T dephasing rates (Supporting Information) and the experimental spectra are simulated by kSTD ) 2 × 109 s-1. This rate constant is consistent with the value predicted analytically by eq 21. Very broad MARY spectra are explained well by S-T dephasing, which enhances S-T mixing at out-ofresonance magnetic fields, where mixing by the hyperfine interaction is inefficient. This mechanism is intrinsically the same as that previously discovered in micellar systems to explain

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the broadening of MARY spectra and incoherent S-T mixing at low magnetic fields.67,68,70,75 In these systems, the dephasing is considered to be due to diffusional fluctuations of 2J by the frequent re-encounter process of the two radicals in the micellar cage.71 This begs the question as to why such a dephasing process occurs in fixed-distance RPs, where the re-encounter process is completely suppressed by the rigid molecular bridges. It has been suggested from the temperature dependence of superexchange mediated electron transfer that torsional motions about the single bonds joining oligo-aromatic bridges modulate the electronic coupling between the donor and the acceptor.34 Namely, superexchange mediated electron transfer can be gated by conformational dynamics, the so-called non-Condon effect.45-47 This indicates that 2J is also modulated by torsional motions about the single bonds joining the bridge segments at room temperature in toluene as supported by a decrease in 2J when the solvent is frozen.54 The ground state optimized dihedral angle between the π systems of the FL units within the FL oligomers is 40-60°, which is not favorable for strong bridge-bridge coupling according to the dependence of the coupling on the dihedral angle θ

VBB ) VBB0 cos θ

(22)

where VBB0 is the largest coupling for θ ) 0°.21,45 Temperature dependence studies on the thermally activated CR rates in related PTZ+•-Phn-PDI-•, where n ) 3 and 4 have shown that the Ph-Ph torsional activation barriers in these RPs are 1290 and 2030 cm-1, which match closely with theoretically predicted and experimentally observed barriers for the planarization of terphenyl and quaterphenyl.34 It is thus likely that 2J modulation by these torsional motions induces the spin dephasing process. For the total electronic coupling that governs 2J,torsionalmotionsaroundthedonor-bridgeandbridge-acceptor linkages need to be considered as well. These observations strongly indicate that there is a possibility that one can extract the torsional dynamics that gate superexchange-mediated CR from the spin dynamics of the RPs. Spin dynamics simulations that directly take into account the bridge dynamics and 2J fluctuations are underway.45 Conclusion We have analyzed MFEs on CR within PTZ+•-FLn-PDI-• and have obtained the spin-selective CR and S-T relaxation rates for n ) 2-4. It has been demonstrated for n ) 2 and 3 that when 2J is large, the spin dynamics serves as a bottleneck to CR. The distance dependence of both kCRS and kCRT is explained by the crossover of the electron transfer mechanism from superexchange to thermal hopping. A possible mechanism for S-T relaxation is proposed based on a model for S-T dephasing, which is a consequence of 2J fluctuations caused by torsional bridge dynamics. The results presented here indicate two possible approaches toward assessing electron transfer processes in terms of spin dynamics. First, as has been demonstrated previously, spin dynamics is a powerful probe to determine spin-selective CR rates and the electronic coupling matrix elements. In addition, the present analysis of MARY spectra suggests that the RP molecular dynamics gate CR reactions. Second, spin dynamics can control RP lifetimes. For example, energy wasting triplet formation can be inhibited by slightly increasing the electronic coupling or rDA so that S-T interconversion becomes very slow.

