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Picosecond study of energy transfer. Deviations from Förster theory — evidence for an inhomogeneous spatial distribution of molecules
229
Chemical Physics 102 (1986) 229-240 worth-Holland, Amsterdam
PICOSECOND STUDY OF ENERGY TRANSFER. DEVIA~ONS FROM FORSTER THEORY - EVIDENCE FOR AN INHOMOGENEOUS SPATIAL DIS~RU~ON
OF MOLECULES
Michael KASCHKE and Klaus VOGLER Sektion Physik, Friedrich - Schiller-Universitiit Jena, Max - Wien - Platz I, 6900 Jena, GDR Received 14 June 1985
We have investigated long-range dipole-dipole energy transfer between several laser dyes in solution. Time-resolved absorption and fluorescence experiments on a picosecond excite-and-probe-beam spectrometer and a single-photon counting equipment have been carried out to measure the excitation depletion of the donor (Rh 6G) and the rise of the excitation of the acceptors (DOTC, cresylviolet, oxazine) as a function of the acceptor concentration. An unexpected very-rapid and effective energy transfer has been been found, even at very low acceptor concentrations. F&ster radii R, have been determined for each dye ~mbination. At high acceptor con~n~ations these R, values agree well with those calculated from spectral data, but rather large deviations have been found at low acceptor concentrations. The observed deviations from the common Fijrster law are tentatively explained in terms of a modified Forster model which takes into account an inhomogeneous distribution of molecules in solution.
1. Inaction
Intermolecular energy transfer processes play a major role in various fields in nature. In photosynthesis, for instance, optical energy is transferred via radiationless transfer mechanisms to the reaction center [l], whereby a simultaneous transfer of the optical energy from one spectral region to the other takes place. In addition, the technical use of energy transfer mechanisms is gaining an increasing importance in a number of industrial and scientific applications. A few examples are exciton migration in solids [2] or the sensibilization of photophysi~ f3] and phot~he~c~ [4] processes. Dye lasers are widely used as spectroscopic tools for photochemical and photophysical investigations. Energy transfer in laser dye mixtures can extend the lasing range, increase the output efficiency and can also have considerable influence on the temporal characteristics of the output pulses of energy transfer dye lasers [5,6]. Thus, in order to understand the processes proceeding in this case of application, as well as to gain principal knowledge on energy transfer
processes in condensed matter the investigation of energy transfer mechanisms between dye molecules in solution is of substantial interest. There are several possible mechanisms for energy transfer between dye molecules in solution: radiationless dipole-dipole (or higher multipole) interaction [7], diffusion-controlled exchange interaction between nei~bo~ng molecules [g], and radiative transfer [9]. Two or more of these mechanisms can also act simultaneously [lO,ll]: the predominant mechanism is determined mainly by the intermolecular distances between the solved molecules. An exact distinction between these mechanisms is only possible expe~mentally by means of time-resolved spectroscopic methods. Up to now, most of the experimental investigations of energy transfer between dyes in solution have been timeresolved fluorescence studies, using either singleshot 1121 or synchroscan [13] streak cameras. In order to check the differences between improved theoretical models experimentally a high time resolution and accuracy are necessary, since the mechanisms differ mainly in the time domain shortly after excitation. The picosecond excite-and-probe-
0301-0104/g6/$03.50 0 Elsevier Science Publishers B.V. (North-Holl~d Physics ~blishing Division)
M. Kaschke, K. VogIer / Picosecond study of energy transfer
230
beam absorption spectroscopy fulfills these requirements and offers the advantage of being able to observe simultaneously the kinetics of both the donor and the acceptor. The purpose of our study was to obtain detailed information on the range of validity of different energy transfer models by means of time-resolved absorption and fluorescence spectroscopy. Particular interest was paid to the question of the validity of the common Forster model [7], since experimental verifications [14] as well as deviations [4,13] from the predicted law have been reported recently. In our experiments we used various dye mixtures, rhodamine 6G (Rh 6G) as donor and cresylviolet (CV), DOTC iodide (DOTC), and oxazine 725 (Ox) as acceptors. The concentration ratio of donor and acceptor covered three orders of magnitude. The experiments revealed an unexpectedly rapid and effective transfer of excitation energy to the acceptor molecules, even at low acceptor concentration. The experimental findings were tentatively evaluated in terms of a modified Forster model, which takes into account an inhomogeneous spatial distribution of the dye molecules in solution.
2. Experimental arrangement Most of our experiments have been performed on a picosecond probe-beam spectrometer, driven by an Nd-phosphate-glass laser, and a single-pho-
ps-N&phosphate
Fig. 1. Experimental
glass
laser
prolc-beam
arrangement of the picosecond
ton counting equipment. A streak camera and a picosecond Nd-YAG absorption spectrometer, which consists of an optical parametric amplifier, have been used in additional measurements. 2.1. Excite-and-probe-beam
spectrometer
In fig. 1 the picosecond Nd-phosphate-glass apparatus is shown [15]. A Pockels cell [2] selects a single pulse (or = 5 ps) from the leading part of the pulse train emitted by a passively mode-locked Nd-phosphate-glass oscillator, 1. After a two-stage amplifier, 3 the main part (90%) of the amplified single pulse (h = 1054 nm, E = 20 mJ) is frequency converted to the second harmonic (X = 527 nm, E = 5 mJ) or alternatively to other frequencies, 6: third harmonic, Raman shifted lines. This light pulse serves as excitation beam. The remaining part of the split fundamental wave pulse, 4, after a further amplification, 9, generates the picosecond broad-band continuum in a 10 cm D,O cuvette, 10. The picosecond continuum covers a wavelength range of = 300-900 nm and is used as interrogation beam after splitting into a probe and reference beam, 11, which probe the transmission of the excited and unexcited sample volume, 12. Both the light pulses are focused onto the entrance slit of a spectrograph, 13, and detected by a vidicon, 14, of an optical multichannel analyzer, 15, PAR OMA II. The spectral resolution of this arrangement amounts to 2 nm in an overall spectral extension of 250 nm. The excitation beam can
spectrometer
Nd-phosphate-glass
laser and the excite-and-probe-beam
spectrometer.
M. Kaschke, K, Vogier / Picosecond study of energy iransfer
properly be delayed, 7 (maximum to = 900 ps) and altered in polarization with respect to the interrogation pulse 16. The automatic registration by the OMA contains a special procedure for the discrimination of the excitation pulse energy, 8, and pulse duration. The time resolution of this equipment is better than 6 ps. A registration accuracy of 2% in transmission changes within the whole observable spectral region of 250 nm and a dynamic range of more than 200 is achieved. Thus we have been able to observe the donor population depletion and the acceptor excitation, simultaneously and with high accuracy, by changes in the optical density of the sample. Care has been taken to prevent a depletion of excited donors by stimulated emission. 2.2. Single-photon counting equipment (SPC) This apparatus consists of an acousto-optically mode-locked Ar-ion laser as excitation source (X = 514 run, rp = 80 ps) and an usual fluorescence detection unit: monochromator, photomultiplier, and the electronic sampling and control unit (SPC 100, ZOS). Excitation as well as fluorescence light can be attenuated by suitable filters. Alternatively a synchronously pumped picosecond dye laser ( rp = 2 ps, X = 560-620 nm) can be used for excitation [16]. The time resolution of the SPC 100 unit amounts to = 100 ps (without deconvolution), the dynamic range within the recorded time interval of 8 ns (given by the repetition rate of the laser) is better than 104. Samples of small thickness (100-200 pm) and front face excitation and detection had to be used in order to prevent reabsorption effects at high concentrations of solved molecules. 2.3. Picosecond Nd- YAG laser spectrometer This equipment is composed of a passively mode-locked Nd-YAG laser, a single-pulse selection unit, an amplifier stage, and an SHG driven optical parametric generator (OPA), as well as an excite-and-probe-beam interaction part [17]. An SHG pulse (X = 532 nm, rp = 30 ps, E = 5 mJ) was used for exciting the samples (cuvette thickness: 1 mm). Whereas in the Nd-phosphate-glass
231
laser spectrometer a broad-band interrogation of the sample by means of the picosecond continuum was achieved, here monochromatic interrogation and reference beams probed the transmission of the excited and unexcited sample volume. The probe beams were obtained from the frequencydoubled OPA signal wave (X = 560 nm or X = 660 nm). The laser pulse repetition rate was 1 Hz. Accuracies of smaller than 1% in transmission changes can be achieved by summing up 50 laser shots. A microcomputer (Robotron K1520) controlled the complete measuring process, including discrimination of the excitation pulse energy. In another experiment the Nd-YAG laser apparatus was used in combination with a streak camera (Agat, Mashpriborintorg) to observe the fast fluorescence radiation decay of the SHG excited samples. A SIT-vidicon in connection with an optical multichannel analyzer (OMA I, PAR) recorded the resulting streak traces. In both experiments the laser pulse duration (or = 30 ps) determined the time resolution of the measurement.
