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Instantaneous fluorescence quantum yield of organic molecular systems: information content of its intensity dependence
JOURNAL OF
LUMINESCENCE
--
ELSEVIER
Journal of Luminescence 59 (1994) 125—133
Instantaneous fluorescence quantum yield of organic molecular systems: information content of its intensity dependence S. Oberlãndera,*, D. Leupoldb Institute of Applied Analysis and Stochastic, Hausvogteiplatz 5-7, D-1O1 /7 Berlin, Germany ~Max-Born-Institute for Nonlinear Optics and Short Pulse Spectroscopv, Rudower Chaussee 6, D-12489 Berlin, Germani’ a
(Received 12 May 1993; revised 22 September 1993; accepted 3 November 1993)
Abstract In pulse laser investigations of systems of fluorescent organic molecules (e.g. aggregates, light harvesting antennas) it is usual to derive from the intensity dependence of time-integrated fluorescence (fluorescence yield) conclusions on deactivation channels, bimolecular interaction and sizes of interacting units (“domains”), though the content of information of this type of measurement is restricted because the shape of this function is hardly structured. Especially when further processes take place (e.g. excited singlet absorption) problems of ambiguity arise. Based on analytical investigations of the adapted rate equation system of noncoherent nonlinear light-matter interaction, especially of the asymptotic behaviours of its solution with respect to the excitation intensity, we propose a new measuring quantity “instantaneous fluorescence quantum yield”, which in its intensity dependence reflects far more sensibly the several excited state processes and therefore should allow the characterization of absorption, relaxation, and annihilation quantitatively and in a unique way.
1. Introduction In fluorescent systems of interacting uniform organic molecules with it-electron systems (aggregates, clusters, light harvesting antennas) the fluorescence quantum yield may depend on excitation intensity for two principal causes: excited singlet absorption and excited singlet annihilation (cp. Fig. 1). In the usual experimental situation, the dependence of the time-integrated fluorescence intensity on the pulsed excitation fluence is measured. The fluorescence yield function ~F(I) obtained in this way is identical to the fluorescence quantum yield only as long as the corresponding absorption of the *
Corresponding author,
molecular system measured under the same pulsed excitation conditions is a linear function of the excitation fluence.
The typical shape of a fluorescence yield function, obtained e.g. with the well-known pseudoisocyanine aggregates [1] as well as with antennas from photosynthetic bacteria [2] and from higher plants [3,4], is shown in Fig. 2. It is worth mentioning that this typical monotonously decreasing curve can be observed in cases of either excited state absorption or exciton annihilation alone, as well as under conditions of simultaneous operation of both processes; there is no specific “fingerprint” Though relatively unspecific in its general ~.
Concerning the relevance of the question of excited state absorption versus exciton annihilation, cf. e.g. Ref. [5].
0022-2313/94/$07.00 © 1994 — Elsevier Science B.V. All rights reserved SSD! 0022-231 3(93)E0488-J
1 26
S. ()her/dus/e,, I). Leopold
I ~23
Joioiio/ of Liu,i9u oenee 59
questionable because of very simplifying assumptions of the parameter identification procedure k
32
2
______________ - --
~
125 /33
In many cases, the uniqueness of these results is
ç
032
1994
~2l
1
____________
-
b) Fig. I. Three level model of interacting uniform (a 6) fluorescent organic molecules. The interaction is represented by singlet singlet annihilation ): besides this, the excitations and deactivations are characteriied by induced processes ~ absorption and emission I and spontaneous relaxation processes k,. whereat each k~,in principle may consist ofa radiative as well as a nonradiative part. For simplicity, in the following it ~ assumed that k2, is the only pure radiative relaxation. For the sake of clearness. oa processes are indicated only in the left part al and k5 processes are indicated only in the right part (h)
-
~Ff
~
Fig. 2. Typical shape of an experimentally determined floorescence yield function ~. from systems of interacting organic molecules (aggregates, photosynthetic antennas) at pulsed cxcitation with variable intensity I.
shape, these fluorescence yield curves are used e.g. as one of the main sources of information on the photophysics of photosynthetic antenna systems )for reviews see Refs. [6,7]).
(two level systems, s-shaped excitation pulse). But even if more adequate models and the real excitation pulse shape are used in the numerical simulation of the experimentally determined functions. the ambiguity remains, and this is even true if in addition the nonlinear absorption is measured. which means that the intensity dependence of fluorescence quantum yield is known [1] 2 In addition. photosynthetic objects seldom fulfill the supposition of a uniform sample (spectral heterogeneity of photosynthetic antennas, antennas with reaction centers). Attempts to simulate fluorescence yield and fluorescence decay functions of these samples on the basis of more adapted models showed the necessity of additional experimental quantities to reduce these ambiguities [13.14]. Of course, the danger of over-interpretation of the fluorescence yield functions is not restricted to photosynthetic objects. Considering this, one may ask whether the usual 5F(i) really exhausts the experimental quantity ~ principal content of information of fluorescence quantum yields. In other words: is there a fluorescence quantum yield-like quantity which is more informative and also experimentally accessible’? As is shown in the following, the answer is yes: based on analytical investigations of the differential equation system describing noncoherent light-matter interaction, adapted to the above mentioned ~-eIectron systems, an instantaneous fluorescence quantum yield will be defined, which represents in different parts of the function a high sensitivity to excited state absorption, to relaxation processess, to annihilation, and offers a specific annihilation fingerprint.
2. Model, rate equations and definition of instantaneous fluorescence quantum yield ~,
Quantities derived
from these fluorescence yield functions are (cp. Fig. l)the relaxation rates k 2~,k5~,k32, the bimolecular decay (annihilation) rate ;‘, the excited state absorption cross section a23 as well as the number of interacting photosynthetic units (size of domains) (see e.g. Refs. [1.8—Il]).
Each molecule of the fluorescent system is represented by a three level model shown in Fig. I. It 2
‘fhc same problems of ambiguity of parameter identification hold for the second standard experimental quantity of photosynthetic objects, the excitation intensity’ dependence of tluorcsccnce decay [121.
