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Numerical investigations on the experimental accessibility of “instantaneous fluorescence quantum yield”
JOURNALOF
LUMINESCENCE
--
EL.SEVIER
Journal of Luminescence 60&6l (1994) 895 898
Numerical investigations on the experimental accessibility of “instantaneous fluorescence quantum yield”t J. Eh1ert~,D. Leupold, S. Oberländer’, K. Teuchner Mav-Born-Institut fur Nichtlineare Optik und Kurzzeitspektroskopie, Rudower Chaussee 6, D-12474 Berlin, Germany
Abstract By numerical investigations it is shown that the content of information of the instantaneous fluorescence quantum yield, recently defined at the end of a rectangularexcitation pulse, is conserved if this quantity is related to the maximum of a proper chosen Gaussian pulse.
1. Introduction Recently, it has been shown that the intensity dependence of the so-called instantaneous fluorescence quantum yield, 4~, introduced on the basis of analytical investigations, is a very sensitive function of excited state processes in systems of fluorescent organic molecules [1]. Adapted to a three-level model with ground state (a12) and first excited state (a23) absorptions, induced emissions (a21 a12 and a32 a23) as eous =radiative relaxation ratewell k as the spontan25, the quantity ~ is defined at the end (t = t) of a rectangular pulse of intensity I in the following manner: i
‘YlflSk
/
—
I[a12(S1
—
k21 S2 S2) + a23(S2
—
53)]
‘,
J
(S1, S2, S3 are the normalized dimensionless population densities). Besides, the molecular level model used in [1] includes relaxations k31 and k32, resp., as well as bimolecular interaction of first excited singlets, which results in population of the levels 1 and 3, and which is characterized by an annihilation constant y. (Due to the normalization of the population densities, y is connected with the usual annihilation rate T 1via the volume density of molecules; y NT.) Depending on the values and relations of these molecular parameters, at least eight fundamentally different shapes of the ~ versus I-curves can be expected. Points of their difference are combinations of the principal possibilities lim 4~
*
Corresponding author, Dedicated to Prof. Wolfhard Rüdiger on the occasion of his 60th birthday. Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39, D-10117 Berlin, Germany.
~~ins,
u ~
lim ~
=
Extensions of the model effect simple extensions of the ~ term, e.g. inclusion of radiationless relaxation of the first excited state causes only a multiplicative constant.
0022-2313/94/507.00 C 1994 Elsevier Science B.V. All rights reserved SSD1 0022-2313(93)E045l-3
1/96
.1 EIili’i t it at
./ournal at 1 uni,neii
as well that with increasing I these limit values can be left and reached, resp.. on a decreasing or increasing curve each [I]. This manifold characterizes the high content of information of the new quantity ~~.jI): the various shapes with maxima. minima and or inflection points reflect the sensitiv ity of q~~5(I)to all excited state parameters and offer specific fingerprints, e.g. on annihilation [1]. This is in remarkable contrast to the insensitive, uniform shape of the usually measured intensity dependence of the fluorescence yield (cf. below). In principle, the numerator of (I) is available from a measurement of the instantaneous fluorescence intensity from the entrance plane of the sample at the moment t r of the exciting rectan gular pulse. The term in brackets in the denominator can be obtained from a transmission measurement at t r in the entrance plane, orthogonal to the direction of the exciting-pulse propagation. In the following, possible changes of these re quirements will be studied numerically, which simplify the experiment conserve of information as much asbutpossible. To the this content end. fluorescence measurements at the maximum of a Gaussian pulse (of variable intensity and width) and instantaneous transmission measurements with that exciting pulse at an optical thin layer have
60&6 /
ins
1994
~9
591/
Table I The rate ~quatiuns for the underlying model and the procedure for their numerical solution Rati’ l’i/iiaIioii
d
di
Shill
41(1 + B/SO)
SO)
vS~)t S/U)
+
5(h)
(.S )r), S~O)S ~)i0’ sector
1(1)
!,,~t)1).