3. Samples The dyes in our experiments (rhodamine 6G, cresylviolet, DOTC iodide, and oxazine 725) are of practical interest for application in dye lasers and were purchased from Exciton Company. In appropriate mixtures these dyes are potential candidates for a long-range dipole-dipole interaction because of the good spectral overlap of absorption and fluorescence spectra (fig. 2). The dyes were dissolved without further purification in spectral grade, doubly distilled ethanol (EtOH) and dimethylsulfoxide (DMSO). Rhodamine 6G has been chosen as donor molecule because of its absorption characteristics. The other laser dyes act as acceptors (table 1) which receive their excitation via energy transfer from the originally excited donor. Two fixed values of the donor concentration have been used: Nn = 1 x 10e3 mol/d and N, = 1 X 10e4 mol/d, whereas the acceptor concentration NA has been varied over three orders of magnitude from some 10e6 mol// up to some 10e3 mol//. The solutions were con-
M. Kaschke, K. Vogler / Picosecond sturdy of energy transfer
232 Fluorescence
Absorption
The critical transfer distance R,, which is a measure of the strength of the dipole-dipole interaction and indicates the distance at which deactivation by energy transfer has the same probability as all other deactivation processes (in particular fluorescence) of the isolated donor, can be calculated from spectral data of the donor and acceptor molecule by [ll]:
cv
t
LOO
500
600
700 A lnml
R6,=
l/3
Absorption Fluorescence
I, = [(Do -.
400
I
500
I
600
I
700
‘.
-._
I
-
600 h
[ml
Fig. 2. Absorption and fluorescence spectra for donor and acceptor dye molecules. (a) Rhodamine 6G (solid line) and cresylviolet (dashed $ne), (b) rhodamine 6G (solid line) and DOTC iodide (dashed line), (c) rhodamine oxazine 725 (dashed line).
dF
(I)
Here fo(C) is the normalized spectral distribution of the donor fluorescence, /to< E) dfi = 1, and f*(C) the molar decadic extinction coefficient in units of cm’/mol. i is the wavenumber in cm-‘. N denotes Avogadro’s number (in mol-‘), n denotes the refractive index of the solvent, and qo is the fluorescence quantum yield of the isolated donor (i.e. without energy transfer). K’ is a factor given by the spatial average over all relative orientations of the donor transition moment with respect to that of the acceptor. The molecular orientation relaxation time of Rh 6G in ethanolic soluof the tion is 70r = 320 ps [18]. Thus the orientation donor can be assumed to be fixed at times t Ed T,,~ yielding u2 = 0.476 [19]. It should be noted explicitly that the expression for R, [eq. (l)] does not contain the acceptor concentration NA. Taking into consideration the estimated donor and acceptor diffusion coefficient in ethanolic solution: D, = DA = 2 x lop6 cm2/s [14] the diffusion length
h Pm1
214
9 In 10~~~11~ m.fd%d~) Jo S4 128n5n4N
6G (solid line) and
tained in optical path cuvettes of various thicknesses (d = 100 pm-2 mm) to have an optimum optical density near one in the spectral region of interest.
+ D+,)Q]“~
= 13 zk
is small compared to R, and the molecules can be regarded to be fixed in space during the initial transfer process. For the given concentration ranges of our samples the average distances between two donors and between donor and acceptor molecules amount to R ,,>74A and R,, > 43-400 A, respectively. Thus energy transfer by direct collision (diffusion controlled) can be neglected in the given concentration range if the criteria r&l, < Ri [20] and hold, where rM is the. collision radius, ‘, =Ro, which is of the same magnitude as the molecular radius of Rh 6G (6 A).
M. Kasrhke, K. YogIer / Picosecond study of emergv&xtsfer
233
Table 1 Spectroscopic parameters of the investigated dyes a) Abbr.
Dye
M (9)
ia (nm)
(a (G mol-’ cm-‘)
Tf W
RO (A,
A, Mm)
donor rhodamine 6G (perchlorat)
Rb
543
530
1.1 x 10s
4.2
-
556
acceptors cresylviolet (perchlorat) DOTC iodide oxazine 725
cv DOTC ox
362 512 424
606 680 645
0.1 x 10s 1.5x105 1.2x105
2.9 2.3 1.0
55 48 51
632 725 680
a) M = molecular weight, f, = maximum decadic extinction coefficient at wavelength X,, h, = wavelength at maximum of fluorescence, 7r = fluorescence decay time, R, = Fiirster radius calculated from eq. (1) and Rh as donor.
4. Experimental results 4.I. ~e~~rernent~ spectrometer
at the excite-end-probe-beam
Fig. 3 shows a typical time-dependent spectrum of changes in the optical density AOD, derived from the spectral transmission of the picosecond broad-band continuum at various delay times. The negative values of the optical density indicate an optical gain of the picosecond probe continuum pulse by stimulated emission in the ranges of the fluorescence bands (marked for CV and Rh 6G) and represent a measure of the actual population Cresylvidett
Rhodomin
6 G
SHG
?_. i
-1 -.-
---AOD
-..-
density in the excited state of donor and acceptor molecules. The transfer of excitation energy from the short wavelength range of Rh 6G to that of CV is clearly seen by this spectral broad-band registration. Fig. 4 depicts the changes in optical density versus the delay time at fixed wavelengths in the fluorescence region of Rh 6G and CV, respectively. The curve of Rh 6G shows the immediate response to the light pulse, while the rise of acceptor excitation via energy transfer is somewhat delayed. It can be seen from these two pictures that energy transfer is fairly effective and that the
excitation
0 - 2
6
3
12
._ ‘
24
Fig. 3. Spectral changes in optical density at various delay times after excitation by a picosecond SHG pulse (T,, 7 5 ps, Rh/CV: N, = NA= 1 X 10m4 moI/d). The polarization angle of the p &second probe ~ntinuum was at magic angle with respect to the excitation polarization to exclude orientational effects.
-623
CV,,
Fig. 4. Excitation density of donor and acceptor molecules versus delay time tr, (Rh/CV: N, = 1 X 10e4 mol/C, NA = 5 X 10m5 mol/Q. The expected rapid decay of the donor excitation which should correspond to the rise of the acceptor excitation is somewhat obscured in the AOD spectra because in the considered wavelength range the CV ground state bieacbing increase is of the same amount as the decrease of the induced Rh emission.
234
M. Kaschke, K. Vogler / Picosecond stu& of energy transfer
Table 2 Experimental
parameters of different dye solutions used in picosecond
Dye mixture
No (mot/d)
Rh/cv
1 x10--’
0.1 0.1 0.3 0.3 0.7 0.2 0.2 _ 0.4
0.7 0.9
EtOH EtOH EtOH EtOH EtOH EtOH EtOH EtOH EtOH DMSO
60 67 48 53 47
0.1 0.06 0.3 0.3 0.3
_ _
0.09 0.01
40 24
0.5 0.5
0.01
40
-
1 x1o-3 1 x10-4 2x10-3 1x1o-3 5x10-4 1x10-4 5x10-s 1x1o-5 5x1o-6 1x10-4
0.02 0.02 0.48 0.19 0.09 0.02 0.01 0.01 0.01 0.02
13 20 lib’ 20 19 23 19 16 15 c’ 26
1x10-s 1x10-4 1x1o-4 1x1o-5 1x10-4
0.01 0.01 0.01 0.01 0.01
1x10-4
5x10-4 1x10-s
1x10-4
1x1o-4
1x10-s
1 x10-4
Rh/ox
solvent
f;,
1x1o-4
+ DOTC
AOD, -AOD,
aAN a caoaDN,
1 x 1o-4
Rh/CV
N~exc
(mot/d)
1x10-4
Rh/DOTC
energy transfer experiments
NA
NA
1.0 1.0 1.0 0.3 0.1 0.1 1.0
EtOH EtOH EtOH EtOH DMSO EtOH
0.6
EtOH
eaD, e$ donor and acceptor a) The ratio is a measure of direct acceptor excitation, No, NA donor and acceptor concentration, absorption cross section at 527 nm AOD,, AOD, changes in optical density in the donor and acceptor band. b, Disturbed by direct excitation of acceptor molecules. ‘) Superimposed by the long wavelength fluorescence tail of Rh.