S. Oherlander, D. Leupold / Journal of Luminescence 59 (1994)125—133
127
consists of the ground state (1), the fluorescent first excited singlet state (2) and a higher excited singlet (3). To demonstrate the progress with the new quantity ~. as lucidly as possible, the vibronic substructure of these levels as well as Franck—
This pulse may traverse the sample orthogonal to the entrance plane. For this entrance plane we define the instantaneous fluorescence quantum yield ~r~s at t = t in the following manner:
Condon levels are neglected (but there is in principle no mathematical restriction with respect to more extended models). When interacting with a monochromatic radiation field 1(t) with photon energies equal to the (equal) differences between the energy levels, these levels are connected by absorption as well as induced emission processes (characterized by cross sections ~ and a~respectively, a~= afi), by relaxation processes (described by overall rate constants k~1,which may contain radiative as well as radiationless contributions) and by bimolecular interaction of two first excited singlets which results in one ground state and one higher excited singlet
~~.(I)
(characterized by the annihilation constant y, compare below). If this light-matter interaction can be described by the rate equation approach, one gets the following differential equation system for the population densities S~: =
52 =
—
ai~1(S~ — S2) + k2~S2 + k3153 +
cri2l(Si
+ k32S3 S3
=
—
—
S2)
—
a23 I(S2
—
S3)
—
~2’
k2~S2
2yS~,
a23 1(S2 —S3)
—
(k3~+ k32)S3 + yS~,
with the normalization S~+ S2 + S3 what follows
=
1, from
N’F,
whereby N is the volume density of molecules and F the usual annihilation constant in units volume/time. 3, the time course of the excitaIn isthe following tion assumed to be a single rectangular pulse of width ‘r, i.e. Iconst.
1(t)
~o,
>
0,
if t E [0, else.
t]
With respect to the following definition of ~,, the time course ofexcitation at t > s has no relevance. We use this rectangular pulse description only for the sake of simplicity,
=
I[a12(Si
—
k21S2 S2) + a23(S2
—
S3)]
Notwithstanding its definition at a definite (but variable) instant of time t = t (where the population densities are the result of “history”), the quantity ~. fulfills the convention of quantum yield definitions in case of multiphoton processes [15]. The large content of information of this new quantity is demonstrated in the following section with the help of mathematical analysis.
3. Asymptotic behaviour of ~ to excited state processes
and its sensitivity
If the exciting radiation field enters the matter at the time t = 0, we have in the entrance plane the initial conditions S~(0)= 1, S~(0)= S3(0) = 0. To these initial conditions exists one solution of the rate-equation system. At first we shall investigate with help of mathematical analysis the asymptotic behaviour of this solution with respect to I for = t >0. Secondly we shall obtain from this the asymptotic behaviour of ~ Thirdly we shall see unexpectedly that at least eight different types of ~ curves exist. Such a way, successful by many problems, is provoking two questions by experimentalists: (i) Why asymptotic investigations? One cannot measure for I 0 or I cxc. Answer: knowledge the asymptotic behaviour of a The function leads toofinformation about their global behaviour including measurable ranges. (ii) Why mathematical analysis? One can get each desired curve most quickly with any computer. Answer: One can. May be one finds even some
t
—~
—~
different types of curves. But it is not easy, to find out the parameter ranges for any type. (In this paper, after some normalizations, one has to find
1 2)~
S. Oherldnder. D. Leupo/sl / Journal of Luniine,rcenc’e 59 (1994, /25- /33
ranges in a five-dimensional field for eight types of curves.) And it is impossible after calculation of a finite set of curves to clear up the behaviour of the remaining infinite set. It is impossible to get any general structural information, Therefore we make use of mathematical analysis. Therefore we investigate the asymptotic behaviour of interesting functions, at first of the S,. To describe their asymptotic behaviour, we try an ansatz of a polynomial of the order p in I for low and in J for high intensities, p ~ 0. If we can show that the asymptotic behaviour of S has such a structure, p should be chosen as small as possible but great enough for getting the desired information. We start with the ansatz -
asymptotic representation for ~.
=
i
=
a
a231 el~~r + f(’r)~— + k2~
—
S,
~
/a231\’ a~
--0
~
=
=
~ hi., ~=0
+
+ R5~, for high intensities (2) .
.
1,
0(12)
(3)
2
22
—
/ (1 ~
—
2ai2~(1 a,~/
—
e
kilt)
From asymptotic representations (2) for S~,S2. and S3 with p = 2 one gets
l~=
k21 + 2k31 + k32 (2k31 + k32)(k3~ + k32
R~~1for low intensities. (I)
-
=
with
+
5,
with p
~,
+
—
-h’)
k,1
~
(k2~+ 2k3~+ k32)” O(I~2)
a731 (4)
Though p =it 2. the derivation of (instead this expression is easier than seems to be because of the h,,) the differences h,1 h22, h2-, are needed only. From Eqs. (3) and (4) one can take limit values —
with polynomial coefficients a~1,~ and error terms ~ ~ One can show that a~,h~,exist so that
~-,
=
lim~~. = I
—
e
I ‘0
Rapi ,/ I”
—*
0
R5~1V’_3O
if I ~
0,
and =
~‘
~
-
k,~+ 21<31
-,
Therefore with such coefficients the polynomials of the order p describe the asymptotic behaviour of s, with an error less than terms of the order p + I, that means, with an error term denoted usually and in the following as Q(JP~ for low and as O(I~ i) for high intensities. It holds i)
a1,1 = S,(O), b01
=
1/3.