‘(1).
of
S
(1.0. (II’
population densities
l~, excitation pulse peak intensity
temporal pulse shape, max °‘~
(t)
I
°‘
0 0 B
/.
1)
k,
~
0
k11
k~
II
\‘u,ni’rica/ procedure ti l)t~)with b~ Gtxen Fo calculate
k
Sk.
k/i and step sue/i fork
the numerical
approximation
0. ol S/i
1,2.
Trapezoidal rule CS5 C
E
d
+
e[St]’ with
Al 5
been found to be a good compromise.
B (E identity matrix),
1
d
2. Numerical procedure, definitions To calculate physical quantities as (1) numer ically, the rate equation system for the population densities has to be solved. Because these differential equations may be stiff, the trapezoidal rule is used which is A-stable but implicit. Table I shows a short description of the numerical procedure for the underlying model. At the end of this procedure one obtains approximated population densities as a function of the pulse peak intensity ‘max and the temporal pulse shape. If in the following, rectangular and Gaussian pulses will be compared, we denote ~ for a rectangular pulse (pulse duration rR, I Ig) and ~ for a Gaussian pulse (pulse duration r6, t at the maximum). To make these quantities comparable,
(E v \ 6 + ~41~ ~ B)S~ I + , v[S~ l]2 e Solving Ca dand Ch cone obtains Sk a + hFSfl’ and St
(I +
S~ a1
+
.,,
I
4a2h2) 2h2 (choose physical solution),
2
h~[St]’, S~
a1 +
h1fS~]
the time-integrated pulse shapes must be equal, this means 1R 0.53 i~. Moreover, for the Gaussian pulse the quantum yield r
S~dt ~(1max)
IT
I[a 1,(s,
S-,) + a25(S2
S5)]dt
0
(2)
J. Ehlert er at.
10
—
Journal of Lunnne3cence 60&61 (1994) 895 898
897
1,0
.-.—_..~,
\.
\
.5
5’ ‘/.‘
\~\\\\\\
05
1019
1022
0,5
1025
2s1
In,ox lcm
]
1028
1021
1024
~
[cm2s1]
Fig. I. Calculated Im~, dependencies (Gaussian pulse. — 1.5 x 10 iO~) of the quantities ~ and ~“ resp., for a molecular three-level model with 7i2 — 02i 10 i3Cm2 023 — 032 10 iSCm2 k 2~ l0~s I, k32 10~~ ~, k3~ 0, ~‘ 5 x 10~s (ci. text for further explanations).
Fig. 2. Calculated ‘ma, dependencies (Gaussian pulse, r 9s) of the quantities ~ and ip, resp., for a 0 1.5 X 10three-level model with 0i2 molecular 02i 10 ~ cm2 7cm2, k 9s ~, 10~‘ 1.4 x ~21 032 21 1.4 x io~ s ~, k32 k3~ 2.~x 10 10~~5 (cf. text for further explanations).
and the fluorescence yield
pulse (with pulse duration tG = (0.53) ‘tR). In search for a further simplification of the experimental conditions, we calculated in all cases also the quantity 4 according to (2), but in general this time in integration in a underlines distinct losstheofspecial structure the curve results shape. This feature of the new quantity As an example, for an i~ 1~9(I)-curve of type 7 of the nomenclature of [1] ~ o > 4~, ~ 4~,~ left increasing; q5~,~ reached decreasing) in Fig. 1 are shown both the 4~s(Imax)-curve(which is quasi identical with the 4~~(I)-curve) and the corresponding 4~(1max) curve. Also shown is the corresponding curve for the (usually measured) fluorescence yield tP(’rnax) according to (3). Here as in all cases investigated, ~P(’rnax) has by far the least content of information (the lesser the structure of the curve, the lesser is the information). It is worthy to note that there are few cases of an acceptably small difference between çb~~(I~~,) and 4~(’max)’ An example is shown in Fig. 2 which represents type 4 of the above-mentioned nomenclature ~ o > /~,~ 4~, left decreasing; ~
j
(3) and (P(Irnax) will be norare discussed. malized to t/(O) 1 and q(0) = 1, resp., with normalization constants of proper dimension. The ~PUrnax)
=
j
S2 dt
4~(’max) —
upper bound T must be chosen sufficiently large to ensure that all molecules have been returned to the ground state at this time.