main transfer takes place within rather short time intervals after excitation. After this initial rise a nearly constant value of the acceptor excitation density is reached. The depletion via fluorescence is too slow to be significant in the considered time interval. The rise time t, of the acceptor excited state population is characterized by the 10 to 90% increase of the corresponding AOD values. Surprisingly no significant dependence of t, neither on the acceptor concentration nor on the solvent has been found (table 2). A similar rapid energy transfer has been observed in dye mixtures with a smaller Forster radius R, (tables 1 and 2), though the characteristic rise time has been determined to be somewhat increased. The ratio of the changes in optical density AOD,/AOD, is a measure of the quantum of excitation energy transferred. 4.2. Single-photon counting measurements A number of experimental curves obtained with the SPC are displayed in fig. 5 for illustration. The
fluorescence decay of the donor is considerably shortened by the influence of energy transfer at high acceptor concentrations. Because of the limited time resolution of the SPC conclusions from the time evolution of the donor fluorescence can be drawn only at times greater than 200 ps. The SPC, however, offers the advantage of being able to deliver accurate data on the deviations of the donor fluorescence temporal evolution in the presence of acceptor molecules from the normal fluorescence decay at comparatively late times (some ns). According to Forster [7] the time behavior of the excited donor ensemble density is given by: D(t) = D(0) exp( - t/-rD) Xexp[ - jv3/2RiNA(t/7,)“2],
(2)
where T,, is the fluorescence lifetime of the donor in absence of energy transfer, NA is the acceptor concentration and R, is the FSrster radius calculated by eq. (1). Thus picosecond time-resolved
235
M. Kaschke, K, Vogter / Picosecond study of energy transfer
0
d 0
t
1
2
3
1
5
6
Fig. 5. SPC measured fluorescence decay curves of the excited donor mole&e at different acceptor ~n~ntmtions (Rh/CV: No = 1 X 10m3 mol/r?). Curves 1: NA = 2 X10-j mol/C, Q, = 1.8 ns, curve 2: N,, - 1 X 10W3 mol/C, 7err= 2.4 ns, curve 3: N* = 5 X 10m4 mol/d, ec,r = 3.1 ns, curve 4: NA = 2x 10S4 mol/t, Q = 3.5 ns, curve 5: N, = 0, T = 4.2 ns (T=@is taken as time of l/e decrease of fluorescence intensity.
spectroscopy allows to determine the Fijrster radius R, expe~ent~ly by fitting experimental data to eq. (2). In doing this, we plotted lo&l,,(t)/J,(t)] versus t’/* (fig. 6), where IA(f), lo(t) are the donor fluorescence decays with and without the presence of acceptors in the solution. The data yield a straight line (at least at times greater than I ns), from the slope of which the critical transfer distance R, can be determined using eq. (2) for the various solutions (table 3). We obtained a dependence of the transfer distance R 0 determined in this way on the acceptor concentration NA at smaller acceptor concentrations. This dependence is, however, not explainable in terms of eqs. (1) and (2). Similar results have been reported by other authors [4,13]. They also observed a virtual dependence of R, on NA at acceptor concentrations smaller than the donor concentration. The experimental results obtained by the monochromatic measurements at the Nd-YAG spectrometer have confirmed with a smaller time reso-
I
I
2
1
7 thsl
,
3
I
L
t Inal
5
Fig. 6. _h$rmalized fluorescence intensity log[l,( t)/lA(t)] versus t”‘. f,,(t) is the decay of the donor fluorescence without acceptors in the solution, l,,(f) is the same in presence of acceptor molecules.
lution but higher accuracy in changes of the optical density our results of the picosecond broadband excite-and-probe-beam spectrometer (fig. 7). The single-shot streak camera was only used for qualitative m~surements. Because of the low dynamic range ( < 10) and relatively high noise levels the single-shot streak camera proved to be inferior in accuracy to all the other methods applied. normalized
A00 I
i3=660ml I
k=-+;_-___:_.
i
x
Ft=SMlnm
0.5 I’
200
100
600
BOO
c
1000 tips1
Fig. 7. Normahid excitation density of donor and acceptor molecules measured by a Nd-YAG probe-beam spectrometer N* = 1 X low4 mol/C). X = 660 mn probing CV fluores(NDcence, X = 560 nm probing Rh fluorescence and CV bleaching.
236
M. Kmchke, K. VogIer / Picosecond stuo)~ of energy transfer
Table 3 Critical transfer radius Ra calculated from the decay of the donor fluorescence in dependence on the acceptor concentration NA. The table is completed by synchroscan streak camera data from refs. 14,131. NA
R0
@01/O
(4
Rh/CV in EtOH Nn=1X10W3 mol/L theory b, 4x10-3 2x10-3 1x10-3 5x1o-4 3x10-4 5x10-s
56 50 52 61 70 13 94
1x10-3 5x10-4 1x1o-4
49 45 46 64
5x10-3 2x10-3 1 x10-3 5x10-4 2x10-4 lxlo-4 5x10-5 1 x10-5
59 45 46 48 69 84 91 120 152
4x10-3 4x10-4 3x10-s
24 33 50 75
Rh/DOTC in EtOH No=l~lO-~ theory b,
DODC/MG Nn=l~lO-~ theory b,
a/b
‘)
(3)
_ 1 2 9 18 22 57
in EtOH ‘)
HPD monomer/dimer in Hz0 d, ND=NA theory b,
The rate equation for the deactivation of the excited single donor differs from that of a completely isolated donor by an additional relaxation term, describing the deactivation due to long-range dipole-dipole energy transfer to the surrounding acceptors.
a) a/b is the ratio of raised to bulk density of acceptors as explained in section 5. b, Values calculated from spectral data using eq. (1). ‘) From ref. [13]. d, From ref. (41.
Here rk is the distance of the k th acceptor from the donor of interest. Spatial averaging over the acceptor distribution around the donors leads to the well-known t 1/2 law [eq. (2)] of Forster [21] which describes the deexcitation behavior of an ensemble of excited donor molecules after excitation by a S-like pulse. Some restrictive assumptions are made in deriving this decay function: (i) The acceptor molecules are homogeneously distributed around each donor of the ensemble. (ii) Donor and acceptor molecules are at fixed locations in space during interaction. (iii) Mutual influence between donor molecules is not taken into account. Assumptions (i) and (iii) imply directly that the donor concentration No is much smaller than that of the acceptors. It is worthy to note that because of N,, +z NA eq. (2) does not contain the donor density. In order to obtain the temporal evolution of the excited acceptor ensemble we follow the concept of Heber [22]. In treating the problem of excitation diffusion he introduced mean statistical rate equations for the donor and acceptor ensemble. These coupled differential equations can be derived from eq. (2) by derivation with respect to time. The mean statistical transfer rate XDA, responsible for the decrease of the mean donor excitation o(t) due to energy transfer, is assumed to be the same as the rate, which increases the mean acceptor excitation A(t).
5. Discussion In a first attempt we tried to explain our experimental findings in terms of the well-known Forster model [7]. In this model an isolated excited donor is considered, which interacts with a number ( NA) of surrounding acceptors, each in the ground state.
