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Sub-picosecond transient grating measurements of the resonant energy transfer in cresyl violet solutions
2 December 1994
ELSEVIER
CHEMICAL PHYSICS LETTERS
Chemical Physics Letters 230 ( 1994) 343-350
Sub-picosecond transient grating measurements of the resonant energy transfer in cresyl violet solutions. Evidence for non-diffusive energy transport Siegfried Schneider,
Rudolf Bierl, Michael Seischab
Institut ftir Physikalische und Theoretische Chemie, Universitiit Erlangen-Niirnberg, Egerlandstrasse 3, D-91058 Erlangen, Germany Received 28 July 1994; in final form 29 August 1994
Abstract Three exponentials are needed to fit the decrease in scattered light intensity with increasing delay time between pump and probe pulses. The decay time of the fastest component varies between 270 and 120 fs if the grating fringe spacing is reduced from 15.5 to 7.8 ,um at constant dye concentration (4x lop3 M) or if the dye concentration is increased from 2X 10e3 to 5~ 10e3 M at constant fringe spacing ( 10 pm). The results are discussed as evidence for non-diffusive energy transport.
1. Introduction
Electronic excitation transfer in disordered systems, such as solutions or amorphous solids, is of great importance in a large majority of fields, e.g. in the sensitization and quenching of photophysical and photochemical primary processes, or the collection of light energy in technical or natural antenna systems (photosynthesis). Therefore, the dynamics and spatial mobility of excited-state energy transport in disordered systems has received considerable attention both with respect to experimental determination and theoretical description. Assuming a dipole-dipole interaction and modelling the disorder by a cubic lattice with the lattice period given by the average intermolecular separation, Fijrster [ 1 ] could describe the spatial mobility of the excitation by a diffusion constant D which depended on the reduced concentration C, the so-called Fijrster radius R. and the excited state lifetime 7. in the absence of any quenching process,
D=AC4'3R;z<', A=0.409.
(1)
Later, the full problem of energy transport was treated taking into account the random distribution of intermolecular separations present in disordered systems [ 2-6 1. The general conclusion of these models is that energy transport is non-diffusive in the low concentration or short-time limit while becoming diffusive only at high concentration or at long time. The diffusion constant derived in the more general treatments for long times is identical to Eq. (1) except that the prefactor A adopts different numerical values [7]. Gochanour et al. already pointed out that the picosecond transient grating technique previously applied by Phillion et al. [ 8 ] to study orientational relaxation times and S, singlet lifetimes for rhodamine 6G molecules was ideally suited to measuring the resonant energy transfer and thereby verifying the predictions of the more sophisticated models. GomezJahn et al. [ 71 applied the picosecond transient grating technique to measuring directly the diffusion
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDIOOO9-2614(94)01186-9
constant for the excited state energy transport of highly concentrated solutions of disodium fluorescein in ethanol ( 3 x 10p3-6 x 1Om2M ) . The concentration dependence of the long time diffusion constant D was found to be in relatively good agreement with the theoretical prediction by Gochanour et al. [ 31. At higher concentrations, a significant deviation from linear C4’3 scaling was found and attributed to the finite volume of the interacting molecules which is not included in the theoretical model. Weiner et al. [ 91 used the three-pulse scattering technique to study femtosecond dephasing in dye molecules in liquids and in a polymer host. Cresyl violet in PMMA at 15 K exhibits, like oxazine 720, a partial decay of the scattering efficiency on the timescale of 700 fs. It was assigned to intraband excited-state relaxation since similar rapid excited-state relaxation had been observed for several dyes in liquid solution at room temperature. In this contribution we report the results of three-putse grating experiments with 100 fs pulses, varying both the dye concentration as well as the fringe spacing. By variation of the fringe spacing, one can separate spatial diffusion from pure lifetime effects in the grating relaxation whereas variation in concentration should eventually monitor the effect of concentration on energy transfer (cf. Eq. ( 1) )_
2. Experimental
procedures
Pulses with about 100 fs fwhm and an energy of about 50 pJ at 625 nm are generated by a two-prism rhodamine 6G/DODCI CPM dye laser oscillator [ lo] and ampli~ed in a four-stage excimer laserpumped RhlOl amplifier [ 111 with a gain of 106. In order to avoid the effects of long-term drifts, averaging was done over only live laser shots per data point. In order to improve the signal-to-noise ratio of the measured curves, the scanning of the variable delay line was repeated up to 10 times and only those measurements were included in the averaging procedure, in which the pulse intensity was within a predetermined window. In the three-pulse transient grating technique, a short laser pulse is split into three fractions, two of which act as pump pulses and the third provides the probe pulse. The path of the two pump pulses is arranged such that they intersect simultaneously in the
sample (see Fig. 1a). Due to interference between the two coherent pump pulses, an optical fringe pattern is created in the region where the beams overlap (see Fig. lb}. The spacing between the fringes (A) is determined by the angle between the two pump beams (0~ O1 + 0,) and by their wavelength (A) in air, A=L/2nsin(0/2),
(2)
where n is the index of refraction in the solution. Since the frequency of the exciting light coincides with the absorption band of the molecules under investigation (cresyl violet) there will be a spatial distribution in the number density of excited molecules according to the optical interference pattern. Since the excited molecules contribute differently to the complex index of refraction, the probe pulse sees a diffraction grating which causes it to split into several diffracted beams of different order. If a completely symmetrical arrangement (0, = 0,) were used, several of the diffracted beams of different origin would be superimposed. In order to avoid this superposition. a slightly unsymmetrical a~angement was chosen; this means the angles 0, and 0, differ. With this arrangement about ten different spots can be observed in an image plane behind the sample (Fig. la). First of all, one observes the intersections of the two pump beams, named Pul and Pu2, and the probe beam Pr. The diffracted pulses of order + 1 and - 1 intersect the image plane at equal distance to the left and right of the image of the probe pulse (spots called D(1) (Pr(PulPu2)) and D(2) (Pr(PulPu2)) in Fig. 1). Because both pump pulses intersect simultaneously in the sample, one observes in addition the spots caused by self diffraction of the two pump beams, namely SD( 1) (Pul(PulPu2)) and SD(7) (Pu2 (Pul Pu2 ) ). For zero delay between probe and pump pulses, self diffraction of the probe beam by pump beam 2 gives rise to the spot termed SD( 5 ) (Pr(PrPu2)). By means of a CCD camera, the integrated intensity of each of these spots can easily be measured as a function of delay time. Alternatively, the intensities of the undiffracted probe beam or of the beams scattered in the direction of order + 1 or - 1 are monitored by diodes whose signal is fed into a gated integrator; the output of the latter is transferred to a computer. Cresyl violet perchlorate was purchased as laser dye