For v > 0 the a~1depend on t. These functions a,1(t) can be calculated recursively as solutions of linear differential equations using the initial conditions a~~(0) = 0. The b,,1 are independent of ‘r for v > 0 too. These values can be calculated recursively as solutions of linear algebraic equations using the additional conditions + h,,2 + h,3 = 0. Fromh,,~ asymptotic representations (1) for S~with p = I and for S2 with p = 2 one can derive an
+ 1<32
and one is led to Statement >
1:
~,
~ <
as well as
~
~
/~, is possible
From Eq. (4) one is led to Statemenf 2: With increasing intensity I the limit value i~ will be reached for < 3(k31 + k32) on a decreasing q’3~~~(I) curve. for ;‘ > 3(k31 + 1<32) on an increasing ç~~,(I) curve For the function f(t) in Eq. (3) it holds f(0)
=
0.
(
a~’
‘~
/
~~~)<°‘ limf(t) = I + a23~)~ With the last, one is led from Eq. (3) to —
S. Oberlander, D. Leupold/ Journal of Luminescence 59 (1994) 125—133
129
Statement 3.1: With increasing intensity I the limit value t1~,~ will be left on a decreasing ~iflS(i) curve for sufficiently long pulses.
meters (in the parameter range of a type, conditions see once again in the Appendix), the existence of an even number of further extrema cannot be excluded. Numerical examples have shown that, for
In the case of c~2~ a23 it holds f(t) < 0 for all t > 0. If a12 > a23, in the case of 2at2 ~ 3a23 it holds either < 0 for all tof>sign, 0, ororf(’r) f(t) has one positive zero f(’r) without change has two positive zeros with changes of sign. In the case of 2ai 2 > 3a~3f(r) has one and only one positive zero with change of sign. With the last, one is led from Eq. (3) to
instance, the types 5, 6 and 7 (dashed in Fig. 3(b)) can exist also with two extrema more, see4~~~(I) Fig. 3(c). are In this sense at least even eleven types of possible.
Statement 3.2: In the case of 2ai2 > 3a~ with increasing intensity I the limit value q5~~ will be left on an increasing ~ curve for sufficiently short pulses.
By comparison of the variety of ~ curves (Fig. 3) with the only representative type of curve obtained with the usual time integrated quantity tI3F(I) (Fig. 2), the progress with the new quantity 4~,(I)is obvious: more structure means more information. As is indicated by the analytical expressions given above, çb1~,(I)depends in a very sensible manner on the several excited state processes and on the width of the exciting rectangular pulse. Therefore it can be expected that this new quantity, if experimentally accessible, should give an essential aid in the unique characterization of processes like excited state absorption, exciton annihilation, and excited state relaxation of uniform molecular systems. Whereas e.g. a fingerprint of excited state absorption is available in the initial-range of
The statements 3.1 and 3.2 can be combined to Statement 3: With increasing intensity I the limit value 4~,~ can be left as well on a decreasing as on an increasing /~,(I)curve, The statements 1,2 and 3 suggest that at least eight different types of 4~,,(I)curves exist. One can show that all these types exist indeed. In Figs. 3(a) and (b) they are sketched under the supposition that they have the minimal number of extrema. That is true for suitable parameters. But, changing the para-
~ins,~i~
4. Discussion
__
__
Fig. 3. Types of ~ curves sketched (a) and (b) under the supposition that they exist with the minimal number of extrema, all cases are numerically realized, (c) for numerically realized cases with more than the minimal number of extrema.
‘S. Oherliinder. D. Leupold - Journal of Lu,nine.si-enc’e 59 (1994
130
the intensity dependence of J~~~(I), in the final range there is a fingerprint of annihilation ~ > 3(k
3~+ k32) if ~jns approaches ~,,, in an increasing way, e.g. in stationary case if a minimum exists (type 4; in the stationary case only types 3 and 4 of Fig. 3 are possible). What about the experimental accessibility of In its present analytical formulation, the quantity requires the measurement of the instantaneous fluorescence intensity from the entrance plane of the sample at the moment t = t of a rectangular pulse of width r (numerator), as well as measurement of the quantity [at2(Si S2) + a23(S2 S3)], also at t = t, in the entrance plane (denominator). The latter is available from a transmission measurement orthogonal to the direction of the exciting pulse propagation. ~jfl5’?
~,
/25 /33
We are currently investigating possible changes of these requirements which simplify the experiment but conserve as much as possible of the content of information. Preliminary results with e.g. fluorescence measurements at the maximum of a gaussian pulse of variable width and with instantaneous transmission measurement with that exciting pulse at an optical thin layer are encouraging [16].
5. Conclusions
—
—
~i
~•
~
(c)
#ini
(e)
I #in.
~
The new quantity ~ the instantaneous fluorescence quantum yield, defined at the end of a rectangular pulse of width t, is a very sensitive measure of excited state processes in fluorescent
:.
~‘
~
—~~-‘
~‘
(d)
#~ (f)
I~.o ~
~
(g)
~
#ini
(h)
I~ Fig. 4. The 8 different shapes of
~,,,(!).
sketched.
S. Oberldnder, D. Leupold / Journal of Luminescence 59 (1994) 125—133
131
Table 1 The 8 different shapes of ~,,,(I) and their general conditions General conditions
Parameter example
Type I, see Fig. 4(a) sufficient for decrease at the left
053 ~ ~i2
< 3(k 3~+ k32(
t
in (I + \~
sufficient for decrease at the right
k2~ 2k3~+ k32
)
necessary and sufficient for ~,,
o<
~
ai~= a a 2i 23 = a32 k21 k31 k32 ;.