d,(t)/dt
= -r,%(l)
-k,,(r)@t),
dx((r)/dt
= -pi%
+k,,(t)~(t),
j&,,(t) Here
= &$JAR;Q3/27;‘/2t-‘/2 the mean
statistical
= pt-‘/=. transfer
rate
(4) is time
M. Kaschke, K. Vogier / Picosecond study of energy transfer
dependent in contrast to eq. (3). rA is the fluorescence decay time of the acceptor molecule. The ~nte~a~on of the second differential equation yields the solution for the time behavior of the acceptor density in the excited state [22]: X(t) = D(0)2@y-“2
exp( - t/r,)
[ w( py-li2)
+w((yt)*‘z
- /3y-1/2)
X exp( - g/r0 - 2/B */2 )] , where
w(x) = exp( - x2)ixexp( u”) du is the so-called Karpov integral and y = TA1 - 7; ‘. Eiq. (5) represents the ~o~te~~t to the common donor decay law [eq. (2)] and is a somewhat more complicated function, which describes the rise and the decay of the acceptor density in the excited state. Fig. 8 shows the time behavior of excited donor and acceptor molecules calculated from eqs. (2) and (5) for various acceptor concentrations and a fixed Forster radius of R, = 50 A. The maximum of the acceptor excitation density is reached not earlier than after a few hundred picoseconds, even
0
too
I
b
I
I
t
1
b
s
200
300
400
500
600
700
800
900
237
at the highest acceptor concentrations. Thus the solution (5) is not capable to explain our experimental findings of a fast rise of the acceptor excitation density (fig, 4). Further it should be noted that in terms of the Fijrster donor decay law (2) the observed concentration dependence of the critical transfer radius R. and NA (table 3) cannot be understood, because the quantum-theoretically derived Fbrster radius R, only depends on molecular and solvent parameters, but not on the acceptor concentration [cf. eq. (l)]. We propose a model for the long-range dipole-dipole interaction which in contrast to assumption (i) of the common Fiirster theory allows for an inhomogeneous distribution of the acceptor molecules around the donors. A similar model was developed in ref. [23] to explain succesfully experimental meas~ements of energy transfer between F-centers in solids. For large and charged molecules there exists a certain probability to aggregate and to form weakly bounded clusters [24]. Assuming a simplified stepfunction-like and spherosymmetric ~homogeneous dist~bution of the acceptor molecules around the donor (fig. 9) spatial averaging of eq. (3) over the inhomogeneously distributed acceptor ensemble (cf. ref. 1231) yields the following expression for the donor decay function, which corresponds to the donor fluorescence de-
1
1000 M
Fig. 8. Time behavior of donor and acceptor molecules after a S-function-like excitation pulse for several acceptor concentrations NA and R, = 50 A. (a) Donor decay (F&ster’s t ‘j2 law) [eq. (Z)], (b) solution of eq. (4) for the acceptor density in the excited state [eq. (5)]. Curve 1: NA = 1X1O-5 mol/d, curve2: NA=1~10W4 mol/Lp, carve 3: NA=1x10T3 mol/t”, curve4: N,=1~10-~ mol/e.
hf. Kaschke, K. Vogler / Picosecond study of energy transfer
238
cay: B(t)=B(O)
exp
- $ i
1
where
Q(d)>
= 1 -@(xi(t))+
is a somewhat complicated the error function +(-xi(t))
= 2v-‘/‘/x’(‘)exp(
1 - exp( _x,‘(t)) n,,zx_(t)
I function
-u’)
and includes
du,
0
with Xi( t ) = (R3,/R;7)(
t/7,)1’2.
The radii R, and R, denote the inner and outer radius of the sphere of raised acceptor concentration a. b is the bulk acceptor concentration in the solution. As schematically shown in fig. 9 such a spatial acceptor distribution may be induced by a static van der Waals interaction between solved molecules. For the limiting case R, --* 0 and a + b eq. (6) coincides with the usual Fijrster formula [eq. (2)]. The appearance of the Q-functions in eq. (6) yields a rather rapid decay of excited donors as long as the unequality xi(t) = Rz/Rj(t/rD)“* +z 1 holds after excitation by a S-like pulse (fig. 1Oa). If the time t is of the order of the fluorescence decay time (i.e. xi(l) 2 5) the time dependence of the function Q(xi(t)) can be neglected (Q(+) = QC = 0.1 and for R, CR,: Q(x,)= 0) yielding again the common Fiirster law, but with a modified transfer radius:
&f = This expression effective Fijrster
Fig. 9. Spatial distribution of acceptors around one excited donor. (a) Homogeneous distribution (conventional Fiirster model), (b) inhomogeneous distribution and a corresponding intermolecular interaction potential. a, b: increased and bulk density of acceptor molecules, RI, R,: radii of sphere of raised acceptor concentration (sphere 0 --) R, represents the forbidden volume), R,: critical transfer radius.
was set equal to the measured radii R, from table 3. For rea-
sonable values of R, and R, the ratio a/b is calculated from (7) and listed in table 3. Thus the concentration dependence of R, on NA (or b) can be reduced on a corresponding dependence of the ratio of the raised acceptor density to the bulk density. Another possible approach in explaining the observed deviations from the Fiirster law eq. (2) will be given elsewhere [25]. In order to obtain the expression for the time behavior of the excited acceptor ensemble in the case of an inhomogeneous distribution of molecules the same procedure as before is applied for deriving the mean statistical rate equations from the donor decay function (6). This results in the following coupled differential equations:
In comparing these rate equations with those of the homogeneous distribution [eq. (4)] the rather difficult time dependence of the mean transfer rate is obvious. Taking the solution (6) for the donor decay the time behavior of the excited acceptor
M. Kaschke, K. Yogler / &&second stuay of energy transfer
239
a
0
tbni)
0
100
200
300
400
500
600
700
600
900
1000
Fig. 10. Time behavior of donor and acceptor molecules after a b-function-Iike excitation pulse spatial distribution of acceptor molecules (N, = b =l X lo-’ mol/C). (a) Donor decay function density [integral of eq. (7)]. Parameters for (a), (b) curve I: a/b = 5, curve 2: a/b = 10, curve R,=lOA, R,=25& R,==50A. For (c) curve 1: R,=15ik, curve 2: R, ==2OA, curve 3: a/b = 15, R, = 50 A, R, = 10 A {curve 0 corresponds to the homogeneous case).
ensemble has been calculated by numerical integration. The resulting time functions of the excited acceptor density after a S-function-like excitation of the donors are shown in figs. lob and 1Oc in dependence on R, and a/b, respectively. The dependence on R, is not si~ifi~t, if the forbidden volume is small enough (i.e. R, -c10 ;i>.
in the case of an inhomogeneous [eq. (S)], (bf, (c) excited acceptor 3: a/b = 15, curve 4: a/b - 25, R,=25& curve 4: R,=30A,
As seen from these calculated curves much shorter rise times of the mean acceptor excitation density than in the original Fiirster case (fig. 8b) are obtained. Thus the rapid rise in the population of the excited state of the acceptor can be fitted fairly well by variation of the parameters R, and a to the experimentally observed data (fig. 4). An
hf. Kaschke, K. Vogler / Picosecond study of energv transfer
240
Acknowledgement We would like to thank Professors M. Schubert and B. Wilhelmi for their stimulating interest in this work and for many helpful discussions.
References
L
10
0
10
20
30
40
50
60
70
80
90
‘00
&I
Fig. 11. Fitting of the numerical solution of eq. (7) for the acceptor rise function to the experimental curve (example). Solid line experimental curve, dashed line calculated curve (a/b = 25, R2= 20A, R, =lOA, R,=50&.
example of this fitting is depicted in fig. 11. Taking the value for R, fixed at 10 A we get very good agreement between the theoretical and the experimental curve for R, = 25 A and a/b = 10. This corresponds to a depth of the Lennard-Jones potential of 2.3 kT which seems quite reasonable.
6. Summary The investigation of several laser dye mixtures by means of time-resolved absorption and fluorescence spectroscopy revealed a rapid and effective energy transfer to the acceptor after picosecond excitation of the donor. The experimental results deviate significantly from the predictions of the common Forster model. Similar observations have been reported earlier [4,13]. Our experimental findings are tentatively explained in terms of a modified Fiirster model, in which one of the basic assumptions of the Fiirster model has been extended to include an inhomogeneous distribution of molecules in solution. However, the range of validity of this modified Forster model is still restricted by the assumption NA % No, which is a severe limitation because in many applications the opposite case NA -=z No is to be found. The abandonment of this assumption and the consequences therefrom will be a matter of investigation in a forthcoming paper [25,26].