S. Schneider et al. / Chemical Physics Letters 230 (1994) 343-350
345
,, SD(l)(Pu2(PulPu2))
(4
I
+
SD(2)(Pul(PulDI)I
Pr -
Fig. 1. (a) Experimental arrangement for three-pulse grating experiment. Pul, Pr2 and Pr denote the pump and probe beams which intersect in the sample with angles 8, and &. In the plane of observation (CCD camera) one observes the undiffracted incoming beams as well as several scattered beams. D (Pl (P2, P3) ) denotes the diffracted part of beam Pl due to the transient grating induced by beams P2 and P3. (b) The calculated variation in number density of excited states for one set of experimental conditions (cell thickness d= 100 wm, crossing angle of pump beams ( 8, + @, ) = 5‘, pump-beam diameter 100 pm)
from Lambda Physik (Germany) further purification.
and used without
3. Results and discussion Fig. 2 displays the output of the CCD camera during an experiment, in which the intensity of the probe
beam was chosen to be higher than usual in order to clearly demonstrate the effect of self diffraction of the probe beam. On one hand, one can see the relative location (pixel number) of the various undiffracted and diffracted beams as discussed above. On the other hand, one can quantify the relative intensities of the various scattered beams as a function of the delay time
346
S. Schneider et al. / Chemical Physics Letters 230 (1994) 343-350
PUl
Pixelnumber Fig. 2. Dependence of intensity (logarithmic scale) of the undiffracted pump and probe beams and the various (self) diffracted beams on delay time between the two pump beams and the probe beam. The recording has been performed by imaging the various beams onto a gatedCCD camera. (Labelling is in accordance with Fig. la.)
between probe and pump pulses. Because of the large differences in signal level, a logarithmic scale is chosen for better presentation. Signals caused by self diffraction of or by the probe beam are easily recognized since they appear only for time delays around zero. Among these, the two strongest features connected with pump beam 2 are found around pixel number 66 (SD(4)) and 48 (SD(5)). Their width in time reflects the (higher-order) cross correlation between the pump and probe pulse and therefore allows an estimate of the laser pulse width. (The corresponding signals due to interaction between probe beam and pump beam 1 are expected around pixel numbers 104 and 38, but in the unsymmetrical arrangement they are too weak to be detected.) Based on its displacement, the feature observed around pixel number 76 (SD (3) ) must originate from the interaction of the diffracted beam D( 1) with pump beam 1, the one around pixel number 42 (SD (6) ) to the interaction of the self-diffracted beam SD( 5) with pump beam 2. The weak features around pixel numbers 28 (SD(7)) and 108 (SD( 1)) are due to self diffraction of the pump beams with each other and therefore independent of the delay time between pump and probe beams. One can also see that the intensity of the probe beam rises sharply when the value of the delay time approaches zero. Furthermore, it is apparent that the
probe intensity is not significantly decreased for positive delay times below 2.5 ps. It may be surprising that the intensities of the pump beams also vary slightly in the region of delay time 0. The reason is that in this experiment the probe beam was chosen to be rather intense and therefore can act as a bleaching pulse with the effect that the transmission for the pump pulses is increased for negative delay times. In actual pump probe experiments the intensity of the probe pulse was lower by a factor of about 100. The intensity of the light scattered from the probe beam into order + and - 1 (signals D ( 1) and D (2) ) rises sharply at delay time 0. The decrease in scattered intensity is sensitively dependent on experimental conditions as will be discussed in detail. In Fig. 3, the dependence of the scattered light intensity on delay time between probe and pump pulses is shown for various experimental conditions (intensity measurements with diodes as described above). In each case the energy of the two pump pulses was about 10 uJ focused into a spot of about 100 pm diameter. The probe pulse energy was about 1 uJ in the case of the scattering experiments and 0.1 uJ for the pump-probe experiment (Fig. 3~). The four traces shown in Fig. 3a are recorded with different crossing angles of the pump beams but with the cresyl violet concentration kept constant (4 x 10W3 M in methanol). The three traces displayed in Fig. 3b are moni-
347
S. Schneider et al. / Chemical Physics Letters 230 (1994) 343-350
Fq.3a
Ilo Zl.U.] 0 0
Flg.3b
Flg.3~
transm. Intens. [ZLU.] Absorption
-10
-05
00
05
10 Delay
15 in
2.0
25
30
relation between the probe and the two pump pulses. It can be clearly seen that the maximum of the scattered light intensity is reached only about 300-400 fs after the start of the signal increase. A similar delay is observed until the maximum bleaching is observed in the pump-probe experiment (Fig. 3~). This is in accordance with the fact that in both techniques the maximum signal is approximately proportional to the time integral over the pulse shape or its autocorrelation, respectively. The common feature of all measurements displayed in Figs. 3a and 3b is that the signal decay is not mono-exponential. In a good approximation, the part after the maximum can be fitted by a tri-exponential function (solid lines), with the lifetime of the second and third component being large compared to the time window of this measurement. In order to get better estimates for the longer decay times, pumpprobe and scattering experiments were performed with lower temporal resolution, but longer time window (Fig. 4). A least-squares fit of the pump-probe curve yields one lifetime of 1.5 ns. The bi-exponential fit of the scattering curve produced one decay time with 1.3 ns and a second decay time with 7 ps (A=0.40). The latter two decay times were fused in the three-exponential fit of the curves displayed in Figs. 3a and 3b. The variable lifetimes of the fastest component and the relative amplitudes produced by the least-squares fit are summarized in Table 1. It is
ps
Fig. 3. Dependence of the variation in intensity of the diffracted beam (a, b) and transient absorption (c) on delay time At between the pump beams and the probe beam. Sample: solution of cresyl violet in methanol. (a) Influence of fringe spacing (c=4x 10m3M). (b) Influence of dye concentration (A= 10.0 pm). (Circles are experimental values, solid lines are tri-exponential fits.) (c) Corresponding pump-probe experiment (c=4~10-’ M) (curves are normalized to equal maximum scattering intensity).
