J. Phys. Chem. 1989. 93. 6073-6079
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CELSIUS TEMPERATURE
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The classical diffusion theory first used by Perrin to describe the depolarization by molecular rotations implicitly assumes the existence of an “infinite” number of possible oscillator orientations. As the time allowed to the diffusion process lengthens the probability that an oscillator will have at emission the orientation that it had at excitation continuously decreases and becomes vanishingly small for a sufficiently long fluorescent lifetime. On the other hand if the number of possible orientations is finite the probability of finding the oscillator in the orientation that it had at excitation cannot fall below a certain value however lively the motions or however long the time allowed for the diffusion. There is little doubt that the latter is the case in media which introduce appreciable differences in the probability of orientation of the fluorophore with respect to its surroundings, and this may even be the case for fluorophores dissolved in some molecular liquids. By a study of depolarization of the fluorescence at sufficiently high temperatures it should be possible to determine whether the diffusion theory or the parametric approach applies best in this instance.
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LOG (TIME/LIFETIME) Figure 5. Plots of the anisotropy against the decimal logarithm of the time, in units of the fluorescence lifetime, computed from eq 24. Anisotropy and thermodynamic parameters are those of Figure 2, with the addition of kl27 = k237 = 10 and thermal coefficients 6 of the 1 2 and 2 3 transitions, respectively O.O7/OC and O.O3/OC. The three curves are for -20, 0, and 20 O C .
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at a wavelength for which a. = 0.2’ The plots of Figure 5 explain the origin of the thermal dependence of the long-time constant polarization observed in studies of the decay in real time of the fluorescence polarization of probes included in membra ne^.^ Besides they indicate that observation of the decay of the anisotropy corresponding to the stationary polarization shown in Figure 2 demands a time resolution of 1/ lo0 of the fluorescence lifetime, which in protein fluorophores roughly corresponds to 30 ps. Access to these short decay times is now becoming possible.22 Continuous and Discontinuous Reorientations
Figure 4. Y plot showing thermal repolarization. Same parameters as “C) = 0.2. those of Figure 3 exceptfi/f2(0 “C) = l,fi/h(O
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favors the growth of one fractional orientation at the expense of the other two. Here the repolarization has a very different origin than in the one case in which this phenomenon has this far been observed: the motions from asymmetric free rotors on excitation
Acknowledgment. The present work has been supported by the USPH through grant GM-11223 to the author. The author thanks Dr. Suzanne Scarlata for the many discussions that gave rise to the present treatment of motional depolarization of fluorescence. (21) Weber, G. In Time Resolved Fluorescence Spectroscopy in Biochemistry and Biology; Cundall, R. B., Dale, R. E., Eds.; Plenum: London, 1983; pp 1-19. (22) Lackowicz, J. R. In Modern Physical Methods in Biochemistry; Neuberger, A,, Van Deenen, L. L. M., Eds.; Elsevier: New York, 1988; Part B, pp 1-26.
Calculation of the Distribution of Donor-Acceptor Distances in Flexible Bichromophorlc Molecules. Application to Intramolecular Transfer of Excitation Energy Bernard Valeur,* Jacques Mugnier, Jacques Pouget, Jean Bourson, and Fransoise Santi? Laboratoire de Chimie CZnZralet and Laboratoire d’lnformatique, Conservatoire National des Arts et MZtiers, 292 rue Saint- Martin, 75003 Paris, France (Received: December 28, 1988) The distribution of center-to-center distances between chromophores linked to the ends of a short flexible chain can be calculated by using the rotational isomer theory together with the statistical weights of the conformations of short sequences of three and four bonds. The distance distributions are calculated for five coumarin bichromophoric molecules in which the spacer is a short polymethylene chain with a variable number of methylene groups, or a chain containing C-C and C-0 bonds. Good agreement between the predicted and experimental values of the efficiency of excitation energy transfer supports the validity of the calculations.
Introduction
Intramolecular excited-state processes in bichromophoric molecules1 form a subject of considerable interest because of their
’Unite Laboratoire d’lnformatique. associee au CNRS no. 1103 “Physicc-chimie organique appliquk”. f
0022-3654/89/2093-6073$01.50/0
implications in numerous fields: photophysics, photochemistry, biOlogY, polymers, ... For the Processes that require close approach of the two chromophores (formation of excimers Or exciPlexes, photochemical reactions) a certain flexibility of the spacer is (1) De Schryver, F. C.; Boens, N.; Put, J. Adu. Phorochem. 1977, IO, 359.
0 1989 American Chemical Society
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The Journal of Physical Chemistry, Vol. 93, No. 16, 1989
Valeur et al.
Figure 1. Schematic representation of the conformations of a three-bond sequence.
necessary. In contrast, excitation energy transfer2 and electron transfer3 can be observed at rather long distances; chromophores have been linked by a variety of spacers, either rigid (or partially rigid) or f l e ~ i b l e . ~When the spacer is flexible, the distribution of interchromophoric distances plays an important role. In principle, this distribution can be recovered from energy-transfer experiments by analyzing the time-resolved emission of the donor moiety; in the investigations reported so far,5i6 an a priori assumption of the shape of the distribution is made, generally a Gaussian or other bell-shaped distribution, and the mechanism of energy transfer is assumed to be only of the Forster type (long-range dipole-dipole interaction). A Gaussian distribution can describe polymer chains of more than about 50 bonds,' but such a distribution may not be achieved for short chains ( 20
where
4i)
Figure 5. Distribution of distances for bichromophore DXA. The broken
line represents the Gaussian having the same mean (9.2) and standard deviation (3.33) as those of the histogram. corresponding to the interchromophoric distance Ri, the transfer efficiency is @Ti and the rate constant is kTP The average transfer efficiency is then
where pi is the probability of the distance RP Let &,be the value of Ro for K~ = 2/3. In the case of dynamic isotropic averaging, we have simply
whereas, in the case of static isotropic average, only an average transfer rate ( kT1)valid for substitution into
can be obtained in the following way. According to eq 9 and the definition of Ro, we have
Ro6= and replacing
K'
by
(K2),ff,
2~~Ro6 2
we obtain from eq 8
Then, by using eq 14, and eq 10 rewritten as ( K 2 ) e f f ZZ
2 5'1 - @TI)
( k T , )can finally be written as
and it should be noted that ( kT,) is not proportional to the reciprocal of the sixth power of the distance. By using eq 12 and 13 (for the dynamic averaging regime) or 12 and 16 (for the static averaging regime) together with the distribution of distances, it is possible to calculate the average transfer efficiency and to compare it to the experimental value. The decay of the donor emission following excitation by a 6-pulse of light can also be calculated:
with the normalization conditions
L is the average Bohr radius and K is a constant which is not related to any spectroscopic data; therefore, experimental characterization of the exchange mechanism is difficult.z' In principle, for those conformations corresponding to very short distances (