[l] S. Kudzmauskas, L. Valkunas and A.Y. Borisov, J. Theoret. Biol. 105 (1983) 13. [2] H. Haken and G. Strobl, Z. Physik 262 (1973) 135. [3] A. Her-z, Advan. Chem. Colloid Interface 8 (1977) 237. [4] M. Yamashita, M. Nomura, S. Kobayashi, T. Sato and K. Aizawa, IEEE J. Quantum Electron. QE-20 (1984) 1363. [5] M. Teroda and Y. Ohba, Japan. J. Appl. Phys. 22 (1983) 1392. [6] M. Kaschke, U. Stamm and K. Vogler, Appl. Phys. B (1986). to be published. Physik 6 (1948) 55. I71 T. Forster, AM. 181 U.K. Klein, R. Fray, M. Hauser and U. Gosele, Chem. Phys. Letters 41 (1976) 139. and Y.V. Konobeev, Opt. i SpektrosA V.M. Agranovich kopiya 6 (1954). ]I01 M. Yokota and 0. Tanimoto, J. Phys. Sot. Japan 22 (1967) 179. (111 V.M. Agranovich and M.D. Galanin, Electronic excitation energy transfer in condensed matter (North-Holland, Amsterdam, 1982). f121 P.Y. Lu, Z.X. Yu, R.R. Alfano and J.I. Gersten, Phys. Rev. A26 (1982) 3610. 1131 M.C. Adams, D.J. Bradley, W. Sibbett and J.R. Taylor, Chem. Phys. Letters 66 (1979) 428. P41 G. Porter and C.J. Tredwell, Chem. Phys. Letters 56 (1978) 278. 1151T. Damm, M. Kaschke, M. Kresser, F. Noack, S. Rentsch and W. Triebel, Exp. Tech. Phys., to be published. WI K.-D. Eickhoff, D. Schubert and H. Wabnitz, Proceedings of the III Symposium on Optical Spectroscopy Reinhardsbrunn, GDR (1984). 1171 H. Bergner, V. Bruckner and B. Schrijder, Opt. Acta 29 (1982) 1491. WI G. Porter, P.J. Sadkowski and C.J. Tredwell, Chem. Phys. Letters 49 (1977) 416. 1191 N. Nakishima, K. Yoshihara and F. Willig, J. Chem. Phys. 73 (1980) 3553. PO1 U. Gosele, M. Hauser, U.K.A. Klein and R. Frey, Chem. Phys. Letters 34 (1975) 519. I-211U. Gosele, M. Hauser, U.K.A. -Klein, 2. Physik. Chem. NF 99 (1976) 81. I221J. Heber, Phys. Stat. Sol. 48b (1971) 319. 1231 M. Schubert and K. Vogler, Ann. Physik 37 (1980) 312. J.K. Hallemayer and W. Kaiser, Chem. 1241 D. Kopainsky, Phys. Letters 83 (1981) 489. PI M. Kaschke, M. Schubert and K. Vogler, in preparation. of the EPS WI M. Schubert and B. Wilhelmi, Proceedings Condensed Matter Conference, West-Berlin (1985).
Chemical Physics 102 (1986) 229-240 worth-Holland, Amsterdam
PICOSECOND STUDY OF ENERGY TRANSFER. DEVIA~ONS FROM FORSTER THEORY - EVIDENCE FOR AN INHOMOGENEOUS SPATIAL DIS~RU~ON
OF MOLECULES
Michael KASCHKE and Klaus VOGLER Sektion Physik, Friedrich - Schiller-Universitiit Jena, Max - Wien - Platz I, 6900 Jena, GDR Received 14 June 1985
We have investigated long-range dipole-dipole energy transfer between several laser dyes in solution. Time-resolved absorption and fluorescence experiments on a picosecond excite-and-probe-beam spectrometer and a single-photon counting equipment have been carried out to measure the excitation depletion of the donor (Rh 6G) and the rise of the excitation of the acceptors (DOTC, cresylviolet, oxazine) as a function of the acceptor concentration. An unexpected very-rapid and effective energy transfer has been been found, even at very low acceptor concentrations. F&ster radii R, have been determined for each dye ~mbination. At high acceptor con~n~ations these R, values agree well with those calculated from spectral data, but rather large deviations have been found at low acceptor concentrations. The observed deviations from the common Fijrster law are tentatively explained in terms of a modified Forster model which takes into account an inhomogeneous distribution of molecules in solution.
1. Inaction
Intermolecular energy transfer processes play a major role in various fields in nature. In photosynthesis, for instance, optical energy is transferred via radiationless transfer mechanisms to the reaction center [l], whereby a simultaneous transfer of the optical energy from one spectral region to the other takes place. In addition, the technical use of energy transfer mechanisms is gaining an increasing importance in a number of industrial and scientific applications. A few examples are exciton migration in solids [2] or the sensibilization of photophysi~ f3] and phot~he~c~ [4] processes. Dye lasers are widely used as spectroscopic tools for photochemical and photophysical investigations. Energy transfer in laser dye mixtures can extend the lasing range, increase the output efficiency and can also have considerable influence on the temporal characteristics of the output pulses of energy transfer dye lasers [5,6]. Thus, in order to understand the processes proceeding in this case of application, as well as to gain principal knowledge on energy transfer
processes in condensed matter the investigation of energy transfer mechanisms between dye molecules in solution is of substantial interest. There are several possible mechanisms for energy transfer between dye molecules in solution: radiationless dipole-dipole (or higher multipole) interaction [7], diffusion-controlled exchange interaction between nei~bo~ng molecules [g], and radiative transfer [9]. Two or more of these mechanisms can also act simultaneously [lO,ll]: the predominant mechanism is determined mainly by the intermolecular distances between the solved molecules. An exact distinction between these mechanisms is only possible expe~mentally by means of time-resolved spectroscopic methods. Up to now, most of the experimental investigations of energy transfer between dyes in solution have been timeresolved fluorescence studies, using either singleshot 1121 or synchroscan [13] streak cameras. In order to check the differences between improved theoretical models experimentally a high time resolution and accuracy are necessary, since the mechanisms differ mainly in the time domain shortly after excitation. The picosecond excite-and-probe-
0301-0104/g6/$03.50 0 Elsevier Science Publishers B.V. (North-Holl~d Physics ~blishing Division)
M. Kaschke, K. VogIer / Picosecond study of energy transfer
230
beam absorption spectroscopy fulfills these requirements and offers the advantage of being able to observe simultaneously the kinetics of both the donor and the acceptor. The purpose of our study was to obtain detailed information on the range of validity of different energy transfer models by means of time-resolved absorption and fluorescence spectroscopy. Particular interest was paid to the question of the validity of the common Forster model [7], since experimental verifications [14] as well as deviations [4,13] from the predicted law have been reported recently. In our experiments we used various dye mixtures, rhodamine 6G (Rh 6G) as donor and cresylviolet (CV), DOTC iodide (DOTC), and oxazine 725 (Ox) as acceptors. The concentration ratio of donor and acceptor covered three orders of magnitude. The experiments revealed an unexpectedly rapid and effective transfer of excitation energy to the acceptor molecules, even at low acceptor concentration. The experimental findings were tentatively evaluated in terms of a modified Forster model, which takes into account an inhomogeneous spatial distribution of the dye molecules in solution.
2. Experimental arrangement Most of our experiments have been performed on a picosecond probe-beam spectrometer, driven by an Nd-phosphate-glass laser, and a single-pho-
ps-N&phosphate
Fig. 1. Experimental
glass
laser
prolc-beam
arrangement of the picosecond
ton counting equipment. A streak camera and a picosecond Nd-YAG absorption spectrometer, which consists of an optical parametric amplifier, have been used in additional measurements. 2.1. Excite-and-probe-beam
spectrometer
In fig. 1 the picosecond Nd-phosphate-glass apparatus is shown [15]. A Pockels cell [2] selects a single pulse (or = 5 ps) from the leading part of the pulse train emitted by a passively mode-locked Nd-phosphate-glass oscillator, 1. After a two-stage amplifier, 3 the main part (90%) of the amplified single pulse (h = 1054 nm, E = 20 mJ) is frequency converted to the second harmonic (X = 527 nm, E = 5 mJ) or alternatively to other frequencies, 6: third harmonic, Raman shifted lines. This light pulse serves as excitation beam. The remaining part of the split fundamental wave pulse, 4, after a further amplification, 9, generates the picosecond broad-band continuum in a 10 cm D,O cuvette, 10. The picosecond continuum covers a wavelength range of = 300-900 nm and is used as interrogation beam after splitting into a probe and reference beam, 11, which probe the transmission of the excited and unexcited sample volume, 12. Both the light pulses are focused onto the entrance slit of a spectrograph, 13, and detected by a vidicon, 14, of an optical multichannel analyzer, 15, PAR OMA II. The spectral resolution of this arrangement amounts to 2 nm in an overall spectral extension of 250 nm. The excitation beam can
spectrometer
Nd-phosphate-glass
laser and the excite-and-probe-beam
spectrometer.