3. Results Examples of each of the above-mentioned eight types of ~ versus I-curves can be realized numerically by a proper choice of the molecular as well as the pulse parameters. In all cases investigated so far we have found cb~~(1) ‘/~s(Imax), where the difference, if ever, is small. This means that, without loss of information, the experimentally impracticable rectangular pulse can be replaced by a Gaussian ‘~‘
~
1/91/
J. Ehieri eb at
Journal of Lumoue,scenc c’ 60&6 /
reached increasing). Its typical shape with a millimum is even conserved if 41(Imax) is calculated for a certain sample thickness x > 0 (note that (1) is defined at ~ 0), which is of practical interest,
4. Conclusions The numerical investigations have shown that the instantaneous fluorescence quantum yield, originally defined at the end of a rectangular pulse, can he well apprnximated hy a cnrresponding quantity related to the maximum of a Gaussian pulse. The realization in the experiment seems not to be simple but possible. An intensity-variable exciting pulse focused on the sample by means of a cylindrical lens as well as a weak pulse for probing the transmission perpendicular to the excitation (with
/ 994 891/ 89,1/
in x U, e.g., at the edge of the sample cell) have to be exactly synchroniicd in time. The fluorescence has to be measured from the front side of the sample (v 0). Both fluorescence and transmission have to be measured simultaneously at the time of the exciting pulse maximum: the numerator of(l) is determined from the fluorescence and the denomin ator from the transmission.
Acknowledgement This work was supported by a grant from the Deutsche Forschungsgemeinschaft (Le 729-1 1).
References [I] S Oberländei and D leopold. J I umun 1/9 1994/
(15
LUMINESCENCE
--
EL.SEVIER
Journal of Luminescence 60&6l (1994) 895 898
Numerical investigations on the experimental accessibility of “instantaneous fluorescence quantum yield”t J. Eh1ert~,D. Leupold, S. Oberländer’, K. Teuchner Mav-Born-Institut fur Nichtlineare Optik und Kurzzeitspektroskopie, Rudower Chaussee 6, D-12474 Berlin, Germany
Abstract By numerical investigations it is shown that the content of information of the instantaneous fluorescence quantum yield, recently defined at the end of a rectangularexcitation pulse, is conserved if this quantity is related to the maximum of a proper chosen Gaussian pulse.
1. Introduction Recently, it has been shown that the intensity dependence of the so-called instantaneous fluorescence quantum yield, 4~, introduced on the basis of analytical investigations, is a very sensitive function of excited state processes in systems of fluorescent organic molecules [1]. Adapted to a three-level model with ground state (a12) and first excited state (a23) absorptions, induced emissions (a21 a12 and a32 a23) as eous =radiative relaxation ratewell k as the spontan25, the quantity ~ is defined at the end (t = t) of a rectangular pulse of intensity I in the following manner: i
‘YlflSk
/
—
I[a12(S1
—
k21 S2 S2) + a23(S2
—
53)]
‘,
J
(S1, S2, S3 are the normalized dimensionless population densities). Besides, the molecular level model used in [1] includes relaxations k31 and k32, resp., as well as bimolecular interaction of first excited singlets, which results in population of the levels 1 and 3, and which is characterized by an annihilation constant y. (Due to the normalization of the population densities, y is connected with the usual annihilation rate T 1via the volume density of molecules; y NT.) Depending on the values and relations of these molecular parameters, at least eight fundamentally different shapes of the ~ versus I-curves can be expected. Points of their difference are combinations of the principal possibilities lim 4~
*
Corresponding author, Dedicated to Prof. Wolfhard Rüdiger on the occasion of his 60th birthday. Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39, D-10117 Berlin, Germany.