tored with different dye concentrations, but constant crossing angle (fringe spacing). As in pump-probe experiments, in which the pump and probe beams originate from the same source, one observes in the scattering experiments a kind of coherent spike corresponding to a higher-order overlap integral of the excitation and probe pulse shapes [ 8 1. The rising part of the signal is, therefore, governed by the cross cor-
-50
0
SO
150
100
time in
200
250
300
ps
Fig. 4. Comparison between pump-probe and scattering experiments performed with methanolic solutions of cresyl violet (c=4x 10m3M) for longer delay times. Fringe spacing A= 10.0 um.
348 Table 1 Parameters
S. Schneider et al. / Chemical Physics Letters 230 (1994) 343-350
derived by the tri-exponential
least-squares
tit of the time course of scattered
light intensity
displayed
c
A
51
A,
A*
A3
(mol/P)
(pm)
(Ps)
rel.
rel.
rel.
4x 4x 4x 4x
10-3 10-3 10-3 10-3
15.5 12.5 10.0 7.8
0.26 0.14 0.12 0. I 1
0.53 0.61 0.91 0.83
0.28 0.29 0.07 0.14
0.19 0.10 0.02 0.03
2x 10-3 4x 10-X 5x10-3
10.0 10.0 10.0
0.23 0.12 0.08
0.61 0.91 0.91
0.23 0.07 0.09
0.16 0.02 0.00
The decay times of component shown in Fig. 4.
2 and 3 were kept constant
at 7 ps and 1.5 ns. These values originate
evident that there is a clear trend to shorter decay times if the fringe spacing is reduced or the concentration of the dye molecules increased. Furthermore, one can conclude that the relative amplitude of the faster decaying component increases if the fringe spacing is made smaller or the dye concentration raised. Independently of the question of the proper fitting function (decay law), one must infer from the data presented in Fig. 3a that the signal decay is dependent on fringe spacing and must therefore, at least in part, be governed by spatial energy transport. This conclusion is supported by the fact that increasing the dye concentration has basically the same effect as decreasing the fringe spacing. It has been mentioned above that Weiner et al. [ 91 have observed similar curves as shown in Figs. 3a and 3b for cresyl violet in PMMA. Because they did not vary the fringe spacing, they missed the A-dependence and interpreted the fast decay in scattering intensity as being due to intra-band excited-state relaxation of the dye molecules. If such an intra-band relaxation is considered as an intramolecular process it should be independent of dye concentration except at concentrations which are so high that intermolecular interactions contribute to vibrational energy redistribution. On the other hand, it would be difficult to understand why such a basically intramolecular process should depend on the spatial modulation in excited state density. We, therefore, believe that the described effects are related to resonant energy transfer rather than only to intramolecular relaxation processes.
in Fig. 3
from the tit of the scattering
curve
If diffusive energy transfer were the only process leading to the disappearance of the transient grating, then one could expect the diffracted probe signal, s(t), to decay mono-exponentially, since its intensity is proportional to the square of the difference in the concentration of excited molecules at the grating peaks and nulls [ 12 ] S(t)=C[N*(O,
t)-N*(A/2,
t)]‘,
(3a)
or ,S(t)=c’exp(-22t[l/7,+(2x/n)‘D]j,
(3b)
where N*(x, t) denotes the number density of excited molecules along the grating axis at time t. The functional dependence of S( t) on A and D expressed in Eq. (3b) can easily be verified by setting up the differential equation for the change in number density of excited molecules. To this end, the one-dimensional space along the grating axis (Fig. 1) is divided in compartments of length AX, each holding N, molecules. If energy transfer is allowed to unexcited molecules in the neighbouring compartment only with a time-independent rate constant k,, then the change in number of excited molecules per unit time is dN*(i,
1, t)
t)/dt=k,N,[N*(i-
+N*(i+l.t)-2N*(i,t)]-l/z,N*(i,t).
(4)
The initial spatial distribution of excited molecules varies sinusoidally along the grating axis N*(i,O)=f[l+cos(2xAX/n)]N*(O,O).