M. Kaschke, K, Vogier / Picosecond study of energy iransfer
properly be delayed, 7 (maximum to = 900 ps) and altered in polarization with respect to the interrogation pulse 16. The automatic registration by the OMA contains a special procedure for the discrimination of the excitation pulse energy, 8, and pulse duration. The time resolution of this equipment is better than 6 ps. A registration accuracy of 2% in transmission changes within the whole observable spectral region of 250 nm and a dynamic range of more than 200 is achieved. Thus we have been able to observe the donor population depletion and the acceptor excitation, simultaneously and with high accuracy, by changes in the optical density of the sample. Care has been taken to prevent a depletion of excited donors by stimulated emission. 2.2. Single-photon counting equipment (SPC) This apparatus consists of an acousto-optically mode-locked Ar-ion laser as excitation source (X = 514 run, rp = 80 ps) and an usual fluorescence detection unit: monochromator, photomultiplier, and the electronic sampling and control unit (SPC 100, ZOS). Excitation as well as fluorescence light can be attenuated by suitable filters. Alternatively a synchronously pumped picosecond dye laser ( rp = 2 ps, X = 560-620 nm) can be used for excitation [16]. The time resolution of the SPC 100 unit amounts to = 100 ps (without deconvolution), the dynamic range within the recorded time interval of 8 ns (given by the repetition rate of the laser) is better than 104. Samples of small thickness (100-200 pm) and front face excitation and detection had to be used in order to prevent reabsorption effects at high concentrations of solved molecules. 2.3. Picosecond Nd- YAG laser spectrometer This equipment is composed of a passively mode-locked Nd-YAG laser, a single-pulse selection unit, an amplifier stage, and an SHG driven optical parametric generator (OPA), as well as an excite-and-probe-beam interaction part [17]. An SHG pulse (X = 532 nm, rp = 30 ps, E = 5 mJ) was used for exciting the samples (cuvette thickness: 1 mm). Whereas in the Nd-phosphate-glass
231
laser spectrometer a broad-band interrogation of the sample by means of the picosecond continuum was achieved, here monochromatic interrogation and reference beams probed the transmission of the excited and unexcited sample volume. The probe beams were obtained from the frequencydoubled OPA signal wave (X = 560 nm or X = 660 nm). The laser pulse repetition rate was 1 Hz. Accuracies of smaller than 1% in transmission changes can be achieved by summing up 50 laser shots. A microcomputer (Robotron K1520) controlled the complete measuring process, including discrimination of the excitation pulse energy. In another experiment the Nd-YAG laser apparatus was used in combination with a streak camera (Agat, Mashpriborintorg) to observe the fast fluorescence radiation decay of the SHG excited samples. A SIT-vidicon in connection with an optical multichannel analyzer (OMA I, PAR) recorded the resulting streak traces. In both experiments the laser pulse duration (or = 30 ps) determined the time resolution of the measurement.
3. Samples The dyes in our experiments (rhodamine 6G, cresylviolet, DOTC iodide, and oxazine 725) are of practical interest for application in dye lasers and were purchased from Exciton Company. In appropriate mixtures these dyes are potential candidates for a long-range dipole-dipole interaction because of the good spectral overlap of absorption and fluorescence spectra (fig. 2). The dyes were dissolved without further purification in spectral grade, doubly distilled ethanol (EtOH) and dimethylsulfoxide (DMSO). Rhodamine 6G has been chosen as donor molecule because of its absorption characteristics. The other laser dyes act as acceptors (table 1) which receive their excitation via energy transfer from the originally excited donor. Two fixed values of the donor concentration have been used: Nn = 1 x 10e3 mol/d and N, = 1 X 10e4 mol/d, whereas the acceptor concentration NA has been varied over three orders of magnitude from some 10e6 mol// up to some 10e3 mol//. The solutions were con-
M. Kaschke, K. Vogler / Picosecond sturdy of energy transfer
232 Fluorescence
Absorption
The critical transfer distance R,, which is a measure of the strength of the dipole-dipole interaction and indicates the distance at which deactivation by energy transfer has the same probability as all other deactivation processes (in particular fluorescence) of the isolated donor, can be calculated from spectral data of the donor and acceptor molecule by [ll]:
cv
t
LOO
500
600
700 A lnml
R6,=
l/3
Absorption Fluorescence
I, = [(Do -.
400
I
500
I
600
I
700
‘.
-._
I
-
600 h
[ml
Fig. 2. Absorption and fluorescence spectra for donor and acceptor dye molecules. (a) Rhodamine 6G (solid line) and cresylviolet (dashed $ne), (b) rhodamine 6G (solid line) and DOTC iodide (dashed line), (c) rhodamine oxazine 725 (dashed line).
dF
(I)
Here fo(C) is the normalized spectral distribution of the donor fluorescence, /to< E) dfi = 1, and f*(C) the molar decadic extinction coefficient in units of cm’/mol. i is the wavenumber in cm-‘. N denotes Avogadro’s number (in mol-‘), n denotes the refractive index of the solvent, and qo is the fluorescence quantum yield of the isolated donor (i.e. without energy transfer). K’ is a factor given by the spatial average over all relative orientations of the donor transition moment with respect to that of the acceptor. The molecular orientation relaxation time of Rh 6G in ethanolic soluof the tion is 70r = 320 ps [18]. Thus the orientation donor can be assumed to be fixed at times t Ed T,,~ yielding u2 = 0.476 [19]. It should be noted explicitly that the expression for R, [eq. (l)] does not contain the acceptor concentration NA. Taking into consideration the estimated donor and acceptor diffusion coefficient in ethanolic solution: D, = DA = 2 x lop6 cm2/s [14] the diffusion length
h Pm1
214
9 In 10~~~11~ m.fd%d~) Jo S4 128n5n4N
6G (solid line) and
tained in optical path cuvettes of various thicknesses (d = 100 pm-2 mm) to have an optimum optical density near one in the spectral region of interest.
+ D+,)Q]“~
= 13 zk
is small compared to R, and the molecules can be regarded to be fixed in space during the initial transfer process. For the given concentration ranges of our samples the average distances between two donors and between donor and acceptor molecules amount to R ,,>74A and R,, > 43-400 A, respectively. Thus energy transfer by direct collision (diffusion controlled) can be neglected in the given concentration range if the criteria r&l, < Ri [20] and hold, where rM is the. collision radius, ‘, =Ro, which is of the same magnitude as the molecular radius of Rh 6G (6 A).
M. Kasrhke, K. YogIer / Picosecond study of emergv&xtsfer
233
Table 1 Spectroscopic parameters of the investigated dyes a) Abbr.
Dye
M (9)
ia (nm)
(a (G mol-’ cm-‘)
Tf W
RO (A,
A, Mm)
donor rhodamine 6G (perchlorat)
Rb
543
530
1.1 x 10s
4.2
-
556
acceptors cresylviolet (perchlorat) DOTC iodide oxazine 725
cv DOTC ox
362 512 424
606 680 645
0.1 x 10s 1.5x105 1.2x105
2.9 2.3 1.0
55 48 51
632 725 680
a) M = molecular weight, f, = maximum decadic extinction coefficient at wavelength X,, h, = wavelength at maximum of fluorescence, 7r = fluorescence decay time, R, = Fiirster radius calculated from eq. (1) and Rh as donor.
4. Experimental results 4.I. ~e~~rernent~ spectrometer
at the excite-end-probe-beam
Fig. 3 shows a typical time-dependent spectrum of changes in the optical density AOD, derived from the spectral transmission of the picosecond broad-band continuum at various delay times. The negative values of the optical density indicate an optical gain of the picosecond probe continuum pulse by stimulated emission in the ranges of the fluorescence bands (marked for CV and Rh 6G) and represent a measure of the actual population Cresylvidett
Rhodomin
6 G
SHG
?_. i
-1 -.-
---AOD
-..-
density in the excited state of donor and acceptor molecules. The transfer of excitation energy from the short wavelength range of Rh 6G to that of CV is clearly seen by this spectral broad-band registration. Fig. 4 depicts the changes in optical density versus the delay time at fixed wavelengths in the fluorescence region of Rh 6G and CV, respectively. The curve of Rh 6G shows the immediate response to the light pulse, while the rise of acceptor excitation via energy transfer is somewhat delayed. It can be seen from these two pictures that energy transfer is fairly effective and that the
excitation
0 - 2
6
3
12
._ ‘
24
Fig. 3. Spectral changes in optical density at various delay times after excitation by a picosecond SHG pulse (T,, 7 5 ps, Rh/CV: N, = NA= 1 X 10m4 moI/d). The polarization angle of the p &second probe ~ntinuum was at magic angle with respect to the excitation polarization to exclude orientational effects.
-623
CV,,
Fig. 4. Excitation density of donor and acceptor molecules versus delay time tr, (Rh/CV: N, = 1 X 10e4 mol/C, NA = 5 X 10m5 mol/Q. The expected rapid decay of the donor excitation which should correspond to the rise of the acceptor excitation is somewhat obscured in the AOD spectra because in the considered wavelength range the CV ground state bieacbing increase is of the same amount as the decrease of the induced Rh emission.