=
2 cm6cm2 2 x 10 ‘ 10” s lOiOS_i 10~~ s~
=
lOiS s~
=
b—u 5
= = = =
Type 2, see Fig. 4(b) a23 ~ a~2 y ;> 3(k 3~+ k32)
r
~c
In (I + \,
k2
sufficient for decrease at the left
ai2 023
= =
sufficient for increase at the right
k2~ 2k31 + k32
)
necessary and sufficient for
,~
~
au a = 32 = k21 = k3~= k32 = =
=
10-i6
2 lO~°cm 2 2 x b0 i6 cm 10” si 100 5~i 10~~ s~ 1O~~ ~ 2s i0~ X
Type 3, see Fig. 4(c). The only type in the stationary case if ‘,‘ ~ 3(k 31 + k32) a23 ~ a~ ‘,‘ <: 3(k 3~+ k32)
>
In k2~
+
aai2 23
sufficient for decrease at the left
= =
sufficient for decrease at the right
k21 2k3~+
k32
)
a2i = a32 = kk2~= 31 = k32
necessary and sufficient for ~
~
>
~
2 2 1016 cm 2x l0~° 1 cm 10” s 10i0
5—i
=
fljii
5i
=
loio s--i
=
10
s
Type 4, see Fig. 4(d). The only type in the stationary case if y > 3(k3~+ k32) a2, ~ a~ y > 3(k 31 + k32)
aai2 23
sufficient for decrease at the left
= =
sufficient for increase at the right
a2i a32 k21
= = =
2 2 lO_i~cm 2 x IO’~~ cm 10” s_i
k3~ = 1010 =
1 / k21 t>—lnl I + k2~ ~ 2k31 +
‘\
necessaryandsufficientfor4~~,0>4~~,,
10ii s~
c,=5xlOns_i
k32j
lO~ s
r
=
=
aui
=
=
032
=
k21
=
2 l0 ~ cm 2 5 x 10—i7 cm l0~s’
k31
=
10i0 ~I
k32
=
Type 5, see Fig. 4(e). Both variants exist. a23 < 4aiu and r sufficiently small y < 3(k 3~+ k32)
In (i +
r < k2~
k21 2k3~+k32
)
sufficient for increase at the left
aiu a 23
sufficient for decrease at the right
necessary and sufficient for ~
< ~,,
‘,‘ =
~
s
-
lOis s~
132
S. Oherldnder. D. Leupoll
,-
Journal of Luminescence
59 (/994)
/25 /33
Table I (contd.( General conditions
Parameter example
Type 6. see Fig. 4(f). Both variants exist. 02,
>
< /a, 2 and r sufficiently small 3(k, 1
+ k32(
sufficient for increase at the left
012
073
= =
sufficient for increase at the right
In (I +
necessary and sufficient for
~
0, 032
= =
k,1 10” k, = 10° ~ k,,= 101 <
~,,
=
5x
t
=
S x 10
—
02i
=
=
a,,
=
,
Type 7. see Fig. 4(g). Both variants exist. 15i2 023
<
~<
3(k
~ai2
and
t
sufficiently smalla
3, + k32)
>
11n
(I +
k,,
\
sufficient for increase at the left sufficient for decrease at the right
k,, k, k3,
k 2~
2k3, + k32)
necessary and sufficient for
~
,
2 10 ‘“ cm 5 x 10 cm2
>
~/3-
,
-
13
10 ‘“cm2 S x 10 cm2
=
10’s iO’°s
=
10n ~
=
10’”
=
1
i
5
I =
tO -
‘
s
Type 8. see Fig. 4(h) 02i 023
< 4a~2 and c sufficiently small”
sufficient for increase at the left
=
~~I2 =032= 023 =
10 “ cm2 It) iScmi
> 3(k
3, + k32)
>
I / k2, :~ In I I + ~ k21 ~ ~k3, + k321 —
sufficient for increase at the right
.
necessary and sufficient for
k2, = tO’’ s k,, =5.4x10’s = S.4x 10’s
J,,~,
>
~
,
r
= =
3.3 x 10 2 x 10
0
s
Both realizable for instance if k32 sufficiently large. Both realizable under some additional conditions.
molecular systems, especially if the intensity vanation is combined with a variation of pulse width. This quantity should be experimentally accessible at least in an approximate manner.
This work was supported by a grant from the Deutsche Forschungsgemeinschaft (Le 729l/l).
of ~/~~,(I)are given above (section 3), in Table I they are written explicitly for each case, with one parameter example each. The latter is included to demonstrate that these conditions fit usual excited state properties of the molecular systems in question. As is expected generally and was verified especially for the given 8 parameter sets, the corresponding fluorescence yield curves ~F(I) always have the monotonous shape as shown in Fig. 2 (calculated with numerical procedure described in [16]).
7. Appendix
8. References
6. Acknowledgement
Though the general conditions for the molecular parameters to get the 8 different shapes (see Fig. 4)
[1] H. Stiel, S. Dähne and K. Teuchner, J. Lumin. 39 (1988) 351.
S. Oberländer, D. Leupold / Journal of Luminescence 59 (1994) 125—133 [2] A.J. Campillo, R.C. Hyer, T.G. Monger, W.W. Parson and S.L. Shapiro, Proc. NatI. Acad. Sci. USA 74 (1977) 1997. [3] D. Mauzerall, Biophys. J. 16 (1976) 87. [4] A.J. Campillo, S.L. Shapiro, V.H. Kollman, K.R. Winn and R.C. Hyer, Biophys. J. 16 (1976) 93. [5] N.E Geactinov, D. Husiak, T. Kolubajev, J. Breton, A.J. Campillo, S.L. Shapiro, KR. Winn and P.K. Woodbridge, Chem. Phys. Lett. 66 (1979) 154. [6] R. van Grondelle, Biochim. Biophys. Acta 811(1986)147. [7] M. Voss, PhD thesis, University of Leiden (1987). [8] R. Jahnke and D. Leupold, Stud. Biophys. 86 (1981) 227. [9] T. Kolubajev, N.E. Geactinov, G. Paillotin and J. Breton, Biochim. Biophys. Acta 808 (1985) 66.