(5)
A numerical
that
solution
of Eq. (4) demonstrates
S. Schneider et al. / Chemical Physics Letters 230 (1994) 343-350
the sinusoidal variation in number of excited molecules is maintained for all times (see Fig. 5 ) For the calculation of the signal decay S(t) it is, therefore, sufficient to calculate the decay of N*(O, t). Using a power series expansion for small values of AX, one gets the simplified relation dN*(O, t)/dt= xN*(O,
t)
- [k,N,,AX*(27~//1)*+
I/Q] (6)
From Eq. (6) it is obvious that, in the case of a timeindependent rate constant k, for energy transfer, the decay of the grating (efficiency) must be monoexponential. By comparison with the EinsteinSmoluchowski equation, D=d2/2r,
(7)
one can assign 2kqNo to the hopping rate r-’ and AX to the average transfer distance d. Deviation from a mono-exponential decay law for the scattered light intensity is, therefore, evidence for a time-dependent quenching constant k, or energy diffusion constant D(t). One approximation underlying Forster’s theory [ 1 ] is that of a regular lattice of energy absorbers. It has been pointed out in the publications cited above [ 26] that in solutions, in which the molecules have a statistical distribution of intermolecular distances, a distribution of transfer rates and pathways exists. As a consequence, the generalized diffusion coefficient D becomes time-dependent with a sharp increase for very short time (t&O). The time, for which disper-
12000
I
Fig. 5. Decay of grating modulation as calculated by numerical integration of the model characterized by Eq. (2 ) (excited-state lifetime r, is assumed to be infinity). Note that there is no deviation from the sinusoidal form of modulation.
349
sive energy transport converges into diffusive transport depends on the power of the multipolar interaction, the dimensionality of the transport and on the concentration of the dye molecules. In all theoretical treatments [ 2-61, an increase of the concentration pushes the diffusive limit to considerably shorter times. The actual values vary somewhat from model to model, as does the increase of D ( t ) versus D( ca ) . In no case the ratio D(t+O)/D(m) exceeds a value of 10. Gomez-Jahn et al. [ 71 found x lop3 M) the contribution of the diffusive energy transport to the signal decay time was small (i.e. (2rr/ A)2D<< l/r, with rO= 5 ns). If one assumes that diffusive energy transport is of the same order of magnitude in fluorescein and cresyl violet solutions, then diffusive energy transport should modify the disappearance of the grating due to electronic relaxation to the ground state (Q z 1.5 ns) only to a small extent. This expectation is fulfilled for the long-lived component in the scattering experiment (Fig. 4), where the decay time is found to be slightly shorter than in the corresponding pump-probe experiment. Because of the possibly larger error in the decay-time determination, a more elaborate discussion is not meaningful. If the above mentioned factor of 10 also holds for the enhancement ofD( t) versus D( co), then grating decay times of less than 1 ns cannot be explained within the framework of the above models. Another important assumption used by Fiirster and later by other authors is that of a thermally relaxed ensemble of energy donor molecules. Vibrational relaxation (energy transfer to the solvent) in large molecules was shown to occur on timescales between 1 and 100 ps (the fast initial recovery with low amplitude observed in the pump-probe experiment (Fig. 3c) could actually be related to this process). Intramolecular redistribution of the vibronic excess energy was shown to occur on the timescale of several hundred femtosecond. Both processes should introduce an additional time-dependent factor into the calculation of energy transfer rates. In a recent publication, Scholes and Ghiggino [ 131
[ 141 mechanism for singlet-singlet energy transfer at close separation. If orbital overlap effects
350
S. Schneider et al. /Chemical
are significant, then they can dramatically increase the rate for energy transfer as calculated on the basis of the Coulombic interaction. In view of the complexity of the problem, a more elaborate discussion of the experimental data seems impossible for us at this present stage. The tri-exponential fit of the fast decay of the scattered signal intensity is inadequate because one must expect a much more complicated decay law. The tit parameters listed in Table 1 are therefore without direct physical meaning. their systematic variation with fringe spacing and concentration are good evidence for the fact that the decay of the grating signal is governed not only by intramolecular relaxation processes but also by intermolecular energy transfer.
Acknowledgement Financial support by Deutsche Forschungsgemeinschaft and Fonds der Chemie is gratefully acknowledged.
Physics Letters 230 (1994) 343-350
References
[I ] Th. Fiirster, Ann. Physik 2 (1948) 55. [ 21S.W. Haan and R. Zwanzig, J. Chem. Phys. 68 ( 1978) 1879. [ 31 C.R. Gochanour, H.C. Andersen and M.D. Fayer, J. Chem. Phys. 70 (1979) 4254. [4] K. Godzik and J. Jortner, J. Chem. Phys. 72 (1980) 4471. [ 51 A. Blumen, J. Klafter and R. Silbey, J. Chem. Phys. 72 (1980) 5320. [6] J. Nieuwoudt and S. Mukamel, Phys. Rev. B 30 ( 1984) 4426. [ 71 L. Gomez-Jahn, J. Kasinski and R.J.D. Miller, Chem. Phys. Letters 125 (1986) 500. [S] D.W. Phillion, D.J. Kuizenga and A.E. Siegman, Appl. Phys. Letters 27 (1975) 8.5. [9] A.M. Weiner, S. De Silvestri and E.P. Ippen, J. Opt. Sot. Am,B2(1985)654. [ lo] W. Dietel, E. Dope1 and W. Rudolph, Forschungsergebnisse FSU Jena, No. N/88/14 (1988). [ 111 C. Rolland and C. Korkum, Opt. Commun. 59 ( 1986) 64. [ 121 T.S. Rose, R. Righini and M.D. Fayer, Chem. Phys. Letters 106 (1984) 13, and references therein. [ 131 G.D. Scholes and K.P. Ghiggino, J. Phys. Chem. 98 ( 1994) 4580. [ 141 D.L. Dexter, J. Chem. Phys. 21 (1953) 836.
