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J. Phys. Chem. A 2010, 114, 11842–11843
Beyond Fo¨rster Resonance Energy Transfer in Linear Nanoscale Systems William Barford* Department of Chemistry, Physical and Theoretical Chemistry Laboratory, UniVersity of Oxford, South Parks Road, Oxford, OX1 3QZ, United Kingdom ReceiVed: August 5, 2010; ReVised Manuscript ReceiVed: September 28, 2010
The line dipole approximation is used to investigate analytical corrections to the Fo¨rster energy transfer rate, k, derived via the point dipole approximation. It is shown that that for molecules whose conjugation length, L, is much larger than the separation, R, between molecules the line dipole approximation predicts k ∼ (RL)-2 ∼ (RN)-2 (where N is the number of conjugated monomer units). This is in contrast to the point dipole approximation, which predicts k ∼ L2R-6 ∼ N2R-6. Fo¨rster resonance energy transfer (FRET) in an important photophysical process in many molecular systems.1 In polymer photovoltaic devices it is one of many factors that determine the device’s ultimate efficiency. It is the purpose of this short note to amplify an observation made in the conclusions of ref 2 that FRET in polymer systems is potentially much smallersand with different size and separation dependenciessthan that predicted by the point dipole approximation. The same conclusion has also been deduced numerically via quantum chemistry calculations.3-6 The starting point for this prediction is the observation that the Fo¨rster transfer rate between a donor and acceptor is determined by the Fermi golden rule expression,
k)
2π 2 |J| δ(EF - EI) p
(1)
where J is the exciton transfer integral between the donor and acceptor. The δ -function in eq 1 ensures energy conservation, and EI and EF are the initial and final energies, respectively. Now, as shown numerically in refs 7-11 and analytically in ref 2 via the line dipole approximation,2,12,13 for molecular systems whose sizesor more correctly, conjugation length14sexceeds their separation, the exciton transfer integral is a decreasing function of molecular size, L. In fact, if we assume that the excited state (or exciton) center-of-mass wave function is delocalized uniformally along the molecule, then for parallel and collinear molecules J ∼ 1/(RL) for L . R, where R is the distance of nearest separation between the molecules. Specifically, for two linear, parallel polymers of length L separated by a distance R, the exciton transfer integral determined by the line dipole approximation, Jld, is related to the exciton transfer integral determined by the point dipole approximation, Jpd, by
Jld ) fJpd where * E-mail:
[email protected].
Jpd )
() µ2 R3
(3)
and µ is the molecular transition dipole moment for the exciton in question. The prefactor f is the correction to the point dipole approximation and is given by2
f)
(
2R2 R 12 2 L √(R + L2)
)
(4)
For L . R, f f 2(R/L)2; whereas for L , R, f f 1. Thus, inserting eq 2 into eq 1, we conclude that the ratio of the Fo¨rster rates in the line and point dipole approximationsskld and kpd, respectivelysis given by
kld /kpd ) f 2
(5)
where f 2 is plotted in Figure 1. For L . R, kld/kpd f 4(R/L)4; whereas for L , R, kld/kpd f 1. Noting that L ) Nd, where N is the number of conjugated monomer units, and d is the repeat unit distance, we may also express the ratio for L . R as, kld/ kpd ) 4(R/Nd)4.
(2)
Figure 1. The ratio of the Fo¨rster rates determined in the line and point dipole approximations, f 2 ) kld/kpd, as a function of the ratio of their separation and size, R/L. The function f(R/L) is given in eq 4.
10.1021/jp107374r 2010 American Chemical Society Published on Web 10/12/2010
Beyond Fo¨rster Resonance Energy Transfer
J. Phys. Chem. A, Vol. 114, No. 43, 2010 11843
It is also useful to express the Fo¨rster rate in the point dipole approximation as
kpd
( )
pd 1 RF ) τR R
6
(6)
where the Fo¨rster radius, RFpd, is defined by equating kpd(R ) RF) with the radiative rate, τR-1. Similarly, according to the line dipole approximation for L . R,
kld )
( )
ld 1 RF τR R
2
(7)
where
RFFld
)
( )
RFpd
RFpd L
are delocalizedsand not the actual molecular size. For conformationally disordered polymers, the conjugation length of the lowest-lying excitons is typically rather short, being 10-20 repeat units for the vibrationally unrelaxed exciton.15-17 Because the conjugation length is a decreasing function of disorder, we conclude that more extended, less disordered polymers are less efficient at interchain FRET. For vibrationally relaxed (i.e., self-trapped) excitons, the conjugation length (again, defined as the exciton localization length) is, however, very short, typically being ∼5 repeat units.18 Assuming, then, that Fo¨rster transfer occurs after vibrational relaxation,19 the point dipole approximation becomes a much better estimate of the exciton transfer rate. In conclusion, the line dipole approximation has been used to derive the size and separation scaling dependencies for Fo¨rster transfer between parallel extended molecules. This formalism can easily be extended to more general molecular configurations.