~~ins,
u ~
lim ~
=
Extensions of the model effect simple extensions of the ~ term, e.g. inclusion of radiationless relaxation of the first excited state causes only a multiplicative constant.
0022-2313/94/507.00 C 1994 Elsevier Science B.V. All rights reserved SSD1 0022-2313(93)E045l-3
1/96
.1 EIili’i t it at
./ournal at 1 uni,neii
as well that with increasing I these limit values can be left and reached, resp.. on a decreasing or increasing curve each [I]. This manifold characterizes the high content of information of the new quantity ~~.jI): the various shapes with maxima. minima and or inflection points reflect the sensitiv ity of q~~5(I)to all excited state parameters and offer specific fingerprints, e.g. on annihilation [1]. This is in remarkable contrast to the insensitive, uniform shape of the usually measured intensity dependence of the fluorescence yield (cf. below). In principle, the numerator of (I) is available from a measurement of the instantaneous fluorescence intensity from the entrance plane of the sample at the moment t r of the exciting rectan gular pulse. The term in brackets in the denominator can be obtained from a transmission measurement at t r in the entrance plane, orthogonal to the direction of the exciting-pulse propagation. In the following, possible changes of these re quirements will be studied numerically, which simplify the experiment conserve of information as much asbutpossible. To the this content end. fluorescence measurements at the maximum of a Gaussian pulse (of variable intensity and width) and instantaneous transmission measurements with that exciting pulse at an optical thin layer have
60&6 /
ins
1994
~9
591/
Table I The rate ~quatiuns for the underlying model and the procedure for their numerical solution Rati’ l’i/iiaIioii
d
di
Shill
41(1 + B/SO)
SO)
vS~)t S/U)
+
5(h)
(.S )r), S~O)S ~)i0’ sector
1(1)
!,,~t)1).
‘(1).
of
S
(1.0. (II’
population densities
l~, excitation pulse peak intensity
temporal pulse shape, max °‘~
(t)
I
°‘
0 0 B
/.
1)
k,
~
0
k11
k~
II
\‘u,ni’rica/ procedure ti l)t~)with b~ Gtxen Fo calculate
k
Sk.
k/i and step sue/i fork
the numerical
approximation
0. ol S/i
1,2.
Trapezoidal rule CS5 C
E
d
+
e[St]’ with
Al 5
been found to be a good compromise.
B (E identity matrix),
1
d
2. Numerical procedure, definitions To calculate physical quantities as (1) numer ically, the rate equation system for the population densities has to be solved. Because these differential equations may be stiff, the trapezoidal rule is used which is A-stable but implicit. Table I shows a short description of the numerical procedure for the underlying model. At the end of this procedure one obtains approximated population densities as a function of the pulse peak intensity ‘max and the temporal pulse shape. If in the following, rectangular and Gaussian pulses will be compared, we denote ~ for a rectangular pulse (pulse duration rR, I Ig) and ~ for a Gaussian pulse (pulse duration r6, t at the maximum). To make these quantities comparable,
(E v \ 6 + ~41~ ~ B)S~ I + , v[S~ l]2 e Solving Ca dand Ch cone obtains Sk a + hFSfl’ and St
(I +
S~ a1
+
.,,
I
4a2h2) 2h2 (choose physical solution),
2
h~[St]’, S~
a1 +
h1fS~]
the time-integrated pulse shapes must be equal, this means 1R 0.53 i~. Moreover, for the Gaussian pulse the quantum yield r
S~dt ~(1max)
IT
I[a 1,(s,
S-,) + a25(S2
S5)]dt
0
(2)
J. Ehlert er at.
10
—
Journal of Lunnne3cence 60&61 (1994) 895 898
897
1,0
.-.—_..~,
\.