ELSEVIER
CHEMICAL PHYSICS LETTERS
Chemical Physics Letters 230 ( 1994) 343-350
Sub-picosecond transient grating measurements of the resonant energy transfer in cresyl violet solutions. Evidence for non-diffusive energy transport Siegfried Schneider,
Rudolf Bierl, Michael Seischab
Institut ftir Physikalische und Theoretische Chemie, Universitiit Erlangen-Niirnberg, Egerlandstrasse 3, D-91058 Erlangen, Germany Received 28 July 1994; in final form 29 August 1994
Abstract Three exponentials are needed to fit the decrease in scattered light intensity with increasing delay time between pump and probe pulses. The decay time of the fastest component varies between 270 and 120 fs if the grating fringe spacing is reduced from 15.5 to 7.8 ,um at constant dye concentration (4x lop3 M) or if the dye concentration is increased from 2X 10e3 to 5~ 10e3 M at constant fringe spacing ( 10 pm). The results are discussed as evidence for non-diffusive energy transport.
1. Introduction
Electronic excitation transfer in disordered systems, such as solutions or amorphous solids, is of great importance in a large majority of fields, e.g. in the sensitization and quenching of photophysical and photochemical primary processes, or the collection of light energy in technical or natural antenna systems (photosynthesis). Therefore, the dynamics and spatial mobility of excited-state energy transport in disordered systems has received considerable attention both with respect to experimental determination and theoretical description. Assuming a dipole-dipole interaction and modelling the disorder by a cubic lattice with the lattice period given by the average intermolecular separation, Fijrster [ 1 ] could describe the spatial mobility of the excitation by a diffusion constant D which depended on the reduced concentration C, the so-called Fijrster radius R. and the excited state lifetime 7. in the absence of any quenching process,
D=AC4'3R;z<', A=0.409.
(1)
Later, the full problem of energy transport was treated taking into account the random distribution of intermolecular separations present in disordered systems [ 2-6 1. The general conclusion of these models is that energy transport is non-diffusive in the low concentration or short-time limit while becoming diffusive only at high concentration or at long time. The diffusion constant derived in the more general treatments for long times is identical to Eq. (1) except that the prefactor A adopts different numerical values [7]. Gochanour et al. already pointed out that the picosecond transient grating technique previously applied by Phillion et al. [ 8 ] to study orientational relaxation times and S, singlet lifetimes for rhodamine 6G molecules was ideally suited to measuring the resonant energy transfer and thereby verifying the predictions of the more sophisticated models. GomezJahn et al. [ 71 applied the picosecond transient grating technique to measuring directly the diffusion
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDIOOO9-2614(94)01186-9
constant for the excited state energy transport of highly concentrated solutions of disodium fluorescein in ethanol ( 3 x 10p3-6 x 1Om2M ) . The concentration dependence of the long time diffusion constant D was found to be in relatively good agreement with the theoretical prediction by Gochanour et al. [ 31. At higher concentrations, a significant deviation from linear C4’3 scaling was found and attributed to the finite volume of the interacting molecules which is not included in the theoretical model. Weiner et al. [ 91 used the three-pulse scattering technique to study femtosecond dephasing in dye molecules in liquids and in a polymer host. Cresyl violet in PMMA at 15 K exhibits, like oxazine 720, a partial decay of the scattering efficiency on the timescale of 700 fs. It was assigned to intraband excited-state relaxation since similar rapid excited-state relaxation had been observed for several dyes in liquid solution at room temperature. In this contribution we report the results of three-putse grating experiments with 100 fs pulses, varying both the dye concentration as well as the fringe spacing. By variation of the fringe spacing, one can separate spatial diffusion from pure lifetime effects in the grating relaxation whereas variation in concentration should eventually monitor the effect of concentration on energy transfer (cf. Eq. ( 1) )_
2. Experimental
procedures
Pulses with about 100 fs fwhm and an energy of about 50 pJ at 625 nm are generated by a two-prism rhodamine 6G/DODCI CPM dye laser oscillator [ lo] and ampli~ed in a four-stage excimer laserpumped RhlOl amplifier [ 111 with a gain of 106. In order to avoid the effects of long-term drifts, averaging was done over only live laser shots per data point. In order to improve the signal-to-noise ratio of the measured curves, the scanning of the variable delay line was repeated up to 10 times and only those measurements were included in the averaging procedure, in which the pulse intensity was within a predetermined window. In the three-pulse transient grating technique, a short laser pulse is split into three fractions, two of which act as pump pulses and the third provides the probe pulse. The path of the two pump pulses is arranged such that they intersect simultaneously in the
sample (see Fig. 1a). Due to interference between the two coherent pump pulses, an optical fringe pattern is created in the region where the beams overlap (see Fig. lb}. The spacing between the fringes (A) is determined by the angle between the two pump beams (0~ O1 + 0,) and by their wavelength (A) in air, A=L/2nsin(0/2),
(2)
where n is the index of refraction in the solution. Since the frequency of the exciting light coincides with the absorption band of the molecules under investigation (cresyl violet) there will be a spatial distribution in the number density of excited molecules according to the optical interference pattern. Since the excited molecules contribute differently to the complex index of refraction, the probe pulse sees a diffraction grating which causes it to split into several diffracted beams of different order. If a completely symmetrical arrangement (0, = 0,) were used, several of the diffracted beams of different origin would be superimposed. In order to avoid this superposition. a slightly unsymmetrical a~angement was chosen; this means the angles 0, and 0, differ. With this arrangement about ten different spots can be observed in an image plane behind the sample (Fig. la). First of all, one observes the intersections of the two pump beams, named Pul and Pu2, and the probe beam Pr. The diffracted pulses of order + 1 and - 1 intersect the image plane at equal distance to the left and right of the image of the probe pulse (spots called D(1) (Pr(PulPu2)) and D(2) (Pr(PulPu2)) in Fig. 1). Because both pump pulses intersect simultaneously in the sample, one observes in addition the spots caused by self diffraction of the two pump beams, namely SD( 1) (Pul(PulPu2)) and SD(7) (Pu2 (Pul Pu2 ) ). For zero delay between probe and pump pulses, self diffraction of the probe beam by pump beam 2 gives rise to the spot termed SD( 5 ) (Pr(PrPu2)). By means of a CCD camera, the integrated intensity of each of these spots can easily be measured as a function of delay time. Alternatively, the intensities of the undiffracted probe beam or of the beams scattered in the direction of order + 1 or - 1 are monitored by diodes whose signal is fed into a gated integrator; the output of the latter is transferred to a computer. Cresyl violet perchlorate was purchased as laser dye