2
(8)
when RFpd , L. Thus, we see that the corrected Fo¨rster rate for extended linear molecules decreases as R-2 when their size exceeds their separation, in agreement with computational results.4,5 Further, since kld ∼ kpd/L4, whereas kpd is proportional to the spectral overlapsimplying that kpd ∼ L2 from the Kuhn oscillator sum rulesthe corrected Fo¨rster rate for extended linear molecules also decreases as L-2. In summary, for L . R, the line dipole approximation predicts k ∼ (RL)-2 ∼ (RN)-2, whereas the point dipole approximation predicts k ∼ L2R-6 ∼ N2R-6 (where N is the number of conjugated monomer units). We also observe that eqs 2-4 imply that the Fo¨rster rate derived from the line dipole approximation, like that for the point dipole approximation, is only nonzero for bright (i.e., optically active) states. This conclusion is in contrast to the quantum chemical calculations, which indicate Fo¨rster transfer between dark states at close molecular separation.4 The resolution of this contradiction is to note that the line dipole approximation is only valid for molecular separations R g 2d and that for very small separations it breaks down. We do, however, conclude that for relatively small separations, that is, R g 2d, dark states will not participate in Fo¨rster transfer. The results described here, although specifically derived for conjugated polymeric systems, also quantitatively apply to other extended linear systems, such as carbon nanotubes. However, as alluded to above, the physically relevant length is the exciton conjugation lengthsthe length scale over which the excitons
References and Notes (1) Beljonne, D.; Curutchet, C.; Scholes, G. D.; Silbey, R. J. J. Phys. Chem. B 2009, 113, 6583. (2) Barford, W. J. Chem. Phys. 2007, 126, 134905. (3) Scholes, G. D.; Jordanides, X. J.; Fleming, G. R. J. Phys. Chem. B 2001, 105, 1640. Jordanides, X. J.; Scholes, G. D.; Fleming, G. R. J. Phys. Chem. B 2001, 105, 1652. (4) Wong, K. F.; Biman, B.; Rossky, P. J. J. Phys. Chem. B 2004, 108, 5752. (5) Das, M.; Ramesesha, S. J. Chem. Phys. 2010, 132, 124109. (6) Hennebicq, E.; Pourtois, G.; Scholes, G. D.; Herz, L. M.; Russell, D. M.; Silva, C.; Setayesh, S.; Grimsdale, A. C.; Mu¨llen, K.; Bre´das, J. L.; Beljonne, D. J. Am. Chem. Soc. 2005, 127, 4744. (7) Soos, Z. G.; Hayden, G. W.; McWilliams, P. C. M.; Etemad, S. J. Chem. Phys. 1990, 93, 7439. (8) McIntire, M. J.; Manas, E. S.; Spano, F. C. J. Chem. Phys. 1997, 107, 8152. (9) Manas, E. S.; Spano, F. C. J. Chem. Phys. 1998, 109, 8087. (10) Cornil, J.; dos Santos, D. A.; Crispin, X.; Silbey, R.; Bre´das, J. L. J. Am. Chem. Soc. 1998, 120, 1289. (11) Beljonne, D.; Cornil, J.; Silbey, R.; Millie´, P.; Bre´das, J. L. J. Chem. Phys. 2000, 112, 4749. (12) Grage, M. M.-L.; Zaushitsyn, Y.; Yartsev, A.; Chachisvilis, M.; Sundstro¨m, V.; Pullerits, T. Phys. ReV. B 2003, 67, 205207. (13) Beenken, W. J. D.; Pullerits, T. J. Chem. Phys. 2004, 120, 2490. (14) For the moment it will be convenient to regard the exciton conjugation length and molecular size as being the same. This constraint will be relaxed in the concluding paragraphs. (15) Barford, W.; Trembath, D. Phys. ReV. B 2009, 80, 165418. (16) Makhov, D. V.; Barford, W. Phys. ReV.B 2010, 81, 165201. (17) Barford, W.; Lidzey, D. G.; Makhov, D. V.; Meijer, A. J. H. J. Chem. Phys. 2010, 133, 044504. (18) Boczarow, I. Barford, W. Submitted. (19) Dykstra, T. E.; Hennebicq, E.; Beljonne, D.; Gierschner, J.; Claudio, G.; Bittner, E. R.; Knoester, J.; Scholes, G. D. J. Phys. Chem. B 2009, 113, 656.
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