133
[10] M. Vos, R.J. van Dorssen, J. Amesz, R. van Grondelle and C.N. Hunter, Biochim. Biophys. Acta 933 (1988) 132. [II] G. Deinum, S.C.M. Otte, AT. Gardiner, Ti. Aartsma, R.J. Cogdelb and J. Amesz, Biochim. Biophys. Acta, 1060 (1991) 125. [12] B. Wittmershaus, TM. Nordlund, W.H. Knox, R.S. Knox, N.E. Geactinov and J. Breton, Biochim. Biophys. Acta 806 (1985) 93. [13] L. France, N.E. Geactinov, S. Lin, B.P. Wittmershaus, R.S. Knox and J. Breton, Photochem. Photobiol. 48(1988)333. [14] J. Deprez, G. Paillotin, A. Dobek, W. Leibl, H.-W. Trissi and J. Breton, Biochim. Biophys. Acta 1015 (1990) 295. [15] D.N. Nikogosyan, Laser Chem. 7 (1987) 29. [16] J. Ehiert, D. Leupold, S. Oberlflnder and K. Teuchner, J. Lumin., to be published.
LUMINESCENCE
--
ELSEVIER
Journal of Luminescence 59 (1994) 125—133
Instantaneous fluorescence quantum yield of organic molecular systems: information content of its intensity dependence S. Oberlãndera,*, D. Leupoldb Institute of Applied Analysis and Stochastic, Hausvogteiplatz 5-7, D-1O1 /7 Berlin, Germany ~Max-Born-Institute for Nonlinear Optics and Short Pulse Spectroscopv, Rudower Chaussee 6, D-12489 Berlin, Germani’ a
(Received 12 May 1993; revised 22 September 1993; accepted 3 November 1993)
Abstract In pulse laser investigations of systems of fluorescent organic molecules (e.g. aggregates, light harvesting antennas) it is usual to derive from the intensity dependence of time-integrated fluorescence (fluorescence yield) conclusions on deactivation channels, bimolecular interaction and sizes of interacting units (“domains”), though the content of information of this type of measurement is restricted because the shape of this function is hardly structured. Especially when further processes take place (e.g. excited singlet absorption) problems of ambiguity arise. Based on analytical investigations of the adapted rate equation system of noncoherent nonlinear light-matter interaction, especially of the asymptotic behaviours of its solution with respect to the excitation intensity, we propose a new measuring quantity “instantaneous fluorescence quantum yield”, which in its intensity dependence reflects far more sensibly the several excited state processes and therefore should allow the characterization of absorption, relaxation, and annihilation quantitatively and in a unique way.
1. Introduction In fluorescent systems of interacting uniform organic molecules with it-electron systems (aggregates, clusters, light harvesting antennas) the fluorescence quantum yield may depend on excitation intensity for two principal causes: excited singlet absorption and excited singlet annihilation (cp. Fig. 1). In the usual experimental situation, the dependence of the time-integrated fluorescence intensity on the pulsed excitation fluence is measured. The fluorescence yield function ~F(I) obtained in this way is identical to the fluorescence quantum yield only as long as the corresponding absorption of the *
Corresponding author,
molecular system measured under the same pulsed excitation conditions is a linear function of the excitation fluence.
The typical shape of a fluorescence yield function, obtained e.g. with the well-known pseudoisocyanine aggregates [1] as well as with antennas from photosynthetic bacteria [2] and from higher plants [3,4], is shown in Fig. 2. It is worth mentioning that this typical monotonously decreasing curve can be observed in cases of either excited state absorption or exciton annihilation alone, as well as under conditions of simultaneous operation of both processes; there is no specific “fingerprint” Though relatively unspecific in its general ~.
Concerning the relevance of the question of excited state absorption versus exciton annihilation, cf. e.g. Ref. [5].
0022-2313/94/$07.00 © 1994 — Elsevier Science B.V. All rights reserved SSD! 0022-231 3(93)E0488-J
1 26
S. ()her/dus/e,, I). Leopold
I ~23
Joioiio/ of Liu,i9u oenee 59
questionable because of very simplifying assumptions of the parameter identification procedure k
32
2
______________ - --
~
125 /33
In many cases, the uniqueness of these results is
ç
032
1994
~2l
1
____________
-
b) Fig. I. Three level model of interacting uniform (a 6) fluorescent organic molecules. The interaction is represented by singlet singlet annihilation ): besides this, the excitations and deactivations are characteriied by induced processes ~ absorption and emission I and spontaneous relaxation processes k,. whereat each k~,in principle may consist ofa radiative as well as a nonradiative part. For simplicity, in the following it ~ assumed that k2, is the only pure radiative relaxation. For the sake of clearness. oa processes are indicated only in the left part al and k5 processes are indicated only in the right part (h)
-
~Ff
~
Fig. 2. Typical shape of an experimentally determined floorescence yield function ~. from systems of interacting organic molecules (aggregates, photosynthetic antennas) at pulsed cxcitation with variable intensity I.
shape, these fluorescence yield curves are used e.g. as one of the main sources of information on the photophysics of photosynthetic antenna systems )for reviews see Refs. [6,7]).
(two level systems, s-shaped excitation pulse). But even if more adequate models and the real excitation pulse shape are used in the numerical simulation of the experimentally determined functions. the ambiguity remains, and this is even true if in addition the nonlinear absorption is measured. which means that the intensity dependence of fluorescence quantum yield is known [1] 2 In addition. photosynthetic objects seldom fulfill the supposition of a uniform sample (spectral heterogeneity of photosynthetic antennas, antennas with reaction centers). Attempts to simulate fluorescence yield and fluorescence decay functions of these samples on the basis of more adapted models showed the necessity of additional experimental quantities to reduce these ambiguities [13.14]. Of course, the danger of over-interpretation of the fluorescence yield functions is not restricted to photosynthetic objects. Considering this, one may ask whether the usual 5F(i) really exhausts the experimental quantity ~ principal content of information of fluorescence quantum yields. In other words: is there a fluorescence quantum yield-like quantity which is more informative and also experimentally accessible’? As is shown in the following, the answer is yes: based on analytical investigations of the differential equation system describing noncoherent light-matter interaction, adapted to the above mentioned ~-eIectron systems, an instantaneous fluorescence quantum yield will be defined, which represents in different parts of the function a high sensitivity to excited state absorption, to relaxation processess, to annihilation, and offers a specific annihilation fingerprint.