S. Schneider et al. / Chemical Physics Letters 230 (1994) 343-350
345
,, SD(l)(Pu2(PulPu2))
(4
I
+
SD(2)(Pul(PulDI)I
Pr -
Fig. 1. (a) Experimental arrangement for three-pulse grating experiment. Pul, Pr2 and Pr denote the pump and probe beams which intersect in the sample with angles 8, and &. In the plane of observation (CCD camera) one observes the undiffracted incoming beams as well as several scattered beams. D (Pl (P2, P3) ) denotes the diffracted part of beam Pl due to the transient grating induced by beams P2 and P3. (b) The calculated variation in number density of excited states for one set of experimental conditions (cell thickness d= 100 wm, crossing angle of pump beams ( 8, + @, ) = 5‘, pump-beam diameter 100 pm)
from Lambda Physik (Germany) further purification.
and used without
3. Results and discussion Fig. 2 displays the output of the CCD camera during an experiment, in which the intensity of the probe
beam was chosen to be higher than usual in order to clearly demonstrate the effect of self diffraction of the probe beam. On one hand, one can see the relative location (pixel number) of the various undiffracted and diffracted beams as discussed above. On the other hand, one can quantify the relative intensities of the various scattered beams as a function of the delay time
346
S. Schneider et al. / Chemical Physics Letters 230 (1994) 343-350
PUl
Pixelnumber Fig. 2. Dependence of intensity (logarithmic scale) of the undiffracted pump and probe beams and the various (self) diffracted beams on delay time between the two pump beams and the probe beam. The recording has been performed by imaging the various beams onto a gatedCCD camera. (Labelling is in accordance with Fig. la.)
between probe and pump pulses. Because of the large differences in signal level, a logarithmic scale is chosen for better presentation. Signals caused by self diffraction of or by the probe beam are easily recognized since they appear only for time delays around zero. Among these, the two strongest features connected with pump beam 2 are found around pixel number 66 (SD(4)) and 48 (SD(5)). Their width in time reflects the (higher-order) cross correlation between the pump and probe pulse and therefore allows an estimate of the laser pulse width. (The corresponding signals due to interaction between probe beam and pump beam 1 are expected around pixel numbers 104 and 38, but in the unsymmetrical arrangement they are too weak to be detected.) Based on its displacement, the feature observed around pixel number 76 (SD (3) ) must originate from the interaction of the diffracted beam D( 1) with pump beam 1, the one around pixel number 42 (SD (6) ) to the interaction of the self-diffracted beam SD( 5) with pump beam 2. The weak features around pixel numbers 28 (SD(7)) and 108 (SD( 1)) are due to self diffraction of the pump beams with each other and therefore independent of the delay time between pump and probe beams. One can also see that the intensity of the probe beam rises sharply when the value of the delay time approaches zero. Furthermore, it is apparent that the
probe intensity is not significantly decreased for positive delay times below 2.5 ps. It may be surprising that the intensities of the pump beams also vary slightly in the region of delay time 0. The reason is that in this experiment the probe beam was chosen to be rather intense and therefore can act as a bleaching pulse with the effect that the transmission for the pump pulses is increased for negative delay times. In actual pump probe experiments the intensity of the probe pulse was lower by a factor of about 100. The intensity of the light scattered from the probe beam into order + and - 1 (signals D ( 1) and D (2) ) rises sharply at delay time 0. The decrease in scattered intensity is sensitively dependent on experimental conditions as will be discussed in detail. In Fig. 3, the dependence of the scattered light intensity on delay time between probe and pump pulses is shown for various experimental conditions (intensity measurements with diodes as described above). In each case the energy of the two pump pulses was about 10 uJ focused into a spot of about 100 pm diameter. The probe pulse energy was about 1 uJ in the case of the scattering experiments and 0.1 uJ for the pump-probe experiment (Fig. 3~). The four traces shown in Fig. 3a are recorded with different crossing angles of the pump beams but with the cresyl violet concentration kept constant (4 x 10W3 M in methanol). The three traces displayed in Fig. 3b are moni-
347
S. Schneider et al. / Chemical Physics Letters 230 (1994) 343-350
Fq.3a
Ilo Zl.U.] 0 0
Flg.3b
Flg.3~
transm. Intens. [ZLU.] Absorption
-10
-05
00
05
10 Delay
15 in
2.0
25
30
relation between the probe and the two pump pulses. It can be clearly seen that the maximum of the scattered light intensity is reached only about 300-400 fs after the start of the signal increase. A similar delay is observed until the maximum bleaching is observed in the pump-probe experiment (Fig. 3~). This is in accordance with the fact that in both techniques the maximum signal is approximately proportional to the time integral over the pulse shape or its autocorrelation, respectively. The common feature of all measurements displayed in Figs. 3a and 3b is that the signal decay is not mono-exponential. In a good approximation, the part after the maximum can be fitted by a tri-exponential function (solid lines), with the lifetime of the second and third component being large compared to the time window of this measurement. In order to get better estimates for the longer decay times, pumpprobe and scattering experiments were performed with lower temporal resolution, but longer time window (Fig. 4). A least-squares fit of the pump-probe curve yields one lifetime of 1.5 ns. The bi-exponential fit of the scattering curve produced one decay time with 1.3 ns and a second decay time with 7 ps (A=0.40). The latter two decay times were fused in the three-exponential fit of the curves displayed in Figs. 3a and 3b. The variable lifetimes of the fastest component and the relative amplitudes produced by the least-squares fit are summarized in Table 1. It is
ps
Fig. 3. Dependence of the variation in intensity of the diffracted beam (a, b) and transient absorption (c) on delay time At between the pump beams and the probe beam. Sample: solution of cresyl violet in methanol. (a) Influence of fringe spacing (c=4x 10m3M). (b) Influence of dye concentration (A= 10.0 pm). (Circles are experimental values, solid lines are tri-exponential fits.) (c) Corresponding pump-probe experiment (c=4~10-’ M) (curves are normalized to equal maximum scattering intensity).