Acknowledgment. We thank Dr. Randall H. Goldsmith for the synthesis of the PTZ-FLn-PDI molecules and Drs. Wenhau Liu and Emily Weiss for magnetic field effect measurements. This work was supported by the Chemical Sciences, Geosciences, and Biosciences Division, Office of Basic Energy Sciences, DOE under Grant No. DE-FG02-99ER14999. Supporting Information Available: Details on the simulation for the RP spin dynamics using the Liouville equation. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) Parson, W. W. Photosynth. Res. 2003, 76, 81–92. (2) Norris, J. R.; Budil, D. E.; Gast, P.; Chang, C. H.; Elkabbani, O.; Schiffer, M. Proc. Natl. Acad. Sci. U.S.A. 1989, 86, 4335–4339. (3) Hoff, A. J.; Deisenhofer, J. Phys. Rep. 1997, 287, 2–247. (4) Wasielewski, M. R. Chem. ReV. 1992, 92, 435–461. (5) Lukacs, A.; Eker, A. P. M.; Byrdin, M.; Brettel, K.; Vos, M. H. J. Am. Chem. Soc. 2008, 130, 14394–14395. (6) Medvedev, D.; Stuchebrukhov, A. A. J. Theor. Biol. 2001, 210, 237–248. (7) Kunkely, H.; Vogler, A. Angew. Chem., Int. Ed. 2009, 48, 1685– 1687. (8) Barber, J. Chem. Soc. ReV. 2009, 38, 185–196. (9) Magnuson, A.; Anderlund, M.; Johansson, O.; Lindblad, P.; Lomoth, R.; Polivka, T.; Ott, S.; Stensjo, K.; Styring, S.; Sundstrom, V.; Hammarstrom, L. Acc. Chem. Res. 2009, 42, 1899–1909. (10) Gust, D.; Moore, T. A.; Moore, A. L. Acc. Chem. Res. 2009, 42, 1890–1898. (11) Wasielewski, M. R. Acc. Chem. Res. 2009, 42, 1910–1921. (12) Fukuzumi, S.; Kojima, T. J. Mater. Chem. 2008, 18, 1427–1439. (13) Schmidt-Mende, L.; Fechtenkotter, A.; Mullen, K.; Moons, E.; Friend, R. H.; MacKenzie, J. D. Science 2001, 293, 1119–1122. (14) Tang, C. W. Appl. Phys. Lett. 1986, 48, 183–185. (15) Yonehara, H.; Pac, C. Thin Solid Films 1996, 278, 108–113. (16) Straight, S. D.; Andreasson, J.; Kodis, G.; Bandyopadhyay, S.; Mitchell, R. H.; Moore, T. A.; Moore, A. L.; Gust, D. J. Am. Chem. Soc. 2005, 127, 9403–9409. (17) Flood, A. H.; Stoddart, J. F.; Steuerman, D. W.; Heath, J. R. Science 2004, 306, 2055–2056. (18) Campbell, I. H.; Crone, B. K.; Smith, D. L. Semiconducting Polymers: Chemistry, Physics and Engineering, 2nd ed.; Wiley-VCH: Weinheim, 2007; pp 421-454. (19) Walter, M. J.; Lupton, J. M. Highly Efficient OLEDs with Phosphorescent Materials; Wiley-VCH: Weinheim, 2008; pp 99129. (20) Iwasaki, Y.; Osasa, T.; Asahi, M.; Matsumura, M.; Sakaguchi, Y.; Suzuki, T. Phys. ReV. B 2006, 74, 195209. (21) Goldsmith, R. H.; Sinks, L. E.; Kelley, R. F.; Betzen, L. J.; Liu, W. H.; Weiss, E. A.; Ratner, M. A.; Wasielewski, M. R. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 3540–3545. (22) Weiss, E. A.; Ahrens, M. J.; Sinks, L. E.; Gusev, A. V.; Ratner, M. A.; Wasielewski, M. R. J. Am. Chem. Soc. 2004, 126, 5577–5584. (23) Van Vooren, A.; Lemaur, V.; Ye, A. J.; Beljonne, D.; Cornil, J. ChemPhysChem 2007, 8, 1240–1249. (24) Shafirovich, V. Y.; Levin, P. P. Russ. Chem. Bull. 2001, 50, 599– 606. (25) Kobori, Y.; Yamauchi, S.; Akiyama, K.; Tero-Kubota, S.; Imahori, H.; Fukuzumi, S.; Norris, J. R. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 10017–10022. (26) Paddon-Row, M. N.; Oliver, A. M.; Warman, J. M.; Smit, K. J.; Dehaas, M. P.; Oevering, H.; Verhoeven, J. W. J. Phys. Chem. 1988, 92, 6958–6962. (27) Bixon, M.; Jortner, J. J. Chem. Phys. 1997, 107, 5154–5170. (28) Jortner, J.; Bixon, M.; Langenbacher, T.; Michel-Beyerle, M. E. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 12759–12765. (29) McConnell, H. M. J. Chem. Phys. 1961, 35, 508–515. (30) Davis, W. B.; Svec, W. A.; Ratner, M. A.; Wasielewski, M. R. Nature 1998, 396, 60–63. (31) Davis, W. B.; Ratner, M. A.; Wasielewski, M. R. Chem. Phys. 2002, 281, 333–346. (32) Davis, W. B.; Wasielewski, M. R.; Ratner, M. A.; Mujica, V.; Nitzan, A. J. Phys. Chem. A 1997, 101, 6158–6164. (33) Goldsmith, R. H.; DeLeon, O.; Wilson, T. M.; Finkelstein-Shapiro, D.; Ratner, M. A.; Wasielewski, M. R. J. Phys. Chem. A 2008, 112, 4410– 4414. (34) Weiss, E. A.; Tauber, M. J.; Kelley, R. F.; Ahrens, M. J.; Ratner, M. A.; Wasielewski, M. R. J. Am. Chem. Soc. 2005, 127, 11842–11850.

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