2. Model, rate equations and definition of instantaneous fluorescence quantum yield ~,
Quantities derived
from these fluorescence yield functions are (cp. Fig. l)the relaxation rates k 2~,k5~,k32, the bimolecular decay (annihilation) rate ;‘, the excited state absorption cross section a23 as well as the number of interacting photosynthetic units (size of domains) (see e.g. Refs. [1.8—Il]).
Each molecule of the fluorescent system is represented by a three level model shown in Fig. I. It 2
‘fhc same problems of ambiguity of parameter identification hold for the second standard experimental quantity of photosynthetic objects, the excitation intensity’ dependence of tluorcsccnce decay [121.
S. Oherlander, D. Leupold / Journal of Luminescence 59 (1994)125—133
127
consists of the ground state (1), the fluorescent first excited singlet state (2) and a higher excited singlet (3). To demonstrate the progress with the new quantity ~. as lucidly as possible, the vibronic substructure of these levels as well as Franck—
This pulse may traverse the sample orthogonal to the entrance plane. For this entrance plane we define the instantaneous fluorescence quantum yield ~r~s at t = t in the following manner:
Condon levels are neglected (but there is in principle no mathematical restriction with respect to more extended models). When interacting with a monochromatic radiation field 1(t) with photon energies equal to the (equal) differences between the energy levels, these levels are connected by absorption as well as induced emission processes (characterized by cross sections ~ and a~respectively, a~= afi), by relaxation processes (described by overall rate constants k~1,which may contain radiative as well as radiationless contributions) and by bimolecular interaction of two first excited singlets which results in one ground state and one higher excited singlet
~~.(I)
(characterized by the annihilation constant y, compare below). If this light-matter interaction can be described by the rate equation approach, one gets the following differential equation system for the population densities S~: =
52 =
—
ai~1(S~ — S2) + k2~S2 + k3153 +
cri2l(Si
+ k32S3 S3
=
—
—
S2)
—
a23 I(S2
—
S3)
—
~2’
k2~S2
2yS~,
a23 1(S2 —S3)
—
(k3~+ k32)S3 + yS~,
with the normalization S~+ S2 + S3 what follows
=
1, from
N’F,
whereby N is the volume density of molecules and F the usual annihilation constant in units volume/time. 3, the time course of the excitaIn isthe following tion assumed to be a single rectangular pulse of width ‘r, i.e. Iconst.
1(t)
~o,
>
0,
if t E [0, else.
t]
With respect to the following definition of ~,, the time course ofexcitation at t > s has no relevance. We use this rectangular pulse description only for the sake of simplicity,
=
I[a12(Si
—
k21S2 S2) + a23(S2
—
S3)]
Notwithstanding its definition at a definite (but variable) instant of time t = t (where the population densities are the result of “history”), the quantity ~. fulfills the convention of quantum yield definitions in case of multiphoton processes [15]. The large content of information of this new quantity is demonstrated in the following section with the help of mathematical analysis.
3. Asymptotic behaviour of ~ to excited state processes
and its sensitivity
If the exciting radiation field enters the matter at the time t = 0, we have in the entrance plane the initial conditions S~(0)= 1, S~(0)= S3(0) = 0. To these initial conditions exists one solution of the rate-equation system. At first we shall investigate with help of mathematical analysis the asymptotic behaviour of this solution with respect to I for = t >0. Secondly we shall obtain from this the asymptotic behaviour of ~ Thirdly we shall see unexpectedly that at least eight different types of ~ curves exist. Such a way, successful by many problems, is provoking two questions by experimentalists: (i) Why asymptotic investigations? One cannot measure for I 0 or I cxc. Answer: knowledge the asymptotic behaviour of a The function leads toofinformation about their global behaviour including measurable ranges. (ii) Why mathematical analysis? One can get each desired curve most quickly with any computer. Answer: One can. May be one finds even some
t
—~
—~
different types of curves. But it is not easy, to find out the parameter ranges for any type. (In this paper, after some normalizations, one has to find
1 2)~
S. Oherldnder. D. Leupo/sl / Journal of Luniine,rcenc’e 59 (1994, /25- /33
ranges in a five-dimensional field for eight types of curves.) And it is impossible after calculation of a finite set of curves to clear up the behaviour of the remaining infinite set. It is impossible to get any general structural information, Therefore we make use of mathematical analysis. Therefore we investigate the asymptotic behaviour of interesting functions, at first of the S,. To describe their asymptotic behaviour, we try an ansatz of a polynomial of the order p in I for low and in J for high intensities, p ~ 0. If we can show that the asymptotic behaviour of S has such a structure, p should be chosen as small as possible but great enough for getting the desired information. We start with the ansatz -
asymptotic representation for ~.
=
i
=
a
a231 el~~r + f(’r)~— + k2~
—
S,
~
/a231\’ a~
--0
~
=
=
~ hi., ~=0
+
+ R5~, for high intensities (2) .
.
1,
0(12)
(3)
2
22
—
/ (1 ~
—
2ai2~(1 a,~/
—
e
kilt)
From asymptotic representations (2) for S~,S2. and S3 with p = 2 one gets
l~=
k21 + 2k31 + k32 (2k31 + k32)(k3~ + k32
R~~1for low intensities. (I)
-
=
with
+
5,
with p
~,
+
—
-h’)
k,1
~
(k2~+ 2k3~+ k32)” O(I~2)
a731 (4)
Though p =it 2. the derivation of (instead this expression is easier than seems to be because of the h,,) the differences h,1 h22, h2-, are needed only. From Eqs. (3) and (4) one can take limit values —
with polynomial coefficients a~1,~ and error terms ~ ~ One can show that a~,h~,exist so that
~-,
=
lim~~. = I
—
e
I ‘0
Rapi ,/ I”
—*
0
R5~1V’_3O
if I ~
0,
and =
~‘
~
-
k,~+ 21<31
-,
Therefore with such coefficients the polynomials of the order p describe the asymptotic behaviour of s, with an error less than terms of the order p + I, that means, with an error term denoted usually and in the following as Q(JP~ for low and as O(I~ i) for high intensities. It holds i)
a1,1 = S,(O), b01
=
1/3.