tored with different dye concentrations, but constant crossing angle (fringe spacing). As in pump-probe experiments, in which the pump and probe beams originate from the same source, one observes in the scattering experiments a kind of coherent spike corresponding to a higher-order overlap integral of the excitation and probe pulse shapes [ 8 1. The rising part of the signal is, therefore, governed by the cross cor-
-50
0
SO
150
100
time in
200
250
300
ps
Fig. 4. Comparison between pump-probe and scattering experiments performed with methanolic solutions of cresyl violet (c=4x 10m3M) for longer delay times. Fringe spacing A= 10.0 um.
348 Table 1 Parameters
S. Schneider et al. / Chemical Physics Letters 230 (1994) 343-350
derived by the tri-exponential
least-squares
tit of the time course of scattered
light intensity
displayed
c
A
51
A,
A*
A3
(mol/P)
(pm)
(Ps)
rel.
rel.
rel.
4x 4x 4x 4x
10-3 10-3 10-3 10-3
15.5 12.5 10.0 7.8
0.26 0.14 0.12 0. I 1
0.53 0.61 0.91 0.83
0.28 0.29 0.07 0.14
0.19 0.10 0.02 0.03
2x 10-3 4x 10-X 5x10-3
10.0 10.0 10.0
0.23 0.12 0.08
0.61 0.91 0.91
0.23 0.07 0.09
0.16 0.02 0.00
The decay times of component shown in Fig. 4.
2 and 3 were kept constant
at 7 ps and 1.5 ns. These values originate
evident that there is a clear trend to shorter decay times if the fringe spacing is reduced or the concentration of the dye molecules increased. Furthermore, one can conclude that the relative amplitude of the faster decaying component increases if the fringe spacing is made smaller or the dye concentration raised. Independently of the question of the proper fitting function (decay law), one must infer from the data presented in Fig. 3a that the signal decay is dependent on fringe spacing and must therefore, at least in part, be governed by spatial energy transport. This conclusion is supported by the fact that increasing the dye concentration has basically the same effect as decreasing the fringe spacing. It has been mentioned above that Weiner et al. [ 91 have observed similar curves as shown in Figs. 3a and 3b for cresyl violet in PMMA. Because they did not vary the fringe spacing, they missed the A-dependence and interpreted the fast decay in scattering intensity as being due to intra-band excited-state relaxation of the dye molecules. If such an intra-band relaxation is considered as an intramolecular process it should be independent of dye concentration except at concentrations which are so high that intermolecular interactions contribute to vibrational energy redistribution. On the other hand, it would be difficult to understand why such a basically intramolecular process should depend on the spatial modulation in excited state density. We, therefore, believe that the described effects are related to resonant energy transfer rather than only to intramolecular relaxation processes.
in Fig. 3
from the tit of the scattering
curve
If diffusive energy transfer were the only process leading to the disappearance of the transient grating, then one could expect the diffracted probe signal, s(t), to decay mono-exponentially, since its intensity is proportional to the square of the difference in the concentration of excited molecules at the grating peaks and nulls [ 12 ] S(t)=C[N*(O,
t)-N*(A/2,
t)]‘,
(3a)
or ,S(t)=c’exp(-22t[l/7,+(2x/n)‘D]j,
(3b)
where N*(x, t) denotes the number density of excited molecules along the grating axis at time t. The functional dependence of S( t) on A and D expressed in Eq. (3b) can easily be verified by setting up the differential equation for the change in number density of excited molecules. To this end, the one-dimensional space along the grating axis (Fig. 1) is divided in compartments of length AX, each holding N, molecules. If energy transfer is allowed to unexcited molecules in the neighbouring compartment only with a time-independent rate constant k,, then the change in number of excited molecules per unit time is dN*(i,
1, t)
t)/dt=k,N,[N*(i-
+N*(i+l.t)-2N*(i,t)]-l/z,N*(i,t).
(4)
The initial spatial distribution of excited molecules varies sinusoidally along the grating axis N*(i,O)=f[l+cos(2xAX/n)]N*(O,O).