234
M. Kaschke, K. Vogler / Picosecond stu& of energy transfer
Table 2 Experimental
parameters of different dye solutions used in picosecond
Dye mixture
No (mot/d)
Rh/cv
1 x10--’
0.1 0.1 0.3 0.3 0.7 0.2 0.2 _ 0.4
0.7 0.9
EtOH EtOH EtOH EtOH EtOH EtOH EtOH EtOH EtOH DMSO
60 67 48 53 47
0.1 0.06 0.3 0.3 0.3
_ _
0.09 0.01
40 24
0.5 0.5
0.01
40
-
1 x1o-3 1 x10-4 2x10-3 1x1o-3 5x10-4 1x10-4 5x10-s 1x1o-5 5x1o-6 1x10-4
0.02 0.02 0.48 0.19 0.09 0.02 0.01 0.01 0.01 0.02
13 20 lib’ 20 19 23 19 16 15 c’ 26
1x10-s 1x10-4 1x1o-4 1x1o-5 1x10-4
0.01 0.01 0.01 0.01 0.01
1x10-4
5x10-4 1x10-s
1x10-4
1x1o-4
1x10-s
1 x10-4
Rh/ox
solvent
f;,
1x1o-4
+ DOTC
AOD, -AOD,
aAN a caoaDN,
1 x 1o-4
Rh/CV
N~exc
(mot/d)
1x10-4
Rh/DOTC
energy transfer experiments
NA
NA
1.0 1.0 1.0 0.3 0.1 0.1 1.0
EtOH EtOH EtOH EtOH DMSO EtOH
0.6
EtOH
eaD, e$ donor and acceptor a) The ratio is a measure of direct acceptor excitation, No, NA donor and acceptor concentration, absorption cross section at 527 nm AOD,, AOD, changes in optical density in the donor and acceptor band. b, Disturbed by direct excitation of acceptor molecules. ‘) Superimposed by the long wavelength fluorescence tail of Rh.
main transfer takes place within rather short time intervals after excitation. After this initial rise a nearly constant value of the acceptor excitation density is reached. The depletion via fluorescence is too slow to be significant in the considered time interval. The rise time t, of the acceptor excited state population is characterized by the 10 to 90% increase of the corresponding AOD values. Surprisingly no significant dependence of t, neither on the acceptor concentration nor on the solvent has been found (table 2). A similar rapid energy transfer has been observed in dye mixtures with a smaller Forster radius R, (tables 1 and 2), though the characteristic rise time has been determined to be somewhat increased. The ratio of the changes in optical density AOD,/AOD, is a measure of the quantum of excitation energy transferred. 4.2. Single-photon counting measurements A number of experimental curves obtained with the SPC are displayed in fig. 5 for illustration. The
fluorescence decay of the donor is considerably shortened by the influence of energy transfer at high acceptor concentrations. Because of the limited time resolution of the SPC conclusions from the time evolution of the donor fluorescence can be drawn only at times greater than 200 ps. The SPC, however, offers the advantage of being able to deliver accurate data on the deviations of the donor fluorescence temporal evolution in the presence of acceptor molecules from the normal fluorescence decay at comparatively late times (some ns). According to Forster [7] the time behavior of the excited donor ensemble density is given by: D(t) = D(0) exp( - t/-rD) Xexp[ - jv3/2RiNA(t/7,)“2],
(2)
where T,, is the fluorescence lifetime of the donor in absence of energy transfer, NA is the acceptor concentration and R, is the FSrster radius calculated by eq. (1). Thus picosecond time-resolved
235
M. Kaschke, K, Vogter / Picosecond study of energy transfer
0
d 0
t
1
2
3
1
5
6
Fig. 5. SPC measured fluorescence decay curves of the excited donor mole&e at different acceptor ~n~ntmtions (Rh/CV: No = 1 X 10m3 mol/r?). Curves 1: NA = 2 X10-j mol/C, Q, = 1.8 ns, curve 2: N,, - 1 X 10W3 mol/C, 7err= 2.4 ns, curve 3: N* = 5 X 10m4 mol/d, ec,r = 3.1 ns, curve 4: NA = 2x 10S4 mol/t, Q = 3.5 ns, curve 5: N, = 0, T = 4.2 ns (T=@is taken as time of l/e decrease of fluorescence intensity.
spectroscopy allows to determine the Fijrster radius R, expe~ent~ly by fitting experimental data to eq. (2). In doing this, we plotted lo&l,,(t)/J,(t)] versus t’/* (fig. 6), where IA(f), lo(t) are the donor fluorescence decays with and without the presence of acceptors in the solution. The data yield a straight line (at least at times greater than I ns), from the slope of which the critical transfer distance R, can be determined using eq. (2) for the various solutions (table 3). We obtained a dependence of the transfer distance R 0 determined in this way on the acceptor concentration NA at smaller acceptor concentrations. This dependence is, however, not explainable in terms of eqs. (1) and (2). Similar results have been reported by other authors [4,13]. They also observed a virtual dependence of R, on NA at acceptor concentrations smaller than the donor concentration. The experimental results obtained by the monochromatic measurements at the Nd-YAG spectrometer have confirmed with a smaller time reso-
I
I
2
1
7 thsl
,
3
I
L
t Inal
5
Fig. 6. _h$rmalized fluorescence intensity log[l,( t)/lA(t)] versus t”‘. f,,(t) is the decay of the donor fluorescence without acceptors in the solution, l,,(f) is the same in presence of acceptor molecules.
lution but higher accuracy in changes of the optical density our results of the picosecond broadband excite-and-probe-beam spectrometer (fig. 7). The single-shot streak camera was only used for qualitative m~surements. Because of the low dynamic range ( < 10) and relatively high noise levels the single-shot streak camera proved to be inferior in accuracy to all the other methods applied. normalized
A00 I
i3=660ml I
k=-+;_-___:_.
i
x
Ft=SMlnm
0.5 I’
200
100
600
BOO
c
1000 tips1
Fig. 7. Normahid excitation density of donor and acceptor molecules measured by a Nd-YAG probe-beam spectrometer N* = 1 X low4 mol/C). X = 660 mn probing CV fluores(NDcence, X = 560 nm probing Rh fluorescence and CV bleaching.
236
M. Kmchke, K. VogIer / Picosecond stuo)~ of energy transfer
Table 3 Critical transfer radius Ra calculated from the decay of the donor fluorescence in dependence on the acceptor concentration NA. The table is completed by synchroscan streak camera data from refs. 14,131. NA
R0
@01/O
(4
Rh/CV in EtOH Nn=1X10W3 mol/L theory b, 4x10-3 2x10-3 1x10-3 5x1o-4 3x10-4 5x10-s
56 50 52 61 70 13 94
1x10-3 5x10-4 1x1o-4
49 45 46 64
5x10-3 2x10-3 1 x10-3 5x10-4 2x10-4 lxlo-4 5x10-5 1 x10-5
59 45 46 48 69 84 91 120 152
4x10-3 4x10-4 3x10-s
24 33 50 75
Rh/DOTC in EtOH No=l~lO-~ theory b,
DODC/MG Nn=l~lO-~ theory b,
a/b
‘)
(3)
_ 1 2 9 18 22 57
in EtOH ‘)
HPD monomer/dimer in Hz0 d, ND=NA theory b,
The rate equation for the deactivation of the excited single donor differs from that of a completely isolated donor by an additional relaxation term, describing the deactivation due to long-range dipole-dipole energy transfer to the surrounding acceptors.
a) a/b is the ratio of raised to bulk density of acceptors as explained in section 5. b, Values calculated from spectral data using eq. (1). ‘) From ref. [13]. d, From ref. (41.
Here rk is the distance of the k th acceptor from the donor of interest. Spatial averaging over the acceptor distribution around the donors leads to the well-known t 1/2 law [eq. (2)] of Forster [21] which describes the deexcitation behavior of an ensemble of excited donor molecules after excitation by a S-like pulse. Some restrictive assumptions are made in deriving this decay function: (i) The acceptor molecules are homogeneously distributed around each donor of the ensemble. (ii) Donor and acceptor molecules are at fixed locations in space during interaction. (iii) Mutual influence between donor molecules is not taken into account. Assumptions (i) and (iii) imply directly that the donor concentration No is much smaller than that of the acceptors. It is worthy to note that because of N,, +z NA eq. (2) does not contain the donor density. In order to obtain the temporal evolution of the excited acceptor ensemble we follow the concept of Heber [22]. In treating the problem of excitation diffusion he introduced mean statistical rate equations for the donor and acceptor ensemble. These coupled differential equations can be derived from eq. (2) by derivation with respect to time. The mean statistical transfer rate XDA, responsible for the decrease of the mean donor excitation o(t) due to energy transfer, is assumed to be the same as the rate, which increases the mean acceptor excitation A(t).