For v > 0 the a~1depend on t. These functions a,1(t) can be calculated recursively as solutions of linear differential equations using the initial conditions a~~(0) = 0. The b,,1 are independent of ‘r for v > 0 too. These values can be calculated recursively as solutions of linear algebraic equations using the additional conditions + h,,2 + h,3 = 0. Fromh,,~ asymptotic representations (1) for S~with p = I and for S2 with p = 2 one can derive an
+ 1<32
and one is led to Statement >
1:
~,
~ <
as well as
~
~
/~, is possible
From Eq. (4) one is led to Statemenf 2: With increasing intensity I the limit value i~ will be reached for < 3(k31 + k32) on a decreasing q’3~~~(I) curve. for ;‘ > 3(k31 + 1<32) on an increasing ç~~,(I) curve For the function f(t) in Eq. (3) it holds f(0)
=
0.
(
a~’
‘~
/
~~~)<°‘ limf(t) = I + a23~)~ With the last, one is led from Eq. (3) to —
S. Oberlander, D. Leupold/ Journal of Luminescence 59 (1994) 125—133
129
Statement 3.1: With increasing intensity I the limit value t1~,~ will be left on a decreasing ~iflS(i) curve for sufficiently long pulses.
meters (in the parameter range of a type, conditions see once again in the Appendix), the existence of an even number of further extrema cannot be excluded. Numerical examples have shown that, for
In the case of c~2~ a23 it holds f(t) < 0 for all t > 0. If a12 > a23, in the case of 2at2 ~ 3a23 it holds either < 0 for all tof>sign, 0, ororf(’r) f(t) has one positive zero f(’r) without change has two positive zeros with changes of sign. In the case of 2ai 2 > 3a~3f(r) has one and only one positive zero with change of sign. With the last, one is led from Eq. (3) to
instance, the types 5, 6 and 7 (dashed in Fig. 3(b)) can exist also with two extrema more, see4~~~(I) Fig. 3(c). are In this sense at least even eleven types of possible.
Statement 3.2: In the case of 2ai2 > 3a~ with increasing intensity I the limit value q5~~ will be left on an increasing ~ curve for sufficiently short pulses.
By comparison of the variety of ~ curves (Fig. 3) with the only representative type of curve obtained with the usual time integrated quantity tI3F(I) (Fig. 2), the progress with the new quantity 4~,(I)is obvious: more structure means more information. As is indicated by the analytical expressions given above, çb1~,(I)depends in a very sensible manner on the several excited state processes and on the width of the exciting rectangular pulse. Therefore it can be expected that this new quantity, if experimentally accessible, should give an essential aid in the unique characterization of processes like excited state absorption, exciton annihilation, and excited state relaxation of uniform molecular systems. Whereas e.g. a fingerprint of excited state absorption is available in the initial-range of
The statements 3.1 and 3.2 can be combined to Statement 3: With increasing intensity I the limit value 4~,~ can be left as well on a decreasing as on an increasing /~,(I)curve, The statements 1,2 and 3 suggest that at least eight different types of 4~,,(I)curves exist. One can show that all these types exist indeed. In Figs. 3(a) and (b) they are sketched under the supposition that they have the minimal number of extrema. That is true for suitable parameters. But, changing the para-
~ins,~i~
4. Discussion
__
__
Fig. 3. Types of ~ curves sketched (a) and (b) under the supposition that they exist with the minimal number of extrema, all cases are numerically realized, (c) for numerically realized cases with more than the minimal number of extrema.
‘S. Oherliinder. D. Leupold - Journal of Lu,nine.si-enc’e 59 (1994
130
the intensity dependence of J~~~(I), in the final range there is a fingerprint of annihilation ~ > 3(k
3~+ k32) if ~jns approaches ~,,, in an increasing way, e.g. in stationary case if a minimum exists (type 4; in the stationary case only types 3 and 4 of Fig. 3 are possible). What about the experimental accessibility of In its present analytical formulation, the quantity requires the measurement of the instantaneous fluorescence intensity from the entrance plane of the sample at the moment t = t of a rectangular pulse of width r (numerator), as well as measurement of the quantity [at2(Si S2) + a23(S2 S3)], also at t = t, in the entrance plane (denominator). The latter is available from a transmission measurement orthogonal to the direction of the exciting pulse propagation. ~jfl5’?
~,
/25 /33
We are currently investigating possible changes of these requirements which simplify the experiment but conserve as much as possible of the content of information. Preliminary results with e.g. fluorescence measurements at the maximum of a gaussian pulse of variable width and with instantaneous transmission measurement with that exciting pulse at an optical thin layer are encouraging [16].
5. Conclusions
—
—
~i
~•
~
(c)
#ini
(e)
I #in.
~
The new quantity ~ the instantaneous fluorescence quantum yield, defined at the end of a rectangular pulse of width t, is a very sensitive measure of excited state processes in fluorescent
:.
~‘
~
—~~-‘
~‘
(d)
#~ (f)
I~.o ~
~
(g)
~
#ini
(h)
I~ Fig. 4. The 8 different shapes of
~,,,(!).
sketched.
S. Oberldnder, D. Leupold / Journal of Luminescence 59 (1994) 125—133
131
Table 1 The 8 different shapes of ~,,,(I) and their general conditions General conditions
Parameter example
Type I, see Fig. 4(a) sufficient for decrease at the left
053 ~ ~i2
< 3(k 3~+ k32(
t
in (I + \~
sufficient for decrease at the right
k2~ 2k3~+ k32
)
necessary and sufficient for ~,,
o<
~
ai~= a a 2i 23 = a32 k21 k31 k32 ;.