(5)
A numerical
that
solution
of Eq. (4) demonstrates
S. Schneider et al. / Chemical Physics Letters 230 (1994) 343-350
the sinusoidal variation in number of excited molecules is maintained for all times (see Fig. 5 ) For the calculation of the signal decay S(t) it is, therefore, sufficient to calculate the decay of N*(O, t). Using a power series expansion for small values of AX, one gets the simplified relation dN*(O, t)/dt= xN*(O,
t)
- [k,N,,AX*(27~//1)*+
I/Q] (6)
From Eq. (6) it is obvious that, in the case of a timeindependent rate constant k, for energy transfer, the decay of the grating (efficiency) must be monoexponential. By comparison with the EinsteinSmoluchowski equation, D=d2/2r,
(7)
one can assign 2kqNo to the hopping rate r-’ and AX to the average transfer distance d. Deviation from a mono-exponential decay law for the scattered light intensity is, therefore, evidence for a time-dependent quenching constant k, or energy diffusion constant D(t). One approximation underlying Forster’s theory [ 1 ] is that of a regular lattice of energy absorbers. It has been pointed out in the publications cited above [ 26] that in solutions, in which the molecules have a statistical distribution of intermolecular distances, a distribution of transfer rates and pathways exists. As a consequence, the generalized diffusion coefficient D becomes time-dependent with a sharp increase for very short time (t&O). The time, for which disper-
12000
I
Fig. 5. Decay of grating modulation as calculated by numerical integration of the model characterized by Eq. (2 ) (excited-state lifetime r, is assumed to be infinity). Note that there is no deviation from the sinusoidal form of modulation.
349
sive energy transport converges into diffusive transport depends on the power of the multipolar interaction, the dimensionality of the transport and on the concentration of the dye molecules. In all theoretical treatments [ 2-61, an increase of the concentration pushes the diffusive limit to considerably shorter times. The actual values vary somewhat from model to model, as does the increase of D ( t ) versus D( ca ) . In no case the ratio D(t+O)/D(m) exceeds a value of 10. Gomez-Jahn et al. [ 71 found x lop3 M) the contribution of the diffusive energy transport to the signal decay time was small (i.e. (2rr/ A)2D<< l/r, with rO= 5 ns). If one assumes that diffusive energy transport is of the same order of magnitude in fluorescein and cresyl violet solutions, then diffusive energy transport should modify the disappearance of the grating due to electronic relaxation to the ground state (Q z 1.5 ns) only to a small extent. This expectation is fulfilled for the long-lived component in the scattering experiment (Fig. 4), where the decay time is found to be slightly shorter than in the corresponding pump-probe experiment. Because of the possibly larger error in the decay-time determination, a more elaborate discussion is not meaningful. If the above mentioned factor of 10 also holds for the enhancement ofD( t) versus D( co), then grating decay times of less than 1 ns cannot be explained within the framework of the above models. Another important assumption used by Fiirster and later by other authors is that of a thermally relaxed ensemble of energy donor molecules. Vibrational relaxation (energy transfer to the solvent) in large molecules was shown to occur on timescales between 1 and 100 ps (the fast initial recovery with low amplitude observed in the pump-probe experiment (Fig. 3c) could actually be related to this process). Intramolecular redistribution of the vibronic excess energy was shown to occur on the timescale of several hundred femtosecond. Both processes should introduce an additional time-dependent factor into the calculation of energy transfer rates. In a recent publication, Scholes and Ghiggino [ 131
[ 141 mechanism for singlet-singlet energy transfer at close separation. If orbital overlap effects
350
S. Schneider et al. /Chemical
are significant, then they can dramatically increase the rate for energy transfer as calculated on the basis of the Coulombic interaction. In view of the complexity of the problem, a more elaborate discussion of the experimental data seems impossible for us at this present stage. The tri-exponential fit of the fast decay of the scattered signal intensity is inadequate because one must expect a much more complicated decay law. The tit parameters listed in Table 1 are therefore without direct physical meaning. their systematic variation with fringe spacing and concentration are good evidence for the fact that the decay of the grating signal is governed not only by intramolecular relaxation processes but also by intermolecular energy transfer.
Acknowledgement Financial support by Deutsche Forschungsgemeinschaft and Fonds der Chemie is gratefully acknowledged.
Physics Letters 230 (1994) 343-350
References
[I ] Th. Fiirster, Ann. Physik 2 (1948) 55. [ 21S.W. Haan and R. Zwanzig, J. Chem. Phys. 68 ( 1978) 1879. [ 31 C.R. Gochanour, H.C. Andersen and M.D. Fayer, J. Chem. Phys. 70 (1979) 4254. [4] K. Godzik and J. Jortner, J. Chem. Phys. 72 (1980) 4471. [ 51 A. Blumen, J. Klafter and R. Silbey, J. Chem. Phys. 72 (1980) 5320. [6] J. Nieuwoudt and S. Mukamel, Phys. Rev. B 30 ( 1984) 4426. [ 71 L. Gomez-Jahn, J. Kasinski and R.J.D. Miller, Chem. Phys. Letters 125 (1986) 500. [S] D.W. Phillion, D.J. Kuizenga and A.E. Siegman, Appl. Phys. Letters 27 (1975) 8.5. [9] A.M. Weiner, S. De Silvestri and E.P. Ippen, J. Opt. Sot. Am,B2(1985)654. [ lo] W. Dietel, E. Dope1 and W. Rudolph, Forschungsergebnisse FSU Jena, No. N/88/14 (1988). [ 111 C. Rolland and C. Korkum, Opt. Commun. 59 ( 1986) 64. [ 121 T.S. Rose, R. Righini and M.D. Fayer, Chem. Phys. Letters 106 (1984) 13, and references therein. [ 131 G.D. Scholes and K.P. Ghiggino, J. Phys. Chem. 98 ( 1994) 4580. [ 141 D.L. Dexter, J. Chem. Phys. 21 (1953) 836.