5. Discussion In a first attempt we tried to explain our experimental findings in terms of the well-known Forster model [7]. In this model an isolated excited donor is considered, which interacts with a number ( NA) of surrounding acceptors, each in the ground state.
d,(t)/dt
= -r,%(l)
-k,,(r)@t),
dx((r)/dt
= -pi%
+k,,(t)~(t),
j&,,(t) Here
= &$JAR;Q3/27;‘/2t-‘/2 the mean
statistical
= pt-‘/=. transfer
rate
(4) is time
M. Kaschke, K. Vogier / Picosecond study of energy transfer
dependent in contrast to eq. (3). rA is the fluorescence decay time of the acceptor molecule. The ~nte~a~on of the second differential equation yields the solution for the time behavior of the acceptor density in the excited state [22]: X(t) = D(0)2@y-“2
exp( - t/r,)
[ w( py-li2)
+w((yt)*‘z
- /3y-1/2)
X exp( - g/r0 - 2/B */2 )] , where
w(x) = exp( - x2)ixexp( u”) du is the so-called Karpov integral and y = TA1 - 7; ‘. Eiq. (5) represents the ~o~te~~t to the common donor decay law [eq. (2)] and is a somewhat more complicated function, which describes the rise and the decay of the acceptor density in the excited state. Fig. 8 shows the time behavior of excited donor and acceptor molecules calculated from eqs. (2) and (5) for various acceptor concentrations and a fixed Forster radius of R, = 50 A. The maximum of the acceptor excitation density is reached not earlier than after a few hundred picoseconds, even
0
too
I
b
I
I
t
1
b
s
200
300
400
500
600
700
800
900
237
at the highest acceptor concentrations. Thus the solution (5) is not capable to explain our experimental findings of a fast rise of the acceptor excitation density (fig, 4). Further it should be noted that in terms of the Fijrster donor decay law (2) the observed concentration dependence of the critical transfer radius R. and NA (table 3) cannot be understood, because the quantum-theoretically derived Fbrster radius R, only depends on molecular and solvent parameters, but not on the acceptor concentration [cf. eq. (l)]. We propose a model for the long-range dipole-dipole interaction which in contrast to assumption (i) of the common Fiirster theory allows for an inhomogeneous distribution of the acceptor molecules around the donors. A similar model was developed in ref. [23] to explain succesfully experimental meas~ements of energy transfer between F-centers in solids. For large and charged molecules there exists a certain probability to aggregate and to form weakly bounded clusters [24]. Assuming a simplified stepfunction-like and spherosymmetric ~homogeneous dist~bution of the acceptor molecules around the donor (fig. 9) spatial averaging of eq. (3) over the inhomogeneously distributed acceptor ensemble (cf. ref. 1231) yields the following expression for the donor decay function, which corresponds to the donor fluorescence de-
1
1000 M
Fig. 8. Time behavior of donor and acceptor molecules after a S-function-like excitation pulse for several acceptor concentrations NA and R, = 50 A. (a) Donor decay (F&ster’s t ‘j2 law) [eq. (Z)], (b) solution of eq. (4) for the acceptor density in the excited state [eq. (5)]. Curve 1: NA = 1X1O-5 mol/d, curve2: NA=1~10W4 mol/Lp, carve 3: NA=1x10T3 mol/t”, curve4: N,=1~10-~ mol/e.
hf. Kaschke, K. Vogler / Picosecond study of energy transfer
238
cay: B(t)=B(O)
exp
- $ i
1
where
Q(d)>
= 1 -@(xi(t))+
is a somewhat complicated the error function +(-xi(t))
= 2v-‘/‘/x’(‘)exp(
1 - exp( _x,‘(t)) n,,zx_(t)
I function
-u’)
and includes
du,
0
with Xi( t ) = (R3,/R;7)(
t/7,)1’2.
The radii R, and R, denote the inner and outer radius of the sphere of raised acceptor concentration a. b is the bulk acceptor concentration in the solution. As schematically shown in fig. 9 such a spatial acceptor distribution may be induced by a static van der Waals interaction between solved molecules. For the limiting case R, --* 0 and a + b eq. (6) coincides with the usual Fijrster formula [eq. (2)]. The appearance of the Q-functions in eq. (6) yields a rather rapid decay of excited donors as long as the unequality xi(t) = Rz/Rj(t/rD)“* +z 1 holds after excitation by a S-like pulse (fig. 1Oa). If the time t is of the order of the fluorescence decay time (i.e. xi(l) 2 5) the time dependence of the function Q(xi(t)) can be neglected (Q(+) = QC = 0.1 and for R, CR,: Q(x,)= 0) yielding again the common Fiirster law, but with a modified transfer radius:
&f = This expression effective Fijrster
Fig. 9. Spatial distribution of acceptors around one excited donor. (a) Homogeneous distribution (conventional Fiirster model), (b) inhomogeneous distribution and a corresponding intermolecular interaction potential. a, b: increased and bulk density of acceptor molecules, RI, R,: radii of sphere of raised acceptor concentration (sphere 0 --) R, represents the forbidden volume), R,: critical transfer radius.
was set equal to the measured radii R, from table 3. For rea-
sonable values of R, and R, the ratio a/b is calculated from (7) and listed in table 3. Thus the concentration dependence of R, on NA (or b) can be reduced on a corresponding dependence of the ratio of the raised acceptor density to the bulk density. Another possible approach in explaining the observed deviations from the Fiirster law eq. (2) will be given elsewhere [25]. In order to obtain the expression for the time behavior of the excited acceptor ensemble in the case of an inhomogeneous distribution of molecules the same procedure as before is applied for deriving the mean statistical rate equations from the donor decay function (6). This results in the following coupled differential equations:
In comparing these rate equations with those of the homogeneous distribution [eq. (4)] the rather difficult time dependence of the mean transfer rate is obvious. Taking the solution (6) for the donor decay the time behavior of the excited acceptor
M. Kaschke, K. Yogler / &&second stuay of energy transfer
239
a
0
tbni)
0
100
200
300
400
500
600
700
600
900
1000
Fig. 10. Time behavior of donor and acceptor molecules after a b-function-Iike excitation pulse spatial distribution of acceptor molecules (N, = b =l X lo-’ mol/C). (a) Donor decay function density [integral of eq. (7)]. Parameters for (a), (b) curve I: a/b = 5, curve 2: a/b = 10, curve R,=lOA, R,=25& R,==50A. For (c) curve 1: R,=15ik, curve 2: R, ==2OA, curve 3: a/b = 15, R, = 50 A, R, = 10 A {curve 0 corresponds to the homogeneous case).
ensemble has been calculated by numerical integration. The resulting time functions of the excited acceptor density after a S-function-like excitation of the donors are shown in figs. lob and 1Oc in dependence on R, and a/b, respectively. The dependence on R, is not si~ifi~t, if the forbidden volume is small enough (i.e. R, -c10 ;i>.
in the case of an inhomogeneous [eq. (S)], (bf, (c) excited acceptor 3: a/b = 15, curve 4: a/b - 25, R,=25& curve 4: R,=30A,
As seen from these calculated curves much shorter rise times of the mean acceptor excitation density than in the original Fiirster case (fig. 8b) are obtained. Thus the rapid rise in the population of the excited state of the acceptor can be fitted fairly well by variation of the parameters R, and a to the experimentally observed data (fig. 4). An
hf. Kaschke, K. Vogler / Picosecond study of energv transfer
240
Acknowledgement We would like to thank Professors M. Schubert and B. Wilhelmi for their stimulating interest in this work and for many helpful discussions.
References
L
10
0
10
20
30
40
50
60
70
80
90
‘00
&I
Fig. 11. Fitting of the numerical solution of eq. (7) for the acceptor rise function to the experimental curve (example). Solid line experimental curve, dashed line calculated curve (a/b = 25, R2= 20A, R, =lOA, R,=50&.
example of this fitting is depicted in fig. 11. Taking the value for R, fixed at 10 A we get very good agreement between the theoretical and the experimental curve for R, = 25 A and a/b = 10. This corresponds to a depth of the Lennard-Jones potential of 2.3 kT which seems quite reasonable.
6. Summary The investigation of several laser dye mixtures by means of time-resolved absorption and fluorescence spectroscopy revealed a rapid and effective energy transfer to the acceptor after picosecond excitation of the donor. The experimental results deviate significantly from the predictions of the common Forster model. Similar observations have been reported earlier [4,13]. Our experimental findings are tentatively explained in terms of a modified Fiirster model, in which one of the basic assumptions of the Fiirster model has been extended to include an inhomogeneous distribution of molecules in solution. However, the range of validity of this modified Forster model is still restricted by the assumption NA % No, which is a severe limitation because in many applications the opposite case NA -=z No is to be found. The abandonment of this assumption and the consequences therefrom will be a matter of investigation in a forthcoming paper [25,26].
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