=
2 cm6cm2 2 x 10 ‘ 10” s lOiOS_i 10~~ s~
=
lOiS s~
=
b—u 5
= = = =
Type 2, see Fig. 4(b) a23 ~ a~2 y ;> 3(k 3~+ k32)
r
~c
In (I + \,
k2
sufficient for decrease at the left
ai2 023
= =
sufficient for increase at the right
k2~ 2k31 + k32
)
necessary and sufficient for
,~
~
au a = 32 = k21 = k3~= k32 = =
=
10-i6
2 lO~°cm 2 2 x b0 i6 cm 10” si 100 5~i 10~~ s~ 1O~~ ~ 2s i0~ X
Type 3, see Fig. 4(c). The only type in the stationary case if ‘,‘ ~ 3(k 31 + k32) a23 ~ a~ ‘,‘ <: 3(k 3~+ k32)
>
In k2~
+
aai2 23
sufficient for decrease at the left
= =
sufficient for decrease at the right
k21 2k3~+
k32
)
a2i = a32 = kk2~= 31 = k32
necessary and sufficient for ~
~
>
~
2 2 1016 cm 2x l0~° 1 cm 10” s 10i0
5—i
=
fljii
5i
=
loio s--i
=
10
s
Type 4, see Fig. 4(d). The only type in the stationary case if y > 3(k3~+ k32) a2, ~ a~ y > 3(k 31 + k32)
aai2 23
sufficient for decrease at the left
= =
sufficient for increase at the right
a2i a32 k21
= = =
2 2 lO_i~cm 2 x IO’~~ cm 10” s_i
k3~ = 1010 =
1 / k21 t>—lnl I + k2~ ~ 2k31 +
‘\
necessaryandsufficientfor4~~,0>4~~,,
10ii s~
c,=5xlOns_i
k32j
lO~ s
r
=
=
aui
=
=
032
=
k21
=
2 l0 ~ cm 2 5 x 10—i7 cm l0~s’
k31
=
10i0 ~I
k32
=
Type 5, see Fig. 4(e). Both variants exist. a23 < 4aiu and r sufficiently small y < 3(k 3~+ k32)
In (i +
r < k2~
k21 2k3~+k32
)
sufficient for increase at the left
aiu a 23
sufficient for decrease at the right
necessary and sufficient for ~
< ~,,
‘,‘ =
~
s
-
lOis s~
132
S. Oherldnder. D. Leupoll
,-
Journal of Luminescence
59 (/994)
/25 /33
Table I (contd.( General conditions
Parameter example
Type 6. see Fig. 4(f). Both variants exist. 02,
>
< /a, 2 and r sufficiently small 3(k, 1
+ k32(
sufficient for increase at the left
012
073
= =
sufficient for increase at the right
In (I +
necessary and sufficient for
~
0, 032
= =
k,1 10” k, = 10° ~ k,,= 101 <
~,,
=
5x
t
=
S x 10
—
02i
=
=
a,,
=
,
Type 7. see Fig. 4(g). Both variants exist. 15i2 023
<
~<
3(k
~ai2
and
t
sufficiently smalla
3, + k32)
>
11n
(I +
k,,
\
sufficient for increase at the left sufficient for decrease at the right
k,, k, k3,
k 2~
2k3, + k32)
necessary and sufficient for
~
,
2 10 ‘“ cm 5 x 10 cm2
>
~/3-
,
-
13
10 ‘“cm2 S x 10 cm2
=
10’s iO’°s
=
10n ~
=
10’”
=
1
i
5
I =
tO -
‘
s
Type 8. see Fig. 4(h) 02i 023
< 4a~2 and c sufficiently small”
sufficient for increase at the left
=
~~I2 =032= 023 =
10 “ cm2 It) iScmi
> 3(k
3, + k32)
>
I / k2, :~ In I I + ~ k21 ~ ~k3, + k321 —
sufficient for increase at the right
.
necessary and sufficient for
k2, = tO’’ s k,, =5.4x10’s = S.4x 10’s
J,,~,
>
~
,
r
= =
3.3 x 10 2 x 10
0
s
Both realizable for instance if k32 sufficiently large. Both realizable under some additional conditions.
molecular systems, especially if the intensity vanation is combined with a variation of pulse width. This quantity should be experimentally accessible at least in an approximate manner.
This work was supported by a grant from the Deutsche Forschungsgemeinschaft (Le 729l/l).
of ~/~~,(I)are given above (section 3), in Table I they are written explicitly for each case, with one parameter example each. The latter is included to demonstrate that these conditions fit usual excited state properties of the molecular systems in question. As is expected generally and was verified especially for the given 8 parameter sets, the corresponding fluorescence yield curves ~F(I) always have the monotonous shape as shown in Fig. 2 (calculated with numerical procedure described in [16]).
7. Appendix
8. References
6. Acknowledgement
Though the general conditions for the molecular parameters to get the 8 different shapes (see Fig. 4)
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[10] M. Vos, R.J. van Dorssen, J. Amesz, R. van Grondelle and C.N. Hunter, Biochim. Biophys. Acta 933 (1988) 132. [II] G. Deinum, S.C.M. Otte, AT. Gardiner, Ti. Aartsma, R.J. Cogdelb and J. Amesz, Biochim. Biophys. Acta, 1060 (1991) 125. [12] B. Wittmershaus, TM. Nordlund, W.H. Knox, R.S. Knox, N.E. Geactinov and J. Breton, Biochim. Biophys. Acta 806 (1985) 93. [13] L. France, N.E. Geactinov, S. Lin, B.P. Wittmershaus, R.S. Knox and J. Breton, Photochem. Photobiol. 48(1988)333. [14] J. Deprez, G. Paillotin, A. Dobek, W. Leibl, H.-W. Trissi and J. Breton, Biochim. Biophys. Acta 1015 (1990) 295. [15] D.N. Nikogosyan, Laser Chem. 7 (1987) 29. [16] J. Ehiert, D. Leupold, S. Oberlflnder and K. Teuchner, J. Lumin., to be published.