\
.5
5’ ‘/.‘
\~\\\\\\
05
1019
1022
0,5
1025
2s1
In,ox lcm
]
1028
1021
1024
~
[cm2s1]
Fig. I. Calculated Im~, dependencies (Gaussian pulse. — 1.5 x 10 iO~) of the quantities ~ and ~“ resp., for a molecular three-level model with 7i2 — 02i 10 i3Cm2 023 — 032 10 iSCm2 k 2~ l0~s I, k32 10~~ ~, k3~ 0, ~‘ 5 x 10~s (ci. text for further explanations).
Fig. 2. Calculated ‘ma, dependencies (Gaussian pulse, r 9s) of the quantities ~ and ip, resp., for a 0 1.5 X 10three-level model with 0i2 molecular 02i 10 ~ cm2 7cm2, k 9s ~, 10~‘ 1.4 x ~21 032 21 1.4 x io~ s ~, k32 k3~ 2.~x 10 10~~5 (cf. text for further explanations).
and the fluorescence yield
pulse (with pulse duration tG = (0.53) ‘tR). In search for a further simplification of the experimental conditions, we calculated in all cases also the quantity 4 according to (2), but in general this time in integration in a underlines distinct losstheofspecial structure the curve results shape. This feature of the new quantity As an example, for an i~ 1~9(I)-curve of type 7 of the nomenclature of [1] ~ o > 4~, ~ 4~,~ left increasing; q5~,~ reached decreasing) in Fig. 1 are shown both the 4~s(Imax)-curve(which is quasi identical with the 4~~(I)-curve) and the corresponding 4~(1max) curve. Also shown is the corresponding curve for the (usually measured) fluorescence yield tP(’rnax) according to (3). Here as in all cases investigated, ~P(’rnax) has by far the least content of information (the lesser the structure of the curve, the lesser is the information). It is worthy to note that there are few cases of an acceptably small difference between çb~~(I~~,) and 4~(’max)’ An example is shown in Fig. 2 which represents type 4 of the above-mentioned nomenclature ~ o > /~,~ 4~, left decreasing; ~
j
(3) and (P(Irnax) will be norare discussed. malized to t/(O) 1 and q(0) = 1, resp., with normalization constants of proper dimension. The ~PUrnax)
=
j
S2 dt
4~(’max) —
upper bound T must be chosen sufficiently large to ensure that all molecules have been returned to the ground state at this time.
3. Results Examples of each of the above-mentioned eight types of ~ versus I-curves can be realized numerically by a proper choice of the molecular as well as the pulse parameters. In all cases investigated so far we have found cb~~(1) ‘/~s(Imax), where the difference, if ever, is small. This means that, without loss of information, the experimentally impracticable rectangular pulse can be replaced by a Gaussian ‘~‘
~
1/91/
J. Ehieri eb at
Journal of Lumoue,scenc c’ 60&6 /
reached increasing). Its typical shape with a millimum is even conserved if 41(Imax) is calculated for a certain sample thickness x > 0 (note that (1) is defined at ~ 0), which is of practical interest,
4. Conclusions The numerical investigations have shown that the instantaneous fluorescence quantum yield, originally defined at the end of a rectangular pulse, can he well apprnximated hy a cnrresponding quantity related to the maximum of a Gaussian pulse. The realization in the experiment seems not to be simple but possible. An intensity-variable exciting pulse focused on the sample by means of a cylindrical lens as well as a weak pulse for probing the transmission perpendicular to the excitation (with
/ 994 891/ 89,1/
in x U, e.g., at the edge of the sample cell) have to be exactly synchroniicd in time. The fluorescence has to be measured from the front side of the sample (v 0). Both fluorescence and transmission have to be measured simultaneously at the time of the exciting pulse maximum: the numerator of(l) is determined from the fluorescence and the denomin ator from the transmission.
Acknowledgement This work was supported by a grant from the Deutsche Forschungsgemeinschaft (Le 729-1 1).
References [I] S Oberländei and D leopold. J I umun 1/9 1